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THE ASTROPHYSICAL JOURNAL, 557 376õ383, 2001 August 10
No copyright is claimed for this article. Printed in U.S.A.
THE CASE AGAINST COLD, DARK CHROMOSPHERES
WOLFGANG KALKOFEN
Harvard­Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138
Received 2000 January 28 ; accepted 2001 April 10
ABSTRACT
Is the solar chromosphere always hot, with relatively small temperature variations (dT /T D 0.1), or is
it cold most of the time, with temperature ÿuctuations that reach dT /T D 10 at the top of the chromo­
sphere? Or, equivalently, is the chromosphere heated continually or only for a few seconds once every 3
minutes? Two types of empirical model, one essentially time independent and always hot, the other
highly time dependent and mostly cold, come to fundamentally di+erent conclusions. This paper analyzes
the time­dependent model of the quiet, nonmagnetic chromosphere by Carlsson & Stein and shows that
it predicts deep absorption lines, none of which are observed ; intensity ÿuctuations in the Lyman contin­
uum that are much larger than observed ; and time­averaged emission that falls far short of the observed
emission. The paper concludes that the solar chromosphere, while time­dependent, is never cold and
dark. The same conclusion applies for stellar chromospheres. A complete, time­dependent model of the
nonmagnetic chromosphere must describe two phenomena : (1) dynamics, like that modeled by Carlsson
& Stein for chromospheric bright points but corrected for the geometrical properties of shocks propagat­
ing in an upward­expanding channel, and (2) the energetically more important general, sustained heating
of the chromosphere, as described by current time­independent empirical models but modiïed in the
upper photosphere for the formation of molecular absorption lines of CO in a dynamical medium. This
model is always hot and, except for absorption features caused by departures from local thermodynamic
equilibrium, shows chromospheric lines only in emission.
Subject headings : hydrodynamics õ shock waves õ Sun : atmospheric motions õ
Sun : chromosphere õ Sun : infrared õ Sun : UV radiation
1. INTRODUCTION
Empirical models of the temperature structure of the
solar chromosphere have traditionally aimed at repro­
ducing the emergent spectrum. Such modelsõfrom the Bil­
derberg model (Gingerich & de Jager 1968) to the
Harvard­Smithsonian Reference Atmosphere (Gingerich et
al. 1971) and culminating in the models by Vernazza,
Avrett, & Loeser (1981, hereafter VAL81), with improve­
ments in the layers of formation of the emission features in
the calcium H line by Avrett (1985) and an extension to the
solar transition region by Fontenla, Avrett, & Loeser (1993,
hereafter FAL93)õhave in common a monotonic tem­
perature rise in the outward direction. Inhomogeneous
brightness components, such as quiet cell, magnetic
network, and active regions, are represented by separate
spatial models. Intensity variations with time, which can
amount to an order of magnitude in the extreme­ultraviolet
(EUV), are accounted for by temperature variations of a few
hundred degrees.
A new kind of model by Carlsson & Stein (1994, hereafter
CS94) simulates the dynamics observed by Lites, Rutten, &
Kalkofen (1993, hereafter LRK93) at several (Ca II) H
2V
bright­point positions in the quiet, nonmagnetic chromo­
sphere. While the model is successful in matching observed
velocity shifts in the H line, it predicts an emergent H
2V
intensity at maximal brightness that is much higher than
observed, a line core in the H line at maximal redshift that is
much darker than observed, and a time­averaged emission
from all heights in the chromosphere that is much lower
than observed.
In addition to reproducing the dynamics, the simulations
produce a time­dependent temperature structure that is fun­
damentally di+erent from the temperature structure of the
earlier models. Except for the shock wave that causes the
bright points, the CS94 model has a temperature that
decreases in the outward direction and is much colder than
their starting model or a model by Kurucz (1996), both of
which are in radiative equilibrium. The temperature ÿuc­
tuations in this dynamical model are very large, at times
exceeding a factor of 10 at the top of the chromosphere. A
time average of the temperature structure for an interval
shorter than the wave period would show the intermittence
of the heating and would not resemble the VAL81 models.
A previous paper (Kalkofen, Ulmschneider, & Avrett
1999) discussed the likely cause and remedy of the energy
deïcit of the CS94 model. The present paper emphasizes the
observational evidence against a chromosphere that is
heated only intermittently. Section 2 presents the tem­
perature structure of the VAL81 model and shows that the
likely heating mechanism of the solar chromosphere is con­
tinuous shock dissipation ; ° 3 describes the dynamical
simulations of CS94 that provide intermittent shock
heating ; ° 4 predicts features of the emergent line radiation ;
° 5 estimates ÿuctuations of the intensity in the Lyman
continuum ; ° 6 considers constraints on models from obser­
vations of infrared lines of the CO molecule ; and ° 7 dis­
cusses various time averages of the time­dependent
temperature. In ° 8 I draw conclusions.
2. SHOCK HEATING OF THE QUIET
SOLAR CHROMOSPHERE
The several empirical models by VAL81 and FAL93 rep­
resent di+erent brightness components of the solar chromo­
sphere. Near the base of the chromosphere at a height of 0.5
Mm, the spatial resolution of the Skylab observations is
5@@ ] 5@@, and in the middle chromosphere at about 1 Mm,
where the emission in the resonance lines of Ca II arises, the
spatial resolution of the H line observations by Cram &
376

CASE AGAINST COLD, DARK CHROMOSPHERES 377
(1983, hereafter CD83) is 500 km and the
Dame
# (0A. 7)
cadence is 10 s. Both the Skylab and the H line observations
ïgure prominently in the VAL81 models (see Avrett 1985).
The component models A through F all have typical tem­
peratures near 6000 K in the middle chromosphere. At a
height of 1 Mm, models A and F di+er from model C, the
average model, by less than 300 K, and the rise to the fully
ionized chromosphere occurs in all models within a band of
100 km near the height of 2.2 Mm.
The chromosphere of the quiet Sun shows large varia­
tions in the intensity of the emergent radiation ïeld. The
corresponding temperature ÿuctuations are bracketed by
models A and F. Although higher temperatures than
implied by model F and cooler temperatures than implied
by model A occur, these extreme conditions cannot play a
signiïcant role in the energy budget of the chromosphere.
For the coolest conditions, for example, CD83 remark that
they give a darker H line proïle than model A does. But
even the lowest 10% of their observations show relatively
high emission at line center, which implies that cold
H 3 ,
conditions can occur only for much less than 10% of the
time or that the temperature drop below that of model A is
minor.
Andersen & Athay (1989) analyzed the average chromo­
spheric model, VAL81 model C. Instead of pursuing the
usual aim in theoretical modeling, which had been to repro­
duce the height­dependent cooling function of the model
(Stein 1985), they started from the empirical temperature
structure and included all important opacity sources,
among them lines of Fe II and other metals, which doubled
the opacity used by VAL81. The result of this investigation
was the conclusion that the radiative emission rate in the
chromosphere is a linear function of mass density, except in
the layers near the base of the chromosphere, where the
emission increases sharply from its low value in the upper
photosphere to its ïnal value in the chromosphere.
Heat conduction is negligible in the layers under dis­
cussion, 0.5 Mm\ z \ 2 Mm. Therefore, the dissipation
mechanism(s) heating the medium must deliver the energy
powering the radiation at the point of emission.
Consider the properties of plane acoustic waves. The
energy ÿux, is
F wave ,
F wave \ c s ov2 , (1)
where is the speed of sound, o the mass density, and v the
c s
velocity ÿuctuation of the gas. Since the sound speed
depends only weakly on temperature (as and the tem­
JT )
perature varies only weakly in the chromospheric layers
above 1 Mm, we may assume that is constant. We are
c s
then concerned with the dependence of on o and v
F wave
only.
In the low­amplitude limit, the energy ÿux in the
v > c s ,
wave is constant ; hence, the velocity grows exponentially
with height to compensate for the exponential decay of o,
i.e.,
v P o~1@2 P ez@2H, F wave \ const , (2)
where z is height above and H is the density scale
q 5000 \ 1
height of the gasõfor convenience we consider isothermal
conditions and one­dimensional wave propagation.
Because of the exponential increase (eq. [2]) with height,
the velocity grows into the nonlinear regime until
(v Z c s )
dissipation in shocks limits further growth. Then the veloc­
ity remains approximately constant (Ulmschneider 1991),
and the energy ÿux decays with height like the mass density,
i.e.,
F wave P o, v \ ac s , a B const , (3)
and the rate at which shocks dissipate energy is proportion­
al to o,
dF wave
dz P o . (4)
Thus, heating and cooling rates have the same, i.e., linear,
density dependence. Note that this conclusion applies to the
instantaneous heating and cooling rates.
The energy ÿux generated in the convection zone by the
Lighthill mechanism (Stein 1967, 1968 ; Musielak et al.
1994) is large enough to cover the radiation losses of the
chromosphere. Therefore, acoustic waves are able to heat
the chromosphere. This has been demonstrated with mono­
chromatic acoustic waves (e.g., Schmitz, Ulmschneider, &
Kalkofen 1985) for a wave period of 30 s ; we expect that a
continuous wave spectrum achieves the same result. A time
average of such models shows a continuous outward tem­
perature rise, resembling that of the VAL81 models. For a
continuous acoustic wave spectrum we expect the instanta­
neous models to resemble the VAL81 models but also to
show the temperature ÿuctuations inherent in the nonlinear
process of shock heating.
Any heating mechanism other than continuous shock
dissipation would have to meet the requirement demanded
by the VAL81 models, that the dissipation rate is a linear
function of mass density. Our result (eq. [4]) therefore pro­
vides theoretical support for the empirical models built on
observations of the emergent chromospheric radiation.
Thus, one can argue, the empirical VAL81 models correctly
describe the temperature structure of the quiet solar
chromosphere. These models allow temperature ÿuctua­
tions of relatively low amplitude owing to sustained, contin­
ual shock heating, but they do not describe the huge
temperature variations that occur in the CS94 simulations
of chromospheric bright points.
3. THE TEMPERATURE FROM SIMULATIONS
OF THE DYNAMICS
Observations of calcium bright points in the quiet non­
magnetic chromosphere suggest a causal link between
velocity shifts of the Ca II H line in the chromosphere and
Doppler motions of an Fe I line in the photosphere imme­
diately below. CS94 simulated these dynamics by taking the
velocity ÿuctuations of the iron line from a 1 hr observing
run of LRK93 as a condition at the lower boundary in their
radiation hydrodynamics code and determined the sub­
sequent velocity shifts in the H line. Except for a brief time
interval on the order of the 3 minute wave period when the
waves enter an undisturbed, initial atmosphere, the simula­
tions reproduce the characteristic features of H line behav­
ior, such as the occurrence of maximal brightening of the
blue emission peak, simultaneously with maximal red­
H 2V ,
shift of the line center, Discrepancies of the simulations
H 3 .
with the observations of LRK93 in the timing of the velocity
shifts in the line proïle, although they amount to a substan­
tial fraction of the sound travel time, are of minor signiï­
(1 3 )
cance, allowing the conclusion that the underlying physics
of the model is correct ; but major discrepancies in the H 2V
and intensities suggest that the underlying temperature
H 3

378 KALKOFEN Vol. 557
structure and the geometry of wave propagation are di+er­
ent from those of the Sun.
The dynamical simulations also yield a time­dependent
temperature structure (Fig. 1) with extraordinary proper­
ties, from which CS94 and Rutten (1994) concluded that the
solar chromosphere does not have a steady temperature
rise. This was further emphasized in a paper by Carlsson &
Stein (1995, hereafter CS95) titled ```` Does a Nonmagnetic
Solar Chromosphere Exist ? îî followed by a paper (Carlsson
& Stein 1998) titled ```` The New Chromosphere.îî The con­
clusions from this modeling were extended to cool stars in
general by Schrijver (2001), who stated that ```` there is no
such thing as a steady temperature minimum over the top
of their photospheres.îî
The salient features of the dynamical temperature struc­
ture, shown in Figure 1, and of a snapshot, shown in Figure
2, are a lower envelope for the cool background, with the
temperature decreasing monotonically with height ; an
upper envelope for the temperature excursions owing to the
upward­propagating shock wave, with the amplitude of the
temperature ÿuctuations generally increasing with height ; a
time average representing the straight time average
throughout the atmosphere, with the temperature decreas­
ing monotonically with height ; and two monotonically
increasing temperature distributions, one corresponding to
the time­averaged emission of the dynamical CS94 model
(the semiempirical curve in Fig. 1) and the other, to the
time­averaged emission of the Sun (FAL93 model A).
The temperature distributions in Figure 1 allow us to
infer a cooling time for the medium and to construct a
two­component model, consisting of a cool background
and a thin, hot slab moving upward over it (see Fig. 2, from
CS95). We estimate the cooling time from the fraction of the
time the atmosphere is in the high state by noting that the
maximal temperature at the top of the atmosphere is
K, the minimal temperature is K
T max \ 25,000 T min \ 2500
(see CS95), and the average is K. The tem­
T av \ 3800
FIG. 1.õTemperatures from simulations by CS94 of the dynamics
observed by LRK93 : cool background model (lower solid curve), upper
temperature excursion of the hot shock (upper solid curve), time average of
the temperature ÿuctuations (thick solid curve), temperature representing
the emission averaged over the observation time of 1 hr, labeled
```` semiempirical îî (thick dashed curve), and the coolest of the FAL93 models
(dash­dotted curve)õfrom CS94.
FIG. 2.õSnapshot from a dynamical simulation, showing a single tem­
perature spike at 1.4 Mm. At the top (z [ 1.7 Mm), the temperature drops
to 2000 Kõfrom CS95.
perature di+erences and are in the
(T max [ T av ) (T av [ T min )
ratio of 20 : 1, implying, grosso modo, that the times spent
by the atmosphere in the high and low states are in the ratio
of 1 : 20 and that the cooling time must therefore be on the
order of 1/20 of the wave period. The temperature behavior
behind the shock that is suggested by Figure 2 is a drop that
is linear in time and begins as soon as the temperature
reaches its maximal value ; it gives a fraction of 88% for the
time interval in the cold state and 6% for the time interval
during which the temperature is above the halfway point
between maximum and minimum. From the wave period of
3 minutes and the sound speed of 7 km s~1, which is the
wave speed in the linear limit, we estimate a cooling time
of 10 s and a thickness of the slab at the halfway height of
70 km.
The actual simulations give the temperature as a function
of height at a ïxed time, and as a function of time at a ïxed
height : Figure 2, from Figure 1 of CS95, shows a snapshot
of a simulation with the temperature proïle as a function of
height when the shock is at 1.4 Mm. The thickness of the
hot slab, deïned as its width at half maximum, is approx­
imately 80 km. And in Figure 3, from Figure 4 of CS94, the
temperature variation as a function of time at unit optical
FIG. 3.õTemperatures (in 103 K) as functions of time (in seconds) at
optical depth unity at the Lyman absorption edge the solid curve is the
brightness temperature of the emergent intensity at j \ 912 the dotted
A# ,
curve is the source function at the dashed curve is the kinetic
q Lc \ 1,
temperature at the solid horizontal line is the time average of the
q Lc \ 1,
brightness temperature during the time segment of the ïgure, and the
dashed horizontal line is the temperature corresponding to the time
average of the Planck functionõfrom CS94.

No. 1, 2001 CASE AGAINST COLD, DARK CHROMOSPHERES 379
depth at the Lyman absorption edge shows that the tem­
perature peaks have a width of about 10% of the wave
period. The cooling time in the layer of formation of the
Lyman continuum is therefore on the order of 20 s.
We note that the degree of ionization of hydrogen during
the oscillations remains near 10~1 (see Fig. 2 of Carlsson &
Stein 1999), implying a recombination time to the hydrogen
ground state that is long compared to the wave period. The
cooling time we have estimated therefore describes only the
behavior of the kinetic temperature. Although the atmo­
sphere between passages of shocks may achieve approx­
imate balance between radiative heating and cooling, the
state of true radiative equilibrium is not reached because of
the long radiative relaxation time of hydrogen.
All estimates of the cooling time concur in implying a
thin hot layer representing the shock moving outward over
a background in which the temperature decreases mono­
tonically with height. We may therefore represent the time­
dependent kinetic temperature of the chromosphere
schematically by a two­component model, consisting of a
time­independent cool background and a time­dependent
hot slab with a thickness of about 10% of the thickness of
the chromosphere traveling upward over the background,
T CS94 (t, z) B T 0 (z) ] dT (t, z) , (5)
where is described by the lower envelope in Figure 1
T 0 (z)
and the maximal excursions dT (t, z) by the upper envelope.
4. INSTANTANEOUS EMERGENT RADIATION IN LINES
The wave period observed in the calcium bright points is
approximately equal to 3 minutes, and the wave travel time
through the chromosphere (the region extending from 0.5 to
2 Mm) is also approximately equal to 3 minutes. Thus, at
any instant of time, there is typically only one strong shock
traveling through the chromosphere. Furthermore, because
of spatial intermittence (Lites, Rutten, & Thomas 1994),
larger areas in the chromosphere may show no high­
amplitude 3 minute oscillations, and because of temporal
intermittence (von & Kneer 1995), trains of high­
Uexku
# ll
amplitude oscillations may be separated by time intervals of
a few times the wave period. When strong shocks are
absent, only the naked background is found, which in the
CS94 model has a monotonically declining temperature, i.e.,
T CS94 (t, z) BT 0 (z).
An interesting quantity to estimate and compare with
observations is the residual intensity in the core of the H
line, i.e., the ratio of the emergent intensities in the line and
the neighboring continuum. For a typical scattering line the
emergent intensity is approximately equal to the line source
function. Taking 3000 K for the temperature in the upper
layers (see Fig. 1), assumed to be isothermal and static, the
source function in complete redistribution at the top is
given by (3000 K), where v is the scattering
S(0) \JvB l
parameter, with a value of v \ 0.02 in the FAL93 model
(E. H. Avrett 2000, private communication), and is
B l (T )
the Planck function. The assumption that the medium is
approximately isothermal must hold for a layer with a
thickness equal to the thermalization length, which for a
Doppler­broadened line in a static medium is equal to
Since the line center optical depth at the height
q relax \ 1/v.
where the line source function has its maximum, near
1 Mm, is and thus 10 times the thermalization
q 3 \ 500
depth, the emergent intensity should be adequately rep­
resented by the isothermal approximation when a shock is
in the low chromosphere or is absent altogether.
We calculate the intensity of the continuum radiation
near the H line as the Planck function for the observed
brightness temperature of 4677 K near the H and K lines
(E. H. Avrett 2000, private communication), which gives a
residual intensity at the center of the H line of 2 ] 10~3.
Note that this estimate is an upper limit because of the
assumption that the layer above 1 Mm is isothermal at
T \ 3000 K and does not drop to 2000 K at the top. We
should observe this or a lower value of the residual intensity
at line center for any position or time without large­
amplitude oscillations, as well as at positions of bright
points when the shock is in the low chromosphere, below
the peak of the emission ; this occurs for about one­
H 2V
third of the wave period at bright­point locations during
active oscillations and generally for about 50% of bright­
point locations (see Carlsson, Judge, &Wilhelm 1997).
The prediction for the residual intensity at is not
H 3
borne out by the observations of CD83, shown in Figure 4.
The proïles for positions of the shock in the lowest layers of
the chromosphere, or for locations and times without large­
amplitude oscillations, are found in the deciles for the
lowest values of H index and residual intensity. At least six
of the 10 bins should be dominated by proïles with line
radiation from the cold upper layers. But instead of the
predicted value of 2 ] 10~3, Figure 4 shows 0.03 for the
lowest decile and 0.06 for the highest. Our prediction agrees
qualitatively with the actual simulations (Carlsson & Stein
1997, 1998), which show a very deep and dark line core,
much darker than the observations of LRK93, while at the
same time exhibiting an intensity at maximum that is
H
2V
much brighter than the observations. But neither our pre­
diction from the CS94 simulations nor the simulations
themselves agree with the time­resolved observations of the
H line by LRK93 (see Fig. 5 of Carlsson & Stein 1998) or
Kariyappa, Sivaraman, & Anadaram (1994), or of the K line
by Liu (1974, shown in Fig. 5).
The spatial resolution of the CD83 observations of 500
km relative to the size of strong bright points of 1800õ2700
km (CD83, Plate 10) and the cadence of 10 s relative to the
FIG. 4.õResidual intensity in the H line observed by CD83. The pro­
ïles are ordered by the H index (emission in a 1 band about line center)
A#
and placed into 10 equal bins. Scattered light (3.8% of continuum) has been
subtractedõfrom CD83.

380 KALKOFEN Vol. 557
FIG. 5.õEvolution of a bright point at three instants of time. In
K 2V
the relaxed state (t \ 0), the proïle is nearly symmetric and shows a
chromospheric temperature rise that is not associated with bright­point
dynamicsõfrom Liu (1974).
wave period of 200 s are sufficient to observe the cold, dark
top of the CS94 model. The failure to match the obser­
vations allows only one conclusion : the chromosphere is
not predominantly cold.
EUV observations with SUMER (Solar Ultraviolet Mea­
surement of Emitted Radiation) by Carlsson et al. (1997)
lead to the same conclusion. The conditions for the forma­
tion of the resonance lines of neutral and singly ionized C,
N, and O are similar to those of the resonance lines of
ionized calcium. For the carbon line at 1657 for example,
A# ,
the line center optical depth at the source function
maximum, near a height of 1 Mm, is equal to 200 in the
FAL93 model (E. H. Avrett 2000, private communication).
The slit in the SUMER observations measured
1@@ ] 120@@, with sampling every arcsecond along the slit.
Although the slit cut through several network arches, most
of the positions were located in the nonmagnetic
internetwork regions, for a total of about 102 positions. The
observing run covered 4 hr, but each of 10 lines was mea­
sured for only 1 hr, corresponding to about 20 times the
wave period. These observations therefore constitute
2 ] 104 complete line segments covering a full wave period
each. Fifty percent of the chromosphere exhibited the
3 minute oscillations. Thus, 104 segments should show the
complete passage of a shock wave through the chromo­
sphere, and 104 segments, corresponding to locations
without large­amplitude oscillations, should originate in a
chromosphere without a shock. Consequently, half the seg­
ments should show the transition from emission to absorp­
tion lines, and the remaining segments should show only
deep absorption. Carlsson et al. (1997) found only emission
lines.
While it would be conceivable that emission from a hot
canopy might ïll in some of the absorption predicted for the
CS94 model, it is inconceivable that it would ïll in every
single one of the 2 ] 104 segments and, furthermore, turn
deep absorption lines completely into emission lines, even if
the canopy had a ïlling factor of unity (see Jones 1985). The
requirement to turn absorption into emission allows no
exception since, as the authors note, ```` all chromospheric
lines show emission above the continuum everywhere, all of
the time.îî The ïrm conclusion that the chromosphere is
never cold is inescapable.
Finally, the K line observed by Liu (1974), in Figure 5,
shows the emergent intensity at three di+erent phases of a
wave. In the relaxed phase, denoted by t \ 0, the chromo­
sphere is without a wave, and the proïle is symmetric. Since
the intensity represents the mapping of the temperature
structure from depth to wavelength, its increase from the K 1
minima toward line center results from the chromospheric
temperature increase outward while the central absorption
results from scattering, which decouples the source function
from the Planck function. The traditional interpretation of
this proïle is that the temperature rises in the outward
direction. The observed line proïle implies a well­deïned
temperature minimum at the top of the solar photosphere
and thus contradicts the claim to the contrary by CS94 and
Schrijver (2001).
5. INTENSITY FLUCTUATIONS IN THE
LYMAN CONTINUUM
The intermittent heating seen in Figures 2 and 3 and
described by equation (5) leaves its imprint also on the
emergent intensity in the Lyman continuum. Although the
CS94 simulations compute the Lyman radiation only at the
absorption edge, their paper provides sufficient information
to determine the emergent Lyman continuum spectrum also
at other wavelengths. The model predictions can then be
compared with Skylab observations.
Consider the second time interval in Figure 3, which
covers a complete wave period. Although the kinetic tem­
perature reaches 104 K and then drops to 2000 K, the
brightness temperature, which describes the emergent inten­
sity at the Lyman edge, varies by only a relatively small
amount about the average of 6250 K, between 6600 and
5750 K. But in the intensity this temperature variation is
magniïed and amounts to an intensity ratio of a factor of
34, which is about 4 times the value seen in the Skylab
observations that are represented in the empirical VAL81
models AõF (see Fig. 9 of VAL81).
The emergent intensity at shorter wavelengths can be
calculated from the intensity at the absorption edge and the
kinetic temperature of the electrons. Since the time for
thermal relaxation of the electrons by Coulomb collisions is
very shortõit is measured in microseconds (Spitzer 1956)õ
the electrons always satisfy a Maxwellian velocity distribu­
tion at the local kinetic temperature, and the emitted
Lyman photons always follow a Planckian frequency dis­
tribution. The shape of the emergent intensity is therefore
known. It is given by the Planck function at the kinetic
temperature approximately at unit monochromatic optical
depth. Since the opacity in the Lyman continuum is a weak
function of wavelength (Dj3), unit optical depth at j is
close to unit optical depth at 912 if j is close to the
A#
absorption edge. For our estimate we ignore the di+erence
in depth and take the temperature at as a close
q Lc \ 1
approximation to the temperature at q j \ 1.
Early in the second time interval in Figure 3, at time t 1 ,
the curves for kinetic temperature, brightness temperature,
the source function at and the time­averaged
q Lc \ 1,
brightness temperature cross at a temperature of T kin,1 \
6250 K. The emergent intensity at wavelength j is therefore
given by the Planck function at T kin,1 ,
I j (t 1 ) \B j (T kin,1 ) . (6)
Later in the time interval, at time when the brightness
t 2 ,
and kinetic temperatures are and the emergent
T br,2 T kin,2 ,

No. 1, 2001 CASE AGAINST COLD, DARK CHROMOSPHERES 381
intensity at wavelength j is given by
I j (t 2 ) \B 912 (T br,2 ) ] B j (T kin,2 )
B 912 (T kin,2 ) . (7)
For the time we choose the instant when the kinetic
t 2
temperature is K and the brightness tem­
T kin,2 \ 2200
perature is K. Since the source function at is
T br,2 \ 6000 t 2
lower than the emergent intensity, the actual ratio of inten­
sities at and is larger than our estimate.
t 1 t 2
For the kinetic temperatures of 6250 and 2200 K we
predict intensity ratios of 2.9 at 912 1.9 ] 103
I(t 1 )/I(t 2 ) A# ,
at 800 and 1.4 ] 105 at 740 The observed ratios at
A# , A# .
these wavelengths in the Skylab data shown in Figure 9 of
VAL81 are less than a factor of 10. Our estimates are very
much larger than any observed. We note that the tem­
perature variation in the four wave segments shown in
Figure 3 is larger than the variation used in the estimate.
The intensity ÿuctuations of the model therefore exceed the
prediction.
The Skylab observations have a spatial resolution of
5@@ ] 5@@ and a cadence of either 1 minute or 41 ms. Spatial
or temporal averaging can therefore not account for the
di+erence between observed and predicted intensity ÿuctua­
tions in the Lyman continuum. We are led to conclude that
the solar chromosphere su+ers much smaller temperature
variations than the model and is not heated only inter­
mittently by large­amplitude waves.
6. CONSTRAINTS ON MODELS FROM CO LINES
Lines of the carbon monoxide molecule, in emission o+
the limb and in absorption on the disk, have never been
successfully explained in terms of an accepted empirical
model except by invoking a separate model, such as the
COmosphere (Wiedemann et al. 1994). It is interesting to
ask whether the dynamical model of CS94 succeeds where
static models have failed.
Observations of the vibration­rotation bands of CO by
Noyes &Hall (1972) and Ayres (1981) indicate temperatures
for line formation as low as 4100 K at disk center and near
3800 K close to the limb, and Ayres & Testermann (1981)
noted that the CO brightness temperatures are incompati­
ble with a chromospheric temperature rise as in the FAL93
models (see also Avrett 1995). Instead, these observations
are believed to require a temperature that is monotonically
decreasing in the atmospheric layers where current empiri­
cal models place the chromospheric temperature increase.
For the line formation problem in CO, the CS94 model
appears to o+er a solution since the background atmo­
sphere has a monotonically declining temperature proïle.
The observational data to be ïtted by a model are (1) the
formation height of the CO absorption (or emission) lines,
(2) the temperature of the gas at that height, and (3) the
temperature ÿuctuations observed in the line intensity.
Observations of emission in CO o+ the limb by Ayres
(1998) suggest that molecular gas is present up to a height of
900 km, and perhaps up to 1 Mm or more, with a tem­
perature of 3500 K. While this temperature agrees at the
quoted height with that of the background model in the
CS94 simulations, the temperature at the shock front
reaches 6300 K at 900 km and 104 K at 1.1 Mm, virtually
guaranteeing the complete destruction of CO molecules by
the shocks responsible for chromospheric bright points.
With an association time of hours (Avrett et al. 1996), CO
could not exist at such heights. The interpretation of the
limb observations in CO lines is thus incompatible with the
temperature structure of the CS94 model. Modeling by
Uitenbroek (2000) of CO lines on the basis of snapshots of
the dynamical model gives an amplitude of intensity varia­
tions that is higher than observed by a factor of 2.5 and
therefore leads to the same conclusion.
The temperature ÿuctuations of ^300 K observed by
Ayres (1998) in CO emission lines are broadly consistent
with observations of CO absorption lines by Uitenbroek &
Noyes (1994) and Uitenbroek, Noyes, & Rabin (1994), who
note that while there are cold elements that are as much as
200 K colder than the average, the bulk of the temperature
variation remains within ^100K. The height at which a
peak­to­peak temperature variation of 400 K is found in the
CS94 model is 500 km. But the time­averaged temperature
at that height is 4800 K (see Fig. 1), which is much higher
than the values suggested by the observations.
Thus, none of the parameter values obtained from
analyses of CO observations on the basis of static atmo­
spheres ït into the dynamical CS94 model. The remedy of
that model proposed by Kalkofen et al. (1999) cannot help
since its e+ect is to raise the background temperature from
the low values of the cool CS94 background model to the
higher values of the FAL93 model.
It is unlikely that calcium bright points and molecular
CO absorption lines occur in di+erent regions on the
Sunõthe latter, for example, in the isolated, much colder
structures suggested by the so­called COmosphere or by the
two­component model of Ayres, Testermann, & Brault
(1986), which overestimates the UV continuum ÿux at 1400
by a factor of 20 (see Avrett 1995). It is much more likely
A#
that CO absorption lines and bright points occur in the
same regions but at di+erent heights. The reason is that
bright­point oscillations are pervasive, with a ïlling factor
of 50% (Carlsson et al. 1997), and the CO­absorbing regions
are even more pervasive, with a ïlling factor of 50%õ85%
(Solanki, Livingston, & Ayres 1994).
Given these considerations, a plausible scenario for CO
lines is that they are formed in a dynamical medium in the
layers of the upper photosphere. This dynamical atmo­
sphere combines features of both the time­dependent CS94
model and the time­independent semiempirical FAL93
model A, the former describing the temperature ÿuctuations
and the latter the low temperatures in the temperature
minimum region, which may be caused by cooling owing to
granular motion (Stein & Nordlund 1989). In these layers,
important constraints on the temperature structure of the
VAL81 and FAL93 models come from observations of the
calcium lines (CD83 ; Ayres & Linsky 1978) and EUV con­
tinua (VAL81) but not from CO lines. A lower empirical
temperature in the temperature minimum region might be
compatible with the UV data as well as with CO line forma­
tion in a dynamical atmosphere. The minimum temperature
of 4240 K of FAL93 model A with superposed velocity and
temperature ÿuctuations, the latter with an amplitude of
200 K, might be consistent with CO observations at disk
center (E. H. Avrett 2000, private communication), but the
problem of the low­brightness temperatures seen at the limb
remains.
7. TIME­AVERAGED MODELS
The various FAL93 models di+er from one another by a
few hundred degrees in the ```` chromospheric plateau îî

382 KALKOFEN Vol. 557
region, i.e., at heights between 1 and 2 Mm and tem­
peratures between 6000 and 7000 K, where they are based
mainly on the CD83 data. Since the H line proïles are
placed into 10 ordered bins, the highest and lowest observed
intensities are represented only in an average way in the
FAL93 models. Very high and very low intensities do not
play a signiïcant role in the energy budget of the chromo­
sphere, however. Otherwise the top and bottom bins (see
Fig. 4) would di+er signiïcantly from the others. Thus,
apart from the intermittent heating due to the passage of
the strong shock responsible for bright points and apart
from the relatively small temperature ÿuctuations inherent
in the sustained shock heating of the atmosphere, the
FAL93 models as a group correspond to time averages over
the 10 s cadence of the CD83 observations.
CS94 show two kinds of average temperature (see Fig. 1) :
(1) the straight time average of the ÿuctuating temperature
and (2) the temperature of a static model that matches the
time­averaged emission from the time­dependent model.
The latter model is thus analogous to the (semiempirical)
FAL93 models, except that whereas the CS94 model is
designed to match the time­averaged radiation from the
dynamical model, the FAL93 models are designed to match
the time­averaged radiation from the Sun.
If the time averages of the CS94 model, either the straight
average or the average emission (```` semiempirical îî in Fig. 1),
were also taken over 10 s, they would reÿect the intermittent
heating pattern and would therefore depend on the phase of
the shock. If they were taken over a full wave period, they
would show the considerable variation of the peak intensity
in the calcium emission peaks (Figs. 2 and 3 of von Uexku
# ll
& Kneer 1995). The FAL93 models would not have the
same variation since the solar chromosphere always shows
background emission in the calcium lines. It seems likely
that the time averages in Figure 5 of CS94 (Fig. 1 in this
paper) were taken over the observation time of 1 hr,
perhaps excluding the initial, transient behavior.
Another noteworthy feature is that the averages in the
CS94 model are taken over, at most, four very strong bright
points, and possibly over only the most luminous bright
point in Figure 1 of LRK93, which far outshines any
network bright point at that observation time and generally
matches the brightest network emission in the observing
run (Fig. 2 of LRK93). For such a bright point, the appro­
priate comparison model of the FAL93 series would be
model C or a hotter model. Instead, CS94 compare their
model with the coolest of the VAL81 models, FAL93 model
A. In spite of this bias, the CS94 temperature falls below the
FAL93 model A temperature by a considerable margin (see
Fig. 1), implying a considerable deïcit in total energy
emitted from the chromosphere.
One can estimate the deïcit, at a height of 1 Mm, for
example, from the temperature curves for the semiempirical
models in Figure 1. The most important emitters in the
chromosphere are the lines of Ca II and Mg II (see VAL81,
Fig. 49). Therefore, measuring the time­averaged emission
by the Planck functions for the H and K lines of Ca II, which
up to that height are tracked by the line source functions,
the shortfall of the CS94 model relative to model FAL93
model A is a factor of 3, and for the h and k lines of Mg II it
is a factor of 5. The shortfall would be even larger in the
more appropriate comparison with an FAL93 model hotter
than FAL93 model A. This ïnding agrees qualitatively with
the observational result of Hofmann, Ste+ens, & Deubner
(1996), discussed by Kalkofen et al. (1999), that the emission
in during the bright­point phase is only 9% of the total
K 2V emission from the nonmagnetic Sun. But the deïcit
K 2V
apparent in Figure 1 is smaller than the factor of 10 given
by these K line observations ; this supports our supposition
that CS94 have modeled exceptionally bright features.
The lower value of the straight time­averaged tem­
perature of the CS94 model compared with the temperature
of the FAL93 model produces a smaller scale height and,
consequently, a smaller thickness of the chromosphere.
Thus, the upper boundary of the CS94 model is not at 2.2
Mm, as in the FAL93 models, but at the lower height of 1.8
Mm. This di+erence is a consequence of the di+erence in
temperatures and, hence, scale heights. (Note that the
FAL93 A model has been redrawn in Figure 5 of CS94, and
the location of its top has been lowered from its original
height of 2.2 Mm in FAL93 model A down to 1.9 Mm. Note
also that the semiempirical curve in Fig. 1 is not deïned
above 1.8 Mm).
8. CONCLUSIONS
The modeling of the dynamics of bright points in the
H 2V
nonmagnetic solar chromosphere by CS94 yielded as a by­
product a time­dependent temperature that varies between
very low and very high values (2000 and 25,000 K at a
height of 1.8 Mm). Predictions of radiation from this model
are of lines that alternate with the phase of the upward­
propagating shock wave between very strong emission lines
and very deep absorption lines ; for locations and times
without large­amplitude oscillations, which at any instant
of time have a spatial ïlling factor of 50%, only absorption
lines are predicted. But none of the absorption lines, which
should be among the strongest lines in the solar spectrum,
have been observed, either from the ground or from space.
For the emergent Lyman continuum radiation, the intensity
for the single wavelength point (912 given by the model
A# )
agrees with the observations within a factor of 4, but the
decrease of the intensity with decreasing wavelength is
much steeper than observed. As a consequence, the intensity
ÿuctuations predicted for the CS94 model are much larger
than observed, exceeding a factor of 105 at a wavelength of
740 where the Skylab observations show a factor of less
A# ,
than 10.
The temperature characterizing the average emission of
the CS94 model reÿects the intermittence of the heating by a
shock wave if the average is determined for a time interval
shorter than the wave period. For longer intervals the tem­
perature shows a monotonic rise in the outward direction
and therefore agrees qualitatively with the FAL93 models.
Since the CS94 model is constructed for the brightest fea­
tures in the LRK93 observations of the quiet, nonmagnetic
Sun, it corresponds to a FAL93 model that is hotter than
their average model, which is FAL93 model C. But instead
of being hotter, the semiempirical CS94 model is every­
where cooler than even the coolest of the FAL93 models,
their model A. This indicates a signiïcant deïcit in the
radiated power of the CS94 model relative to the Sun. We
estimate this deïcit of the CS94 model on the basis of the H
and K lines of Ca II as a factor of 3, and on the basis of the h
and k lines of Mg II as a factor of 5 relative to FAL93 model
A and larger factors relative to the Sun. This conclusion is
not a+ected by seeing, scattered light, or a magnetic canopy.
Carlsson & Stein have modeled only one aspect of
chromospheric physics, namely, bright­point dynamics in

No. 1, 2001 CASE AGAINST COLD, DARK CHROMOSPHERES 383
nonmagnetic regions. Their model contains the intermittent
shock heating that is responsible for the dynamics but not
the energetically much more important sustained heating
required for most of the radiation emitted by the chromo­
sphere. While their simulations give a valid physical
account of the characteristic velocity signal of bright­point
dynamics, the resulting temperature structure does not
match the temperature structure of the solar chromosphere.
I conclude that the solar chromosphere is never as cold and
dark as they have proposed.
A complete, time­dependent model of the nonmagnetic
solar chromosphere would combine features of the time­
independent models of FAL93 and of the time­dependent
model of CS94. The main component of this model would
describe the sustained heating of the chromosphere and the
corresponding emission but without the contribution of the
highly time­dependent emission due to the shock waves that
cause the calcium bright points. Thus, the underlying model
could be model A of FAL93 but modiïed in the tem­
perature minimum region for the time­dependent formation
of the vibration­rotation lines of the CO molecule. The
dynamics of the combined model would be described by the
CS94 simulations but modiïed to take account of the
geometry of wave propagation for bright points in a strati­
ïed medium (Bodo et al. 2000). This complete model should
rectify two defects of the CS94 model, namely, the deïcit of
total emission and the excess of calcium emission at
maximal brightening.
I thank E. H. Avrett and P. Ulmschneider for discussions
and comments on the manuscript, M. Carlsson for clari­
fying statements concerning his model, and the referee for a
spirited debate that helped to expand the scope and
strengthen the arguments of the paper. I also thank the
Institut Theoretische Astrophysik of the University of
fu
# r
Heidelberg for its hospitality. Support by NASA and DFG
is acknowledged.
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