The Solar Wind
I. Why stars should not have winds
If you write the equation of force balance for a plane-parallel
strip of solar atmosphere, it says the difference in gas pressure
above and below the thin slab in question must balance the weight
of the gas in the slab. That looks like dP/dr = -rho*g, where
rho is the mass density and g is the acceleration of gravity at
the surface of the Sun. Since we are imagining a thin layer, we
treat g as constant, and seek how P(r) and rho(r) depend on distance
r above the surface.
Since that is two variables, we need a second equation, so we use
the ideal gas law, P = rho*k*T/m, where m is the average mass per
particle, and now we need T also. But T will be varying much slower
than P and rho, so we approximate it as constant and just take a
value like 6000 K, typical of the solar surface region. Then
we eliminate P in favor of rho, and get a simple differential
equation that looks like drho/dr = -1/H, where H=kT/mg characterizes
the height a particle at T can fly above the surface before falling
back down (that's the "scale height"). The solution is an
exponential, and the density rho falls by the factor e every time
we look a distance H higher in the atmosphere (just like for the Earth).
Now we must check our plane-parallel assumption, which means we must
test that H/R is small (so the radius R does not produce important
curvature into the problem), and indeed it is of order 10^-4, even
a bit smaller than it is for the Earth's atmosphere.
But this has other consequences-- the solar atmosphere is tightly
held, so the density drops very vast with height, and nothing can
escape. Hence there should not be any solar wind if the T remained
at about 6000 K.
II. The chromosphere
But it does not remain that cool, because the density drops so fast
with height that its cooling (which is by creation of radiation when
free electrons collide with and excite atoms that then radiate) becomes
very inefficient (due to the rarity of collisions). Hence it becomes
easy to heat this gas up to higher T (in the vinicity of 20,000 K, called
the chromosphere) if any kind of additional heating is applied, and
indeed such heating is applied.
It comes from the ability of the convection zone to act like a heat
engine, where entropy is generated when heat is removed from a hot
reservoir deep in the Sun (which reduces entropy) and then that same
heat is returned to the much cooler surface regions (which increases
entropy much more, because the cooler gas "cares more" about getting
that heat then the hot gas cared about losing it). Whenever entropy
is generated by heat transfer, there is the possibility of siphoning
off some of that energy, because there is excess entropy being
generated and the process will happen spontaneously (i.e., without
extra forcing). That siphoned-off energy can be in the form of
mechanical work, which has no entropy consequences (it is
"free energy", a technical term) and can be moved
around and deposited anywhere.
This is how a heat engine works, so
the convection zone works like a heat engine that can do work.
The work comes in the form of creating sound waves (many heat engines
are rather loud, are they not?), and also in twisting up and storing
energy in the copious magnetic fields present in the solar atmosphere.
This is "free energy" is transported all around, but the fraction of
it that is deposited in the low density upper atmosphere has the
greatest impact on the temperature, and is responsible for the
"temperature inversion" above the solar surface, leading to the
hot (about 20,000 K) chromosphere.
III. The corona
The chromosphere still has enough density to be able to radiate
this mechanical heating, and maintain energy balance.
But if we look further up, the density continues to drop, and
eventually radiation is too inefficient to achieve an energy
balance at all. This produces a kind of temperature runaway,
until T reaches millions of K (at which point heat conduction
by the fast electrons is able to carry away the added heat).
This super-hot, super-low-density region is called the corona,
and since the T is so high, we have to reassess whether the
scale height in the corona, H_c, is still much less than R.
In fact, it isn't, which has several important consequences.
First of all, when the scale height approaches R, the spherical
geometry becomes important, and with it, the way g falls off
like 1/r^2. So when we ask if the corona can be in a force
balance, we must redo the above calculation for dP/dr, including
that g falls off like 1/r^2. It is still a fairly simple
differential equation, and it has the same form if we define
a new distance parameter. Instead of using the distance above
the surface in R units, call it x = 1 - r/R in the above derivation,
we can account for the 1/r^2 falloff in g by defining our unitless
distance parameter by x = 1 - R/r. That simple difference means
that dx = dr*R/r^2 instead of dx = dr*R, which makes the integral
solution easy when g falls off like 1/r^2 (try it), but more
importantly, it means that x only goes to 1, not infinity, as r
goes to infinity. This means that the solution for the density
now reaches a finite value (not zero) at infinity, and since T
is being held fixed in the corona (for simplicity, and because
heat conduction is very efficient at high T), this means the
gas pressure in the corona is also nonzero at infinity.
Of course there is no infinity here, but the point is, if you get
very far from the Sun in a solution like this, the density and pressure
stop falling off much-- they kind of level out.
If you compare observations of the solar corona to set the proper
scale for the density and temperature, you find that the gas pressure
that the corona levels off at is still much higher than the pressure
in the interstellar medium. Parker used this to infer that the solar
corona must be expanding-- there must be a "solar wind."
Observations of ion tails being rapidly swept away from comets
quickly confirmed this.
However, the magnetic fields that help heat the corona also constrain
the plasma in the corona and can prevent it from being able to expand,
in regions where the field is strong. These are called "closed field
regions", filled with "magnetic loops", and the corona in these regions
does not join the solar wind and can contain a higher density (and hence
be more visible in the remarkable photos of the solar corona).
So the corona has "open field regions", where the field lines have been
swept out by the solar wind, and closed loops, where the plasma is
forced to stay within the "tubes" created by the magnetic field.
Energy releases that alter the shapes of the fields can then produce
eruptions of this plasma, leading to "coronal mass ejections", which
contributes plasma to the solar wind in a sporadic way. So the solar
wind has more steady components, and more sporadic ones, and Earth
sits in this wind and is affected by it, causing aurorae and magnetic
substorms that can affect our satellites and power grids.
Fortunately we have a relatively strong magnetic field for a rocky
planet, which helps protect us from the solar wind, but many exoplanets
might not be so lucky. Certainly, Mars was not, and this may have
played a role in removing most of its atmosphere.
IV. Why the Sun does have a wind
In the 1950s, Eugene Parker reached an important conclusion--
the observed pressure in the low corona
of the Sun, coupled with the very high
temperature there (mechanically heated via that heat engine in ways that
are still under study),
there is no way the high pressure can fall enough to balance the
pressure in the ISM.
This might seem surprising, given the above argument that density
(and pressure) falls off rapidly above the solar surface, but that
argument applied for the small scale heights implied by the lower
temperatures in the photosphere and even the chromosphere.
At coronal T, the scale height starts to approach the scale of the
solar radius, so we can no longer use a plane-parallel treatment.
If we use spherical coordinates instead, gravity has a limited range
in which to induce a pressure drop (say, a few solar radii, perhaps
10 at the most), so if the heating can keep the corona hot over just
that much distance, the pressure will eventually reach a constant value
that is still much higher than in the ISM. Dropping T outside that
domain where gravity is acting does not help any longer, there will
always be an overpressure relative to the ISM, so the corona must
expand.
This is the solar wind, and because it is due to the high gas
pressure maintained outside the solar surface, it is called a
gas pressure-driven wind, or a "Parker wind."
Parker found a simple constant-T solution that reaches supersonic
speeds of order the escape speed from the solar surface, and
importantly, this solution depends only on the assumed T and not
at all on the mass-loss rate.
The density is decoupled from the velocity because density is the
source of the driving pressure force, but also the target of the
gravity that curtails it, so in the end density does not matter.
So we can understand the velocity of the solar wind without
understanding how dense it is, or how high is the mass-loss rate.
The latter requires knowledge of the rate of heating, which so
far we can only try to observe but is difficult to derive from
first principles.