Stellar coronae
Magnetic activity
in late-type stars
The Presidential Address of 10 February 1995, in which Carole Jordan describes how
observations of stellar chromospheres and coronae may test theories of stellar dynamos.
O
ur knowledge and understanding of
phenomena related to magnetic activity in late-type stars has advanced
considerably in the past 20 years. Although we
have to treat stars in a simple manner, observations of the Sun at high spatial, spectral and
time resolution can guide our interpretation of
stellar spectra. In particular, the solar atmosphere is the best place to test theories for the
heating of chromospheres and coronae. However, any successful theory for the solar atmosphere must also account for the properties of
stellar chromospheres and coronae over a
range of conditions in stars that have different
masses, effective temperatures and gravities.
There have been considerable advances in
numerical simulations of dynamo activity in
the solar convection zone, but predictions of
how dynamos might operate in other types of
star are at an early stage. Here I will concentrate on what can be learned from the observed
behaviour of the properties of stellar chromospheres and coronae. Most of what I say will
concern main-sequence stars with outer
convection zones, but I will briefly mention
evolved stars that are observed to have hot
coronae.
Since I will be treating stellar coronae as
spherically symmetric regions, I should justify
this approach from observations of the solar
corona. Because many X-ray images of the Sun
have been obtained in lines (or broad spectral
bands) which are formed at temperatures
above that of the average corona, and these
images show dramatically the active region
loops, the existence of the “average” corona
tends to have been ignored. Images obtained in
emission lines formed at temperatures near the
average coronal temperature of about
1.6 × 106 K show the presence of more amorphous emission from the corona. Even when
the average corona on the disc is too faint to be
observed clearly, it is apparent at the limb
because of the greater path length (e.g. see
images obtained with the Harvard extreme
ultraviolet spectrometer on Skylab by Reeves et
al. 1976). This amorphous emission could be
10
from large loop structures, but the emission
scale height is far less than that of the structures, and one can model the regions with a
radial geometry. Observations of the spatially
integrated extreme ultraviolet and soft X-ray
emission from the Sun, carried out from early
ate-type stars have begun to
reveal their inner workings
through a combination of spectral
analysis and numerical simulations,
theory and observations of stars and
the Sun. The empirical trends that
emerge from studies of stellar
chromospheres, transition regions
and coronae set the scene for a
confrontation between observation
and theory, particularly dynamo
theory. Overall, many parameters,
such as the coronal temperatures
and magnetic flux, depend on the
Rossby number, although it is not
yet clear why this dependence takes
the form that it does. In the future,
X-ray, infrared and ultraviolet
observations will improve the
quantity and quality of data
available and should point the way
to a more satisfactory understanding of stellar magnetic activity.
L
satellites (e.g. Neupert 1965), show that the
dominant contribution comes from the average
corona at about 1.6 × 106 K, and varies little
with solar activity. Lines formed at higher temperatures show an increasing variation with
activity. We know that other stars do have
active regions, and their contribution to the
spatially integrated X-ray or extreme ultraviolet (EUV) flux could be found in future by
observing rotational modulation in the
stronger emission lines.
Early studies of stellar chromospheres
Prior to observations from satellites, the main
way to study stellar chromospheres was
through the H and K lines of Ca II in the optical spectrum, although the H Balmer α line
was also used. In a now classical paper, Wilson
and Bappu (1957) made a systematic study of
the width (W0 ∆λFWHM ) of the emission components observed within the photospheric
absorption lines (figure 1). They found a
remarkable correlation between logW0 and the
stellar absolute magnitude, Mv, covering a
range of 15 magnitudes, although a considerable amount of scatter was present for the
giants to supergiants. In dwarf stars these emission lines are well defined and narrow, but in
supergiants the line profiles are complex, and
can be affected by absorption in stellar winds
and the interstellar medium. In a later paper
Wilson (1967) published data for 67 stars for
which parallax measurements were available,
and obtained a best-fit relation for stars with
Mv up to about –1.0, given by
Mv = –14.94 logW0 + 27.59
(1)
Here, W0 is in km s–1. Since then, observations
of the Ca II lines have continued to enlarge
the database and improve the measurements
(e.g. Wilson 1976; Linsky et al. 1979; Pasquini
et al. 1988).
The physical cause of this apparently simple
correlation was a subject of immediate interest.
Two possible explanations were proposed:
● that W0 lies in the Doppler core of the line
and thus reflects mainly the local “turbulent”
motions – in this case the turbulent velocities
would have to increase with Mv;
● that W0 lies in the Lorentzian wings and is
controlled mainly by the mass-column-density,
m NH dh, in the region of the chromosphere
where the lines are formed, so that m would
need to increase with Mv (Linsky 1980).
Several authors have explored the dependence of W0 on other stellar parameters, such
April/May 1997 Vol 38 Issue 2
The solar corona (Fe XIV) viewed with the Extreme
ultraviolet Imaging Telescope (SOHO). (Courtesy of
the SOHO/EIT consortium. SOHO is a project of
international co-operation between ESA and NASA.)
as the gravity and metallicity (e.g. Lutz and
Pagel 1982).
Lutz and Pagel also summarize the scalings
between W1, the base width of the emission
line, and W2, the separation of the emission
peaks. Ayres (1979) concludes that W1 is in the
Lorentzian wings, while W2 is just outside the
Doppler core, so that W0 is likely to be controlled by the mass-column-density. On the
other hand, Lutz and Pagel argue that W2 is
within the Doppler core, and since W2 W0,
W0 is also controlled by Doppler motions.
However, the formation of the Ca II line profiles is a result of complex radiative transfer
and depends critically on the degree of coherence of the re-emitted photons following
absorption. Using complete redistribution with
frequency (and making use of model chromospheres), line profiles are produced that have
much broader wings than those observed. Calculations using partial redistribution (PRD)
lead to much steeper wings, closer to those
observed. Basri (1980) has stressed how subtle
photon diffusion effects, which are sensitive to
even small local velocity fields, play an important role in producing the profiles. These are
expected to be particularly important in the
low-density chromospheres of supergiants. As
Lutz and Pagel point out, the result may be
that W0 lies at the transition between the
Doppler and Lorentzian regimes.
The only way of determining the total
Doppler broadening in the chromosphere is to
observe lines that are definitely optically thin.
There are several intersystem lines of neutral
atoms (C I, O I, S I) that occur within the
wavelength range of the International Ultraviolet Explorer (IUE) and which are observable
at high resolution. The line of C I at 1993.6 Å
is observable in stars covering a range of Mv.
This is the 2p3s 3P10 – 2p2 1D2 transition that
shares a common upper level with a transition
to the ground 2p2 3P term, at 1657.4 Å. The
latter transition is optically thick and transfers
photons to the 1993.6 Å line, thus the ratio of
the fluxes in the two lines can be used to estimate the optical depth, τ, in the resonance line,
and hence the mass-column-density (Jordan
1967). Judge (1990) has shown that the two
lines are not formed in precisely the same
region of the chromosphere, so the absolute
value of τ is only approximate. However, the
April/May 1997 Vol 38 Issue 2
method indicates the relative values of τ
between different stars, as discussed by Jordan
et al. (1987). The C I lines are formed roughly
around 5000 K, in the region where the Ca II
lines are formed (Judge 1986a,b). The widths
and fluxes in the C I lines are available for a
range of dwarf and evolved stars from observations with IUE, and more recently at higher
spectral resolution from observations with the
Goddard High Resolution Spectrograph
(GHRS) on the Hubble Space Telescope (HST)
(e.g. Carpenter et al. 1994).
These observations show that the width of
the 1993.6 Å line increases by about a factor of
two over a range Mv from 6 to –2.5, while the
mass-column-density, m, increases by about a
factor of 7. Thus the increase in m appears to
be more important than the increase in the turbulence. However, the ratio of W0 (from the
Ca II lines), to the FWHM of the C I 1993.6 Å
line increases from about 1.2 for the dwarfs to
about 2 for the giants and to about 6 for the
supergiant α Ori. Since the observed C I widths
reflect the largest scale of the turbulence, the
microturbulent velocities that are important in
the Ca II radiative transfer may be smaller, so
the above ratios give the minimum number of
Doppler widths at which W0 occurs. Thus it is
likely that in the main-sequence stars, and perhaps the K giants, W0 is controlled by the
Doppler motions, but as Mv increases, W0
becomes much larger than the Doppler width.
This is consistent with the schematic calculations carried out by Basri (1980), who concludes that turbulence alone is not the cause of
the large values of W0 in the supergiants. However, Basri stresses that because of the subtle
effects of Doppler drifting in PRD one cannot
conclude that the effects of the higher masscolumn density in the supergiants are the only
controlling factor.
Thus although the Wilson–Bappu effect
appears as a simple correlation, the underlying
physical effects are subtle, and a proper understanding of the observations will be obtained
only through detailed modelling of a sample of
stellar chromospheres covering a range of Mv.
The Ca H and K lines
Studies of the Ca II emission in different
regions of the solar chromosphere showed that
the emission line fluxes, FHK, are largest in
regions where the surface magnetic field, Bs,
is strongest; for example in the supergranulation network and in active regions (plages)
around sunspots (Skumanich et al. 1975). The
correlation between the Ca II flux and the
magnetic field in solar active regions has been
studied further by Schrijver et al. (1989),
among others, and can be expressed as
log∆RHK = 0.60(±0.10)logBs fA – 6.0
(2)
FHK – Fmin
(3)
σTeff4
where Bs fA is the magnetic field averaged over
the area of the active region, Fmin is a basal
flux in regions of low magnetic field (Rutten
1987), and Teff is the stellar effective temperature. Work by Saar and Schrijver (1987) and
Montesinos and Jordan (1993) has shown
that, within the uncertainties, the same correlation holds for stellar observations of ∆RHK
and spatially averaged stellar magnetic fields,
Bs fs, where fs is the fraction of the stellar surface area occupied by the mean magnetic field.
Since it is far easier to measure stellar Ca II
fluxes than stellar magnetic fields, the dependence of these lines on other stellar parameters is very useful as a proxy for the magnetic
field. To understand the correlation given by
equation (2) we need to know the relation
between the magnetic fields in the photosphere and the chromosphere, and how the
chromospheric structure depends on the heating of the atmosphere. From simple spatially
uniform chromospheric models we do know
how the Ca II flux depends on parameters
such as the electron pressure at the base of the
transition region.
In an important paper, Noyes et al. (1984)
found that, in main-sequence stars, the chromospheric Ca II flux depends on the stellar
rotation period, Prot, but there is a further
dependence on spectral type (e.g. as described
by (B–V)). They found an improved correlation when Prot was replaced by the Rossby
number, Ro = Prot /τc, where τc is the theoretical
convective turnover time at the base of the convection zone, for a constant value of the ratio
of the mixing length to the scale height, at the
base of the convection zone. Their analysis can
be repeated with ∆RHK, and the result is shown
in figure 2 (from Montesinos and Jordan
1993), which is fitted by
∆RHK =
11
Stellar coronae
log∆RHK = –3.94(±0.03) – 0.50(±0.03)Ro (4)
From the limited data available, both Bs fs
and fs appear to depend on Ro (Montesinos
and Jordan 1993), and so does the magnetic
flux. At present the measurements of magnetic
fields and surface filling factors are not of
sufficient quantity or quality to distinguish
between the latter two alternatives. However,
Bs seems to be determined by the equipartition
value (Saar et al. 1986) and varies slowly with
(B–V). If the surface parameter determined by
the stellar dynamo is the magnetic flux, then fs
would be determined by both Ro and Teff. If
the correlations could be improved, then eventually they could be used to test the predictions
of stellar dynamo theory. So far this has been
explored in only a simple way (Montesinos
and Jordan 1993).
Plots of log∆RHK against (B–V) (e.g. Rutten
and Schrijver 1987) show a range of values of
∆FHK for each (B–V), with an upper limit that
decreases with (B–V). This behaviour is consistent with the form of equation (4). At a given
(B–V) the flux increases as Ro decreases, and
there is a limiting value as Ro tends to 0. For a
given value of Ro, the flux decreases as Teff
decreases.
Equations (2) and (4) also show that the Ca
II flux does not depend only on Teff, as expected if the chromospheric heating were by
acoustic waves. From the work of Stein (1981),
it seems most likely that the chromospheric
heating is by slow mode magnetohydrodynamic (MHD) waves. The energy input required
can be established by making models of the
chromosphere. This needs a detailed treatment
of the radiative transfer, since the photons
observed are not escaping from the atmosphere
at the point where they are created.
The situation for giant stars is less clear.
Work by Rutten (1987) and Strassmeier et al.
(1994) shows that there is a general decrease in
the Ca II fluxes as Prot increases, but with a
large scatter about the mean behaviour. Also,
the faster rotators have (B–V) in the range 0.4
to 0.6, while the slower rotators have (B–V) in
the range 0.8 to 1.2. Unlike the situation for
main-sequence stars, there is a range of stellar
masses for a given (B–V), so that one would
not expect a unique dependence of ∆RHK on
Prot, since the structure of the convection zone
will depend on the mass and state of evolution
of the star.
Figure 3 shows log∆RHK versus Prot for single
giants in the sample of Rutten (1987), using his
sources for the values of Teff. ∆RHK appears to
“saturate” at short periods, but at a smaller
value than for the main-sequence stars. We do
not expect Prad to be the only factor controlling
the level of activity; in addition, Ro may not
be the best parameter to use when comparing
dwarf and giant stars. Gilliland (1985) and
12
Rucinski and Vandenberg (1986) have
suggested that a dynamo number, defined as
ND = (Rc /Hc )1/2Ro–1, where Rc and Hc are the
radius and pressure scale-height respectively, at
the base of the convection zone, may be a more
appropriate parameter than Ro. Jordan and
Montesinos (1991) also found that using ND
brought coronal parameters for evolved stars
on to the same correlations as found for mainsequence stars. Since the ratio (Rc /Hc )1/2 is
almost constant, and is 2.5–3 for mainsequence stars later than about F2, Ro alone
can be used in correlations. However, since for
the slightly evolved stars considered by Rucinski and Vandenberg (1986) the above ratio is
1, ND should be used to put the evolved stars
on the same scale as the main-sequence stars. If
the correlation between RHK and ND has the
same form for all stars, then the factor
τc(Rc /Hc ) can be found for the giants shown in
figure 3. The results depend on the value
adopted for the saturated RHK , but for the
giants with (B–V) in the range 0.90 to 1.2, the
average value of τc(Rc /Hc ) is 100 days, and
for giants with (B–V) in the range 0.4 to 0.5 is
≤ 7 days. Thus it should be possible to compare
these empirical results with the predictions of
models for the structure of giant stars.
UV and X-ray observations
There have been two main types of study using
observations of ultraviolet emission-line fluxes
and X-ray broad-band fluxes. In the first, a few
strong lines have been observed in a large sample of stars, with the aim of establishing correlations (e.g. Ayres et al. 1981; Oranje 1986;
Rutten and Schrijver 1987; Capelli et al. 1989;
Rutten et al. 1991; Ayres et al. 1995). In the
second, a few stars have been observed in
detail with IUE, at both low and high spectral
resolution, to make models of the chromosphere, transition region and corona and to
examine the energy balance and heating
requirements (e.g. Brown et al. 1984; Jordan
and Brown 1981; Jordan et al. 1986, 1987).
Such models also allow the physical basis of
any correlations to be investigated.
Several authors (such as Oranje 1986;
Capelli et al. 1989; Rutten et al. 1991; Ayres
et al. 1995) have found that the fluxes in lines
formed in the transition region between
2 × 104 ≤ Te ≤ 105 K, are directly proportional to
each other. For example, Capelli et al. (1989)
found
0.97
FCIV FCII
(5)
for a variety of stars and the Sun, including
solar observations made at different spatial
resolutions. Similarly, the non-basal Ca II and
Mg II fluxes are roughly directly proportional
to each other.
However, there are nonlinear correlations
between the chromospheric and transition
region fluxes, and the transition region and
coronal fluxes. For example:
and
1.5
FCIV ∆FMgII
(6)
1.3–1.5
FX FCIV
(7)
These correlations are remarkable in that
they hold for all types of stars later than about
type mid-F, which have hot coronae (except
M dwarfs, which are expected to be fully convective). We know from observations of the
Sun that the emission over the surface of the
chromosphere and transition region is not uniform, but is enhanced over supergranulation
boundaries. Thus, to within the scatter
observed, the averaging over different regions
of the surface is either not important, or
behaves in a systematic way, which is contained within the correlations.
In the Sun, the area occupied by the supergranulation network varies little between the
temperatures at which C II and C IV are
formed (Reeves 1976) (2 × 104 K and 105 K
respectively), so this may be one reason for the
linear correlation observed for the fluxes of
these lines. However, the magnetic field lines
diverge from the supergranulation boundaries
into the average corona, so one would expect
the changing fractional area to be a factor in
the correlation between the C IV and X-ray
fluxes.
When making detailed models of individual
stars, the line fluxes are used to derive an emission measure for each line (either an average or
as a function of temperature, Te). The emission
measure, averaged over ∆logTe = 0.30 dex, the
typical temperature range for the line formation, is defined as
Em(0.3) =
∆R
NeNH dr
(8)
In studying five well observed main-sequence
stars covering a range of absolute surface fluxes, Jordan et al. (1987) found that, between
about Te = 2 × 104 and 105 K, the shape of the
emission measure distribution with Te was the
same to within the uncertainties in the line
fluxes. A universal shape for the emission measure distribution would lead to a linear correlation between all transition region lines
formed in the above range of Te.
To understand what determines the shape of
the emission measure distribution we have to
examine the energy balance. Below about
2 × 105 K, the energy carried by classical thermal conduction back from the corona is small.
In the absence of strong flows, the energy
balance is between the energy deposited by
whatever heating process is operating and the
radiation losses. Allowing for the area of the
emitting region, A, this can be expressed as
d(AFm)
dFR
dr
=–
= –NeNHPrad(Te) AdTe
dTe
dTe
(9)
where Prad(Te) is the radiative power loss function.
April/May 1997 Vol 38 Issue 2
Stellar coronae
The observed (spatially averaged) emission
measure can be written as
(10)
(12)
in all stars with hot coronae (with same exceptions as above). This is a remarkable result,
which must eventually be accounted for by any
heating process proposed.
The form of equation (12) can be compared
with that for heating by the dissipation of
acoustic or MHD (e.g. Alfvén) waves. If Alfvén
waves are passing through the transition
region to the corona, the flux Fm is given by the
non-thermal energy density multiplied by the
propagation velocity, VA = B/(4πρ)1/2, so
(14)
ρ can be expressed in terms of Pe and Te, and,
over the temperature range of interest, Pe1/2 is
almost constant. Thus differentiating equation
(13) and using equation (14) leads to:
d(FmA/A∗)
constant Te(x–1/2)(x – 1/2) (15)
dlnTe
which can be compared with the form of relation (12). Solar observations show that the
non-thermal velocities follow the form of equation (14) with x 1/2. If x = 1/2, then there
would be no energy dissipation, which is clearly not the case. However, x is close to 1/2, to
satisfy the observational constraints from the
behaviour of the emission measure distribution
and the line widths (Jordan 1991). If there is an
upwards flux of wave energy, then x ≤ 1/2.
If waves were passing through the transition
region to be mainly dissipated in the corona,
then only a small fraction of the wave energy
would need to be dissipated in the low transiApril/May 1997 Vol 38 Issue 2
log ∆R HK
–0.5
–1.0
log Ro
0.0
0.5
2 log∆RHK plotted against logRo, for
FV–KV stars. The symbols refer to
different methods of measuring Prot
(Montesinos and Jordan 1993). The
best fit is given by equation (4).
–4.0
–5.0
1.4
–6.0
0.0
2.0
1.0
log Prot
3 log∆RHK plotted against logProt (days)
for single giant stars. The symbols
refer to (B–V ) in the following ranges:
green squares (0.40–0.45); red
diamonds (0.46–0.62); purple crosses
(0.63–0.80); blue stars (0.84–1.00);
yellow triangles (1.01–1.20).
1.0
0.6
0.0
0.5
1.0
Ro
1.5
2.0
4 log (B c β1/2 ) plotted against Ro, for
main-sequence stars. B c is in
Gauss. Symbols: green squares FV,
purple crosses GV, yellow triangles
KV. The best-fit line is given by
equation (22).
(13)
where ρ is the density and VT2 is the turbulent velocity derived from observed emission
line widths. Although the area A occupied by
the emission is roughly constant (see above),
this form is useful since we expect that
BA = constant. Solar observations show that
one can write
VT2 = V02(Te /T0)x
–5.0
1 Schematic Ca II H or K line profile,
showing W0 ∆λ FWHM , W1 base
width, and W2 separation of
emission peaks.
9.0
log Fm(Tc ) g ∗1/4
ρ1/2
VT2 BA
FmA = (4π)1/2
λ
log ∆R HK
In the main-sequence stars studied (Jordan et
al. 1987), the product Em(0.3)Prad(Te) is
almost constant in the temperature range from
2 × 104 to 105 K (Jordan 1991, figure 5). Thus
the approximately universal shape of the distribution of Em(0.3) with Te implies that
–4.5
W1
d(FmA/A∗)
1
A
(11)
Em(0.3) = – dlnTe
2 Prad(Te) A∗
log (Bc β1/2 )
so that replacing the temperature gradient in
equation (9) gives
d(FmA/A∗)
constant
dlnTe
–4.0
W0
flux
PePH dr A
Em(0.3) = 21/2Te dTe A∗
W2
8.5
8.0
7.5
7.0
0.0
0.5
1.0
Ro
1.5
2.0
tion region. However, as suggested by Fiedler
and Cally (1990), the energy could be carried
down from the corona, by non-classical turbulent thermal conduction. To determine the
direction of the energy flux one needs to know
the value of x to greater accuracy than at present. This may be possible from solar observations with the SUMER instrument to be flown
on SOHO.
Trends from X-ray observations
Systematic trends in mean coronal emission
measures, Em(Tc), and temperatures, Tc, have
been established (Jordan and Montesinos
1991; Montesinos and Jordan 1993) from
5 logFm (Tc)g∗1/4 (where Fm is the
total coronal energy loss in
erg cm–2 s–1 ), plotted against Ro, for
main-sequence stars. The best-fit
line is given by equation (23). The
stars have the same relative
positions as in figure 4. The dotted
line is that predicted by the
minimum energy loss solution
(Montesinos and Jordan 1993).
observations made with the Einstein Observatory (Schmitt et al. 1990). For all types of stars
with hot coronae there is a broad overall trend
for Em(Tc) to scale as
Em(Tc) g∗Tc3
(16)
Writing
0.8Pc2H
Em(Tc) = (17)
Tc2
where H is the pressure-squared isothermal
scale-height, given by H = 7.1 × 107 Tc /g∗, leads
to
Pc Tc2 g∗
(18)
The scaling given by (18) can be obtained by
setting the energy flux from Alfvén waves
or magnetic reconnection, to be equal to the
13
Stellar coronae
energy lost from the corona by conduction and
radiation, provided the plasma β (= 8πPgas /Bc2 )
at which the energy dissipation occurs is a constant (Bc is the coronal magnetic field), and the
energy density in the wave, or perturbed magnetic field, as a fraction of Bc2 is a constant. In
other words, the magnitude of the flux is
dT
dB2 B3
= κTe5/2 e + Ne NH Prad (Te )dr (19)
B2(4πρ)1/2
dr
Using equations (10) and (17), and substituting for B in terms of β and Pc, this becomes
δB2 Pc Te1/2
Pc2
5/2
(20)
c1 2
3/2 = c2Tc g∗ + c3 B
β
Tc3/2g∗
where c1, c2 and c3 are constants, and a radiative loss function Tc–1/2 has been used. If the
energy dissipation occurs at fixed values of
δB2/B2 and β, then equation (20) can only
be satisfied if Pc scales according to relation
(18), and hence Em(Tc) scales as relation (16).
The minimum energy loss solution (Hearn
1975, 1977) leads to the same scaling. In practice, the dependence on temperature of the
radiative power loss function will vary with Tc
and the assumption of plane parallel geometry
and a spatially uniform corona may not be
appropriate for evolved stars.
Thus the correlations between coronal parameters and the fluxes of lines formed at different temperatures are the result of the energy
balance within the atmosphere (plus any area
factors), which determines the structure and
plasma parameters.
In our study of the sample of stars observed
by Schmitt et al. (1990) we found that, for the
same value of Prot, evolved stars have higher
coronal temperatures than main-sequence stars
(Jordan and Montesinos 1991). This alone
shows that stellar factors other than Prot are
involved in determining the level of stellar
“activity”. For the main-sequence stars, both
Em(Tc) and Tc can be fitted to relations of the
form ae–bRog∗–1/2, where a and b are constants
(see Montesinos and Jordan 1993, for numerical fits).
Using the plasma β to replace the electron
pressure term in Em(Tc), an expression can be
found for the coronal magnetic field, that is
8.6 × 10–10 (Em(Tc)Tc g∗)1/4
Gauss (21)
Bc = β1/2
As shown in figure 4, for a fixed value of β, Bc
depends on Ro according to
log(Bc β1/2 ) = 1.20(±0.04) – 0.39Ro(±0.04)(22)
Thus, if the energy dissipation always occurs at
a fixed value of β, the coronal magnetic field is
also determined by Ro (or ND). However, the
actual dependence on Ro is not consistent with
there being constant magnetic flux from the
photosphere, implying that not all of the surface field extends as far as the corona, and that
14
the fraction that does also depends on Ro.
Finally, the total energy required to balance
the sum of the energy losses by conduction and
radiation, derived from the coronal emission
measures and temperatures, also depends on
Ro, with a small dependence on g∗, according
to
line widths made with instruments on SOHO.
Thus, although there have been considerable
advances over the last 20 years, there is much
still to do before we have a satisfactory understanding of stellar magnetic activity as a function of stellar structure and evolution. ●
log(Fm g1/4
∗ ) = 9.0(±0.1) – 1.0(±0.1)Ro (23)
Prof. Carole Jordan, President of the RAS from 1994
to 1996, works at the Department of Physics (Theoretical Physics), University of Oxford, 1 Keble Road,
Oxford OX1 3NP.
as shown in figure 5 (from Montesinos and
Jordan 1993). Thus for the main-sequence
stars, all the coronal parameters (e.g. Tc, Pc, Bc)
and the heating required depend on Ro.
Future work
In this article I have concentrated on the empirical trends that have emerged from studies of
stellar chromospheres, transition regions and
coronae. These provide a point at which theory can be confronted by the observations.
Although the correlations between the properties of these regions are broadly understood in
terms of the energy balance, in the long term,
dynamo theory will need to explain the dependence of the surface magnetic flux on the simple dynamo parameters. Studies of dynamos in
stars other than the Sun are at an early stage
(Brandenberg et al. 1994).
Because of the dependence of emission line
fluxes on Ne2, the X-ray observations refer to
material quite close to the stellar surface, far
below the region where the maximum heating
probably occurs. The same factors that determine the surface magnetic flux and field configuration must also determine the more
extended magnetic field, as well as the stellar
wind, the rotational evolution of a star and the
fraction of closed magnetic field lines at a given
distance from the star (e.g. Mestel 1968; Mestel and Spruit 1987). At present we do not
understand why the dependence of the various
parameters on Ro have the form that they do.
The surface magnetic fields and filling factors
are difficult to measure and at present are
known for relatively few stars, but the situation may be improved by measurements in the
infrared (e.g. Valenti et al. 1994). Long-term
observations of the rotational modulation of
the Ca II lines (Baliunas et al. 1985, 1995) are
leading to improved stellar rotation rates and
the detection of differential rotation (Donahue
and Baliunas 1992).
Observations made with ROSAT and the
Extreme Ultraviolet Explorer are providing
new information on the form of the emission
measure distribution in the high transition
region and corona, and observations with the
GHRS on the HST should give more accurate
measurements of line widths and Ne, allowing
improved models of the structure and energy
balance for individual stars. The best tests of
particular heating processes should come from
spatially resolved measurements of fluxes and
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April/May 1997 Vol 38 Issue 2