Lecture 17: The solar wind
o  Topics to be covered:
o  Solar wind
o  Inteplanetary
magnetic field
The 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 wind
o  STEREO satellite image sequences of comet tail buffeting and disconnection.
Parker’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:
dP
= −ρg
dr
dP GM S ρ
+
=0
dr
r2
o  If first term >> than second €
=> produces an outflow:
€
dP GM S ρ
dv
+
+
ρ
=0
2
dr
r
dt
Eqn. 1
Eqn. 2
o  This is the equation for a steadily expanding solar/stellar wind.
€
Parker’s solar wind (cont.)
o  As, dv = dv dr = dv v
dt dr dt dr
or
=>
v
dP GM S ρ
dv
+
+
ρ
v
=0
dr
r2
dr
dv 1 dP GM S
+
+ 2 =0
dr ρ dr
r
Eqn. 3
€
€
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/
weight), the 2nd term of Eqn. 3 is:
Isothermal wind => dT/dr
€
0
(R is gas constant;
dP Rρ dT RT dρ
=
+
dr
µ dr
µ dr
1 dP $ RT ' 1 dρ
⇒
=& )
ρ dr % µ ( ρ dr
is mean atomic
Eqn. 4
Parker’s solar wind (cont.)
o  Now, the mass loss rate is assumed to be constant, so the Equation of Mass
Conservation is:
dM
= 4 πr 2 ρv = const ⇒ r 2 ρv = const
Eqn. 5
dt
d(r 2 ρv)
dr 2
dρ
2 dv
= r ρ + ρv
+ r 2v
=0
dr
dr
dr
dr
1 dρ
1 dv 2
=>
=−
−
ρ dr
v dr r
o  Differentiating,
€
Eqn. 6
o  Substituting Eqn. 6€into Eqn. 4, and into the 2nd term of Eqn. 3, we get
€
dv RT # 1 dv 2 & GM S
+
− (+ 2 =0
%−
$
dr µ
v dr r '
r
#
RT & dv 2RT GM S
⇒ %v −
+ 2 =0
( −
µv ' dr µr
r
$
v
o  A critical point occurs when dv/dr
0 i.e., when 2RT = GM2 S
µr
€
o  Setting v c = RT / µ => rc = GM S /2v c2
€
€
r
Parker’s solar wind (cont.)
o  Rearranging =>
1 dv
v c2
(v − v ) v dr = 2 r 2 (r − rc )
2
2
c
Eqn. 7
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:
" v %2
" v %2
"r%
rc
−
ln
=
4
ln
+
4
+C
$ '
$ '
$ '
r
# vc &
# vc &
# rc &
Eqn. 8
Parker’s “Solar
Wind Solutions”
where C is a constant of integration. Leads to five solutions depending on C.
€
Parker’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.
v/vc
Critical point
r/rc
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:
2
# v &2
#r&
v #r&
−ln% ( ≈ 4 ln% ( ⇒ = % (
$ vc '
$ rc ' v c $ rc '
o  Therefore, r2v
o  From Eqn. 5:
rc2vc = const or v ≈ 1
r2
€
const const
ρ = 2 = 2 = const
rv
rc v c
€
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.
Parker’s solutions (cont.)
o  Solution V: From the figure, v >> vc for large r. Eqn. 8 can be written:
" v %2
"r%
"r%
$ ' ≈ 4 ln$ ' ⇒ v ≈ v c 2 ln$ '
# vc &
# rc &
# rc &
o  The density is then:
=>
€
0 as r
ρ=
const
const
≈
r 2v
r 2 ln(r /rc )
∞.
€
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.
∞.
Observed solar wind
o  Fast solar wind (>500 km s-1)
comes from coronal holes.
o  Slow solar wind (<500 km s-1)
comes from closed magnetic
field areas.
o  Figure from McComas et al.,
Geophysical Research Letters,
(2008).
Interplanetary magnetic field
B
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:
Bφ vφ
=
Br v r
Ω(r − r0 )
=
vr
tanψ =
o  Inclined at ~45º at 1 AU ~90º by 10 AU.
€
or v
Br or vr
B
(r0,
r
0)
Alfven radius
o 
o 
Close to the Sun, the solar wind is too weak to modify structure of magnetic field:
B2
2
1/2 ρv <<
8π
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).
B2
1/2 ρv >>
8π
2
o 
Transition between regimes occurs at the Alfven radius (rA), where
€
o 
Assuming the Sun’s field to be a dipole, B =
€
€
M
r3
$ M 2 '1/ 6
=> rA = &
2)
% 4 πρv (
€
B2
1/2 ρv =
8π
2
The Parker spiral
Heliosphere