Download 3. Activity in Stars 3.1 Phenomenology of the Active Sun 3.1.1 The

3. Activity in Stars
3.1 Phenomenology of the Active Sun
3.1.1 The Photosphere
In our previous treatment of stellar photospheres, we assumed them to
be homogeneous in nature. However, on closer inspection we find that
the solar surface is mottled with features that are the visible
manifestation of magnetic activity.
Sunspots (Figure 3.01) are dark patches on the surface that contain
strong magnetic fields (of strength 0.25 to 0.3 T). They appear darker
than the surrounding material because they are cooler: temperatures in
the centre of a spot may be as low as 4000 K, in contrast to the 5777-K
effective temperature of the ambient photosphere. The spots, which
have typical diameters of a few thousand kilometres, usually occur in
pairs or groups; further, the magnetic polarity of the spots in a pair are
of opposite sign. This implies that:
• They are ‘imprints’ where magnetic lines of force leave and then
return beneath the surface.
• These emerging ‘tubes’ of flux are anchored in the convection
zone.
As evidenced by Fig. 3.01, the typical surface area covered by spots is
of the order of only a few per cent. However, as we shall see, this does
vary over time in a systematic manner (as part of what is called the solar
activity cycle).
Figure 3.01: sunspots over the visible photosphere
(Courtesy D. Hathaway, MSFC/NASA)
• The strong-field concentrations we have described aggregate into
active regions over the solar surface. This implies that the dispersal
of regions of magnetic activity is non-uniform: the active Sun is
observed to be non-homogeneous.
• This has important implications for any attempts to model the
generation, emergence and resulting organisation of flux.
3.1.2 The upper atmosphere: the chromosphere and corona
When the Sun is obscured, the chromosphere and corona become
visible around the occulted disc.
• The intensity of the corona in white light is lower than that of the
full solar disc by a factor of at least 106.
• This greatly reduced intensity is indicative of the fall off in density
observed as one passes from the photosphere up through the
chromosphere (a layer roughly 103 km thick) and corona (the
tenuous influence of which extends to the boundary of the solar
system!).
Observations of the corona uncovered several important clues regarding
its structure:
• The spectroscopic analysis of white light from the corona revealed
a rich emission-line spectrum. However, the identity of many of
the lines was a puzzle. Initial speculation during the early part of
the 20th century centred on their origin arising from unknown
chemical elements; this gave way to the realization that the lines
were actually formed by what are termed forbidden transitions.
To illustrate what is meant by a forbidden transition, we consider
the simple electronic energy level diagram shown in Figure 3.02.
3
2
1
Figure 3.02: A simple, 3-level atomic system
There are three levels. Let us suppose that the transition from level
2 to level 1 is forbidden; how might such a state of affairs arise?
The ‘forbidden’ tag implies that there is an extremely small
probability that the transition will occur.
• Ordinarily, atoms will undergo collisional excitation and take
the preferred transition routes (with their higher transition
probabilities) when they de-excite.
• However, if the density of atoms is so low as to reduce
significantly the likelihood of collisions occurring, the transition
from level 2 to 1 has a chance of taking place.
• Here, the relevant competing processes are: excitation from
level 2 to 3, followed by spontaneous decay from 3 to 1; and the
collisional de-excitation from 2 to 1 with no emission of
radiation.
So, the presence of forbidden lines in the emission spectrum of the
corona implies that particle densities are exceedingly low.
• Many of the observed transitions were identified as occurring in
highly ionised atoms (for example Fe XV, Ni XVI and Ca XV).
• Such extreme states of ionisation require very high kinetic
temperatures of the order of 106 K and above.
• This immediately poses the troubling question of what might lead
to such elevated temperatures.
• Several driving mechanisms have been proposed, and it is only
since the advent of the ESA/NASA SOHO mission that physicists
now believe they may be close to the definitive answer: that energy
from the magnetic structures at the solar surface may provide that
needed to power the corona.
The resulting, and somewhat surprising, temperature structure of the
solar atmosphere is illustrated in Fig. 3.03. These data are actually the
prediction of a solar model (Vernazza, Avrett & Loeser 1981; and after
Golub & Pasachoff 1997), but provide a good match to the Sun.
Figure 3.03: The temperature profile of a model atmosphere that
closely resembles that of the Sun (Data from Vernazza et al. 1981)
Instruments on board the ESA/NASA SOHO satellite have produced
spectacular pictures of the corona, as viewed through the emission
produced by highly ionised species.
Figure 3.04 shows the lower solar corona—at temperatures of the order
of a million degrees kelvin—as observed in a highly ionised transition
of iron. Loops (at the solar limb) and active regions of magnetic activity
(over the disc) are clearly visible. As we shall go on to see, the loops
reveal where the hot plasma follows lines of magnetic field that have
erupt through the solar surface. The active regions observed here
correspond to those regions in the photosphere that contain strong
concentrations of magnetic field, e.g., sunspots.
Figure 3.04: The lower solar corona, showing active regions and
loops, as observed by EIT on SOHO (courtesy of the EIT/SOHO consortium.
SOHO is a project of international cooperation between ESA and NASA)
• Notice how the active regions are not evenly distributed over the
whole visible surface, i.e., the ‘active’ solar surface is not
homogeneous.
• The area of the surface typically covered by very active regions is
only of the order of a few per cent.
• Measures of the magnetic activity, such as the line-of-sight
magnetic field, or the amount of emission from spectral lines, are
modulated by the rotation of the solar surface.
• Similar behaviour is observed in other solar-like stars, which
provides evidence for a similar type of magnetic surface structure.
Figure 3.05, again taken with EIT (at the wavelength of the He II
emission line) shows prominences at the solar limb: these consist of
large amounts of solar material suspended above the photosphere by
magnetic field structures. The He II emission line originates in the
transition region between the chromosphere and corona, at temperatures
of between 60,000 and 80,000 K.
Figure 3.05: Prominences in the solar corona (courtesy of the EIT/SOHO
consortium SOHO is a project of international cooperation between ESA and NASA)
3.1.3 Solar wind
The Sun is losing matter continuously via what is called the solar wind.
• As we shall go on to see, lines of magnetic field and the solar
plasma have a strong tendency to be ‘tied’ together, and in the
outer corona matter streams out into the solar system along ‘open’
field lines.
• Coronal holes are where the field lines extend out to very large
distances above the surface, i.e., well into the interplanetary
medium. Although of course they must eventually close, they
extend so far that they give the appearance of being open.
• The flow along the field lines reaches speeds of a few hundred
kilometres per second. The mass-loss rate per year is about 10−14
solar masses, i.e., about 2 x 1016 kg.
3.1.4 Coronal Mass Ejections
Figure 3.06: A CME observed by the LASCO instrument on board
SOHO (courtesy of the LASCO/SOHO consortium. SOHO is a project of
international cooperation between ESA and NASA)
Coronal Mass Ejections (CME) are individual events involve the release
of large amounts of solar material. Most of the energy that is released
goes into the kinetic energy of the bulk motion of the ejected matter.
Figure 3.06 shows a large CME observed by the LASCO coronagraph
on board the SOHO satellite.
• The mass associated with a CME may be up to 1013 kg, with the
matter ejected at velocities of between 10 and 1000 km s-1. They
occur at the rate of about one event per day.
3.1.5 Flares
Figure 3.07: A large flare observed by the EIT instrument on board
SOHO (courtesy of the EIT/SOHO consortium. SOHO is a project of international
cooperation between ESA and NASA)
Flares are short-lived (typically of less than one-hour’s duration)
releases of massive amounts of energy.
The largest flares can result in the release of up to 1025 J.
[Note that solar physicists still tend to measure energy in the old CGS
unit of ergs: the conversion is that 1 J is equivalent to 107 erg.]
Figure 3.07 shows a large flare. This image was taken at the wavelength
of the transition of Fe XII (19.5 nm). Different types of energetic
phenomena are associated with flares, whose origin lies in the corona:
• X-ray and UV emission, characteristic of temperatures in excess of
107 K. The temperatures reached can exceed the normal coronal
temperatures by a factor of up to 10.
• Synchrotron radiation, which arises from electrons accelerated in a
magnetic field.
• Hydrogen-α emission is enhanced greatly in the chromosphere.
This means that the chromosphere is somehow heated from above.
• Gamma radiation from excited nuclei returning to their ground
states.
• White-light emission in the photosphere (again, due to energy
input from above).
Where does the energy come from to power flares?
The energy density in the region of a flare can be of the order of about
102 Jm−3. Is the thermal energy density associated with the coronal
temperatures sufficient to account for this?
We find that the thermal energy density is too low. We require another
energy source: the magnetism associated with structures in the corona.
3.2 The Activity Cycle
3.2.1 Changes to the sunspot number
The phenomena that are the signature of the dynamic, active processes
taking place over the surface of the Sun are observed to vary over time.
The Sun exhibits an activity cycle over which the strength or magnitude
of these phenomena is observed to vary in a systematic manner between
minimal and maximal levels of activity.
In 1843, H. Schwabe discovered that the number of sunspots observed
varies over an 11-year period. Today, solar physicists use the Wolf
sunspot index as a quantitative measure of the sunspot behaviour.
It is defined according to:
R = k (10 g + f ) .
In the above: f is the number of individual spots visible on the solar
disc; g the number of spot groups (i.e., spots show a tendency to
aggregate into groups in magnetic plage regions); and k is a correction
factor that allows for differences in the observational interpretation and
the equipment used to make the observations.
• Figure 3.08 shows variations in R over the last century (smoothed
over a one-month period). These show clear, cyclic behaviour.
• Solar physicists number the 11-year cycles. A cycle begins at a low
level of activity.
Figure 3.08: Variation of the sunspot index, R, over the last century
(data from the National Geophysical Data Centre)
Similar periodic trends are revealed if we plot other proxies of the level
of solar activity as a function of time. Examples include:
• The average line-of-sight magnetic field over the surface of the
Sun;
• The intensity of radio emission from the solar corona; and
• The strength of emission of certain spectral lines.
These all point clearly toward the absolute level of surface activity
showing clear, period behaviour on an 11-year timescale.
• Interestingly, the maximal activity of different cycles does differ.
Indeed, during the latter half of the 17th century very few spots
were observed at all (as discovered from the analysis of suitable
records by Maunder some two hundred years after the event).
Maunder Minimum
Figure 3.09: Variation of the sunspot index, R, over the last four
centuries (data from the National Geophysical Data Centre)
• The reason why such an extended minimum was present is still a
matter of some debate.
• However, low recorded temperatures in the northern hemisphere
over this period point toward a possible period of very low (or
quiescent) solar activity.
• Several sources of data are used as proxies of terrestrial
temperature, e.g., from ice cores, tree rings etc.
• One often-used proxy is the concentration of 14C in ice cores.
•
14
C (a cosmo-nuclide) is produced in the Earth’s atmosphere by
nuclear reactions caused by cosmic rays impacting on atmospheric
species.
• The flux of cosmic rays is influenced by the magnetic field in the
solar-terrestrial environment.
When the solar magnetic field is more intense (i.e., when
there are more spots at times of high activity) fewer
charged particles reach the Earth (hard to cross field lines).
There is therefore a lower rate of production of species like
14
C.
• An anti-correlation therefore exists between cosmo-nuclide
concentration and solar activity.
• Rigozo et al. (2001, Solar Physics, 203, 179-191) used the 14C
records to reconstruct the Sunspot number over the past
millenium.
Figure 3.10: Reconstructed Sunspot number from Rigozo et al.
(2001)
3.2.2 The spatial characteristics of the activity
Next, we consider how the location for these active regions varies over
the surface of the Sun, i.e., the spatial properties of the activity over
time.
Sunspots have a strong tendency to congregate only over certain bands
in latitude on the solar surface.
• At the start of a new cycle (when the level of activity is low) they
appear in bands of latitude in both the northern and southern
hemispheres at about 30 to 35 degrees.
• As the cycle progresses, so the zone where the spots appear
migrates to lower latitudes.
• At the end of the cycle, spots normally lie within 10 degrees of the
equator. This is shown graphically in Figure 3.11. The location of
identified sunspot groups is plotted as a function of time.
• The migration toward the equatorial regions gives the figure its
characteristic butterfly appearance.
Figure 3.11: The location of sunspot groups over time (image courtesy
of Mt. Wilson Observatory)
Sunspots are more often than not located in large regions of intense
magnetic activity, i.e., active regions.
Figure 3.12 shows magnetograms of the photosphere taken by the Kitt
Peak Vacuum Telescope in the US.
• These reveal the line-of-sight magnetic field strength:
o Light regions show magnetic flux emerging from the surface.
o Dark regions show flux returning beneath the photosphere.
The two pictures here were taken during levels of low and high
magnetic activity. One is immediately struck by:
• The change in the absolute level of activity; and
• The tendency for the active regions, when present, to congregate in
certain latitudinal bands.
Figure 3.12: Magnetograms of the solar surface showing regions of
magnetic activity: left-hand panel in February 1996; right-hand
panel in March 2000 (images courtesy of the Kitt Peak Vacuum
Telescope, the National Solar Observatory)
Figure 3.13 shows images of the Sun taken by the EIT instrument on
board the ESA/NASA SOHO satellite. These observations were made
over a range in wavelength centred on a coronal emission line of highly
ionised iron. This reveals active regions and magnetic loops in the
corona at temperatures of over one million degrees kelvin
• Again, the two images were taken at different levels of activity,
here for low and intermediate parts of the solar cycle.
• Once more, the concentration of the activity into bands of latitude
is clearly visible in the right-hand panel.
• Loops of hot plasma that trace out magnetic field lines are most
striking. We will go on to discuss why the plasma is constrained to
lie upon the magnetic field lines.
• Figure 3.12 revealed the ‘footprints’ over the photosphere where
magnetic lines of force leave and return beneath the solar surface.
Higher up in the atmosphere, we should expect to be able to follow
these field lines as they trace out closed loops: this is what we see
here Figure 3.13.
Figure 3.13: The hot solar corona, revealed in observations made by
the EIT instrument on board SOHO (courtesy of the SOHO/EIT consortium.
SOHO is a project of international cooperation between ESA and NASA)
Since magnetic field lines are closed (magnetic monopoles have not
been discovered in nature), we would expect to see an equal amount of
magnetic flux leaving and returning below the surface.
A visual inspection of Figure 3.12 indicates similar areas of positively
and negatively signed field lines (as revealed by the bright and dark
patches). As noted earlier, sunspots tend to occur in pairs with opposite
magnetic polarities. If we take the time to carefully note the polarity in
sunspot groups over long periods of time, we uncover another important
characteristic of the solar cycle:
• During any given cycle, the polarity of the leading spots in a pair
will be of opposite sign in the northern and southern hemispheres.
(The leading spot leads in the direction of the rotation.)
• In subsequent cycles, the polarity of the leading spot in a given
hemisphere changes sign.
This behaviour is displayed in Figure 3.14, where the colour-coded
locations of sunspots are shown over the solar disc (vertical axis lines of
latitude, horizontal axis lines of longitude).
• Yellow-coloured spots have flux emerging from the surface
(positive polarity); and Blue spots have flux returning to the
surface (negative polarity).
• During solar cycle 21 (from approximately 1974 to 1985) the
leading spots had positive polarity; this switched in cycle 22
(approximately 1985 to 1996).
• This effect is called Hales’ Polarity Law. A complete ‘magnetic’
cycle therefore strictly lasts about 22 years, i.e., the period over
which the magnetic characteristics of spot pairs or groups in each
hemisphere vary.
Figure 3.14: The polarity of sunspots over the solar surface during
subsequent solar cycles (courtesy D. Hathaway, MSFC/NASA)
3.3 Activity on other Stars
In order to study magnetic phenomena on other stars, and any cyclic
variations present, precise observations of active phenomena are
required. Certain resonance lines provide the means to do this.
The Calcium H and K lines (Ca II, H & K; 397 nm and 393 nm) are
formed across a range of depths spanning the chromosphere and
photosphere (at temperatures ranging from 4000 to 7000 K).
‘Quiet’ star
Intensity
‘Active’ star
Effect of active
regions
Wavelength
Figure 3.15 Ca K line profile, showing core reversal
Strong magnetic fields from active regions give rise to an emission
feature in the line core (see Figure 3.15).
Since the feature is observed in emission, it must be formed above the
temperature minimum of the photosphere (recall our discussion in
Section 1.6.3).
The amount of emission provides a good proxy for the level of activity
on the surface.
A systematic study of Ca II H&K emissions on other stars (broadly
solar-like) was begun in the late 1970s by Olin Wilson at the Mount
Wilson observatory in California.
3.3.1 The S Index and R’HK
The Mount Wilson survey provides observational estimates of :
S =C⋅
IH + IK
,
I cont
(3.01)
where IH and IK are measured intensities in the emission cores of the H
and K lines, Icont is the continuum intensity measured on both sides of
the lines, and C is a calibration factor. This can be converted into an
equivalent flux (in J m−2 s−1), FHK.
The continuum values depend upon the spectral type (temperature) of
the star. This dependence can be removed by normalising by the total
flux emitted by the star (which is just σ T 4eff). One must also correct for
the ‘quiet-star’, photospheric contribution to the line-core emission,
Fphot. Putting this all together gives a final proxy for activity on a star
that is:
R ' HK =
[ FHK − Fphot ]
σ Teff4
.
(3.02)
Reasonably precise estimates of R’HK can be measured on other stars.
These data provide the means to study variations in levels of magnetic
activity for different ‘solar-like’ stars.
Figure 3.16 shows some examples of real data.
Figure 3.16 Left-hand panel: HR diagram of stars whose data are
used. Right-hand panel: Activity level
• The left-hand panel shows the stars whose data are used (as an HR
diagram of luminosity L (relative to Sun) versus log10 Teff of each
star). The Sun is marked by the large diamond symbol.
• The right-hand panel shows the corresponding measures of activity,
R’HK.
• I have divided the data into two subsets: ‘more active’ stars (with
log10 R’HK > − 4.75, rendered as crosses) and ‘less active’ stars (log10
R’HK < − 4.75; triangles).
• Vaughan and Preston (1980) were the first to draw attention to the
‘gap’ (which now carries their name) which gives two strips of
activity. At lower values of Teff the gap disappears.
In addition to measuring the absolute level of activity, long-term
observations also show that many stars undergo cyclic variation of the
activity, as is observed for the Sun (Figure 3.17):
• These cyclic variations are not seen for stars with log10 Teff > 3.81.
Remember this marks the boundary where efficient sub-surface
convection zones disappear (see Section 2.2.3)
• So, without a convection zone we might not expect to see cyclic
variations in other stars. This is an important clue to what might
explain activity in stars.
Sun
HD 103095
HD 10476
HD 143761
Figure 3.17 Activity cycle in the Sun; and stellar cycles in three
other stars
3.4 A Possible Explanation: Dynamo Action
How are the complex magnetic features observed over the solar surface
generated? From where does magnetic activity in stars originate? What
explains the solar, and stellar, activity cycles?
First, we have to ask the question: where did the magnetic field
originate from in the first place?
• Here, we assume that field lines were swept up from the interstellar
medium as stars formed, and that this gave rise to a very simple,
primordial solar magnetic field. This alone might be expected to
give rise to a fairly simple field configuration (a magnetic dipole)
• We know in the case of the Sun that the surface magnetism, and its
behaviour over time, is however very complicated.
• We therefore require some means of re-generating and reorganising the original field over time. We must sustain the field
on a timescale that is much longer than the natural timescale for
the decay of the field.
• One means of achieving this is through the action of a dynamo.
• Moving electric charge gives rise to a magnetic B field.
• Similarly, a varying magnetic field induces an electric E field.
• So, when a charged conductor moves through a magnetic field, the
electrons experience a force due to the field and this gives rise to
additional motion and additional electric current; this flows
through the conductor and generates an additional magnetic field.
Therefore, the dynamic motion of charge in the presence of a primordial
B field leads to the amplification and evolution of the ‘seed field’. The
manner in which the field evolves depends upon the dynamic nature of
the interaction: this is a dynamo.
In stars with subsurface convection zones, we have conditions
conducive to dynamo action that can mimic the patterns of field, and
cyclic behaviour, observed in stars.
Basic conceptual models of stellar dynamos can give rise to much of the
coarse large-scale behaviour that observation dictates they must
replicate.
3.4.1 Derivation of the Induction Equation
Boardwork
3.4.2 Mean Field Dynamo Theory
Boardwork
3.5 What must stellar dynamos achieve?
Let us take the example of the Sun, for which we have plenty of
observational data on magnetic phenomena. In order to mimic the
features of the solar magnetic field structure:
− The dynamo must convert poloidal field lines into toroidal field
lines. At solar minimum the solar magnetic field is largely dipolar
in nature.
−
Toroidal lines are oriented parallel to lines of latitude (i.e.,
wrapped around the equator). Sunspot field structure is oriented
predominantly in this manner. So, this is the type of structure we
must produce at times of high activity levels.
− The latitude at which the field lines congregate must:
i. Be fairly concentrated (i.e., to mimic the bands of active
latitude observed on the solar surface; and
ii. Migrate slowly toward the equator as the cycle
progresses (i.e., to mimic the ‘butterfly’ diagram). This
should take about 11 yr.
− We then need to regenerate new poloidal field from the toroidal
components, but in an opposite sense to the old poloidal field. This
gets us back to the dipole-like minimum-activity configuration. It
will give the required field reversal and a 22-yr full magnetic
cycle.
In summary: the main requirement is that we convert poloidal field into
toroidal field, which in turn must be converted back to a poloidal
(reversed) configuration and so on…
In the following subsections, we shall describe what is called the α-ω
dynamo.
3.5.1 Differential Rotation: the ω effect
This effect converts poloidal into toroidal field as a result of the
differential rotation.
Helioseismology (see Section 4) allows us to measure the internal
rotation profile of the Sun. We find (Figure 3.18) that the differential
rotation observed at the surfacewhere regions close to the equator
rotate more rapidly than those at the polespersists to the base of the
convection zone
Figure 3.18: Internal rotation profile of the Sun
• Since the equatorial regions rotate faster than the polar regions, the
lines of force become kinked in the longitudinal direction.
• The progressive distortion of the field that results after many
rotations wraps the field lines along lines of latitude, thereby
giving toroidal field.
• Figure 3.19 shows a poloidal component (left-hand side) being
converted into toroidal field after several rotations.
Figure 3.19: The ω effect
• The direction of the field lines reverses between hemispheres: this
accounts for the opposite observed polarities in the leading and
following members of the sunspot pairs in different hemispheres
(Figure 3.20).
Figure 3.20: Opposite polarity
• Since the rotation rate increases toward lower latitudes, the field
lines will become progressively more concentrated at lower and
lower latitudes as the cycle progresses.
• Current models favour placing the ω effect at the base of the
convection zone, where there are strong rotation gradients across
what is called the tachocline, meaning speed slope (see Figure
3.18). Beneath the tachocline, there is no evidence for differential
rotation.
• Magnetic instabilities lift magnetic field into the convection zone,
where it becomes buoyant. This buoyancy arises because regions
containing field are assumed to be in pressure equilibrium with
their surroundings. Inside such a region, we have gas and
magnetic pressure, while outside we have only gas pressure, i.e.,
Pout = Pin + B 2 / 2 µ0 .
Since ρ = µP/ℜT where ρ is the density of the fluid, T its
temperature and ℜ the gas constant, the presence of a non-zero B
implies that ρ in < ρ out , so the region will be buoyant.
• Field rises through the convection zone. How does the field
strength vary? We already know that the gas pressure falls off
exponentially with some characteristic scale height, H, i.e.,
 r −r 
P1 = P0 exp− 1 0  ,
H 

(R2.13)
and from Equation 3.07 we have:
B = 2 µ 0 ( Pout − Pin )1 / 2 .
(3.16)
But this will be height dependent. So, since B depends upon the
square root of the gas pressures, we would expect a relation of the
following form for the fall-off of the field strength with height in
the atmosphere:
 r −r 
B1 = B0 exp− 1 0  .
 2H 
(3.17)
• Buoyant field that is sufficiently buoyant pierces the surface; the
imprints at the photosphere where the field emerges and then
returns appear as sunspots.
• Magnetic field tends to accumulate in the boundaries between the
convection cells. When of sufficient strength the accumulating
field is able to exert enough pressure to react back on the gas and
stop the accumulation. This happens when the magnetic pressure is
comparable to the kinetic energy density, i.e.,
Pmag = B 2 / 2 µ0 ≈
1 2
ρv
2
(3.18)
This gives rise to characteristic, or ‘typical’, field strength at the
surface of the Sun.
3.5.2 Conversion back to poloidal field: the α effect
The means to get the poloidal field back again is far from clear (and
controversial). Here, we consider one possibility, which depends on the
α term from Equations 3.12 and 3.13, and is often called cyclonic
turbulence.
We have buoyant elements that as they rise through the convection zone
will expand. This gives their motion a twisting component as a result of
being subjected to the Coriolis force.
(Think of the analogy of several people at the centre of a roundabout,
facing out in different directions, who all throw a ball at the same time.
Think of the balls as defining the edge of an expanding object. The
trajectories of the balls are deflected, and follow curved paths: we could
also think of our ‘object’ as being twisted.)
The twisting motion also twists the field lines. If the degree of twist is
just right, and twisted field from many small elements accumulates, new
poloidal field of opposite sign to the original N-S field can be
generated.
Recall that in the mean-field kinematic dynamo the driving part
includes a term that depends on:
α =−
1
(u ⋅ ∇ ∧ u ) ,
3
(3.12)
which depends on the helicity. This helicity is what results from the
Coriolis force.
3.6 Further evidence for Dynamo Action on other Stars
Boardwork
Figure 3.21 Left-hand panel: Activity versus rotation period. Righthand panel: Activity versus Rossby number.
3.7 Dynamo Numbers
We may describe the efficiency of stellar dynamos in terms of the socalled dynamo number, ND. For the α-ω dynamo, we have a number to
represent the efficiency of each effect, so that:
N D = Nα Nω .
(3.26)
We have seen that the α effect depends on the Coriolis force. The faster
the rotation, the more effective the effect will be. It should also depend
inversely on ψ. We assume that, as in the Sun, ψ >> η (Section 3.4.2),
so we only need consider the effects of ψ. Thus far we have:
α
ψ.
To get our dimensions correct, we require an extra lD on top. So:
Nα =
α lD
ψ .
(3.27)
We follow a similar dimensional argument for the ω effect. It should
depend on the gradient of rotation, i.e., on the angular velocity ω
divided by a suitable length scale for the dynamo, lD. Our dynamo
number should be inversely proportional to ψ. So, thus far we have:
ω
lDψ .
To get our dimensions correct, we require an extra lD3 on top. So:
ω lD2
Nω =
ψ .
(3.28)
So, this gives:
α ω lD3
ND =
ψ2 .
(3.29)
This shows us that for stars of a given size, the faster is the rotation, the
more efficient is the dynamo.