Lecture 18: The solar wind

PY4A01 - Solar System Science

Lecture 18: The solar wind Lecture 18: The solar wind

o Topics to be covered:

o Solar wind

o Inteplanetary

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The solar windThe solar wind

o Biermann (1951) noticed that many comets showed excess ionization and abrupt changes in the outflow of material in their tails - is this due to a solar wind?

o Assumed comet orbit perpendicular to line-of-sight (vperp) and tail at angle =>

tan = vperp/vr

o From observations, tan ~ 0.074

o But vperp is a projection of vorbit

=> vperp = vorbit sin ~ 33 km s-1

o From 600 comets, vr ~ 450 km s-1.

o See Uni. New Hampshire course (Physics 954) for further details:http://www-ssg.sr.unh.edu/Physics954/Syllabus.html


The solar windThe solar wind

o STEREO satellite image sequences of comet tail buffeting and disconnection.

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Parker’s solar windParker’s solar wind

o Parker (1958) assumed that the outflow from the Sun is steady, spherically symmetric and isothermal.

o As PSun>>PISM => must drive a flow.

o Chapman (1957) considered corona to be in hydrostatic equibrium:

o If first term >> than second => produces an outflow:

o This is the equation for a steadily expanding solar/stellar wind.

€

dP

dr= −ρg

€

dP

dr+

GMSρ

r2= 0 Eqn. 1

€

dP

dr+

GMSρ

r2+ ρ

dv

dt= 0 Eqn. 2


Parker’s solar wind (cont.)Parker’s solar wind (cont.)

o As,

or

o Called the momentum equation.

o Eqn. 3 describes acceleration (1st term) of the gas due to a pressure gradient (2nd term) and gravity (3rd term). Need Eqn. 3 in terms of v.

o Assuming a perfect gas, P = R T / (R is gas constant; is mean atomic weight), the 2nd term of Eqn. 3 is:

€

dv

dt=

dv

dr

dr

dt=

dv

drv

€

=>dP

dr+

GMSρ

r2+ ρv

dv

dr= 0

€

vdv

dr+

1

ρ

dP

dr+

GMS

r2= 0 Eqn. 3

€

dP

dr=

Rρ

μ

dT

dr+

RT

μ

dρ

dr

⇒1

ρ

dP

dr=

RT

μ

⎛

⎝ ⎜

⎞

⎠ ⎟1

ρ

dρ

dr Eqn. 4Isothermal wind => dT/dr 0



o Now, the mass loss rate is assumed to be constant, so the Equation of Mass Conservation is:

o Differentiating,

o Substituting Eqn. 6 into Eqn. 4, and into the 2nd term of Eqn. 3, we get

o A critical point occurs when dv/dr 0 i.e., when

o Setting

€

dM

dt= 4πr2ρv = const ⇒ r2ρv = const

€

d(r2ρv)

dr= r2ρ

dv

dr+ ρv

dr2

dr+ r2v

dρ

dr= 0

€

=>1

ρ

dρ

dr= −

1

v

dv

dr−

2

rEqn. 6

€

vdv

dr+

RT

μ−

1

v

dv

dr−

2

r

⎛

⎝ ⎜

⎞

⎠ ⎟+

GMS

r2= 0

⇒ v −RT

μv

⎛

⎝ ⎜

⎞

⎠ ⎟dv

dr−

2RT

μr+

GMS

r2= 0

€

vc = RT /μ => rc = GMS /2vc2

Eqn. 5

€

2RT

μr=

GMS

r2



o Rearranging =>

o Gives the momentum equation in terms of the flow velocity.

o If r = rc, dv/dr -> 0 or v = vc, and if v = vc, dv/dr -> ∞ or r = rc.

o An acceptable solution is when r = rc and v = vc (critical point).

o A solution to Eqn. 7 can be found by direct integration:

where C is a constant of integration. Leads to five solutions depending on C.

€

v

vc

⎛

⎝ ⎜

⎞

⎠ ⎟

2

− lnv

vc

⎛

⎝ ⎜

⎞

⎠ ⎟

2

= 4 lnr

rc

⎛

⎝ ⎜

⎞

⎠ ⎟+ 4

rc

r+ C

Eqn. 8Parker’s “Solar Wind Solutions”

€

v 2 − vc2

( )1

v

dv

dr= 2

vc2

r2(r − rc ) Eqn. 7


Parker’s solutionsParker’s solutions

o Solution I and II are double valued. Solution II also doesn’t connect to the solar surface.

o Solution III is too large (supersonic) close to the Sun - not observed.

o Solution IV is called the solar breeze solution.

o Solution V is the solar wind solution (confirmed in 1960 by Mariner II). It passes through the critical point at r = rc and v = vc.

r/rc

v/vc

Critical point


Parker’s solutions (cont.)Parker’s solutions (cont.)

o Look at Solutions IV and V in more detail.

o Solution IV: For large r, v 0 and Eqn. 8 reduces to:

o Therefore, r2v rc2vc = const or

o From Eqn. 5:

o From Ideal Gas Law: P∞ = R ∞ T / => P∞ = const

o The solar breeze solution results in high density and pressure at large r =>unphysical solution.

€

−lnv

vc

⎛

⎝ ⎜

⎞

⎠ ⎟

2

≈ 4 lnr

rc

⎛

⎝ ⎜

⎞

⎠ ⎟⇒

v

vc

=r

rc

⎛

⎝ ⎜

⎞

⎠ ⎟

2

€

=const

r2v=

const

rc2vc

= const

€

v ≈1

r2


Parker’s solutions (cont.)Parker’s solutions (cont.)

o Solution V: From the figure, v >> vc for large r. Eqn. 8 can be written:

o The density is then:

=> 0 as r ∞.

o As plasma is isothermal (i.e., T = const.), Ideal Gas Law => P 0 as r ∞.

o This solution eventually matches interstellar gas properties => physically realistic model.

o Solution V is called the solar wind solution.

€

v

vc

⎛

⎝ ⎜

⎞

⎠ ⎟

2

≈ 4 lnr

rc

⎛

⎝ ⎜

⎞

⎠ ⎟⇒ v ≈ vc 2 ln

r

rc

⎛

⎝ ⎜

⎞

⎠ ⎟

€

=const

r2v≈

const

r2 ln(r /rc )


Observed solar windObserved solar wind

o Fast solar wind (v~700 km s-1) comes from coronal holes.

o Slow solar wind (v<500 km s-1) comes from closed magnetic field areas.


Interplanetary magnetic fieldInterplanetary magnetic field

o Solar rotation drags magnetic field into an Archimedian spiral (r = a).

o Predicted by Eugene Parker => Parker Spiral:

r - r0 = -(v/)( - 0)

o Winding angle:

o Inclined at ~45º at 1 AU ~90º by 10 AU.

B or v

Br or vr

Br

€

tanψ =Bφ

Br

=vφ

vr

=Ω(r − r0)

vr

(r0, 0)


Alfven radiusAlfven radius

o Close to the Sun, the solar wind is too weak to modify structure of magnetic field:

o Solar magnetic field therefore forces the solar wind to co-rotate with the Sun.

o When the solar wind becomes super-Alfvenic

o This typically occurs at ~50 Rsun (0.25 AU).

o Transition between regimes occurs at the Alfven radius (rA), where

o Assuming the Sun’s field to be a dipole,

€

1/2ρv 2 <<B2

8π

€

1/2ρv 2 >>B2

8π

€

1/2ρv 2 =B2

8π

€

B =M

r3

€

=>rA =M 2

4πρv 2

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 6


The Parker spiralThe Parker spiral

o http://beauty.nascom.nasa.gov/~ptg/mars/movies/

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HeliosphereHeliosphere

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Lecture 18: The solar wind

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