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A&A manuscript no.
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ASTRONOMY
AND
ASTROPHYSICS
5.2.1996
Letter to the Editor
Mechanical heating levels in ultra­fast rotating dwarfs
J.G. Doyle 1
Armagh Observatory, Armagh BT61 9DG, N. Ireland ... jgd@star.arm.ac.uk
Received date, accepted date
Abstract. Using available observational data we have
estimated the total radiative output from the chromo­
spheric/coronal plasma as a result of magnetic heating for
several F, G, K & M dwarfs. When tabulated as a function
of the bolometric flux, a clear `levelling­off/saturation' ex­
ists for the faster rotators as found previously when us­
ing single spectral lines. However, using recent chromo­
spheric radiative transfer model calculations it is shown
that this `saturation' can be due to a significantly larger
fraction of the losses being radiated by Hydrogen in the
ultraviolet/optical/infrared continuum regions. The im­
plication being that these ultra­fast rotators have a high
density chromosphere, plus a high temperature minimum,
although additional chromospheric model calculations are
required which include better molecular line opacities plus
and an extension of the models for stars with higher effec­
tive temperatures.
Key words: stars: rotation -- chromospheric activity --
late­types -- Rossby number
1. Introduction
In a study of chromospheric activity versus rotation pe­
riods in main sequence stars, Noyes et al. (1984) intro­
duced a parameter log P=Ü where Ü is the turnover time
in the surface convection zone. This parameter reduced
the scatter in their plots of the Ca II H&K surface flux
(representative of the mechanical heating level) as a func­
tion of the bolometric flux. In numerous follow­up papers
involving other spectral features, e.g. Mg II h&k, C IV
1550 š A , the X­ray flux, and including objects other than
main sequence stars, several authors (Simon et al. 1985,
Rucinski 1985, Simon & Fekel 1987, Vilhu & Walter 1987,
Doyle 1987) have drawn attention to the apparent satura­
tion of the line flux in the faster rotators. These previous
Send offprint requests to: J. G. Doyle
investigations were based mostly on the use of a single
spectral line; the objective of the present study was to
show that this `saturation' was an artifact of the limited
flux coverage.
The best method of determining the total radiative
output from the hot chromospheric/coronal layers is via
the emission measure technique. However, this still re­
quires a good selection of lines over the temperature range
10 4 to 10 7 K, plus knowledge of their atomic parameters.
For the Sun, Bruner & McWhirter (1988) showed that for
different solar regions, the total radiated power (exclusive
of hydrogen line radiation) may be estimated knowing the
intensity of only one spectral line (either a line formed in
the transition region or corona). This has now been ex­
tended and shown to be valid for other stellar objects,
including main sequence, sub­giants and giants (Doyle,
1996). Here we use the strong C IV 1550 š A line, however we
still find an apparent saturation. Making use of chromo­
spheric radiative transfer model calculations (Houdebine
et al. 1996), we suggest a mechanism other than the pre­
vious suggested changes in the dynamo and/or complete
plage coverage.
2. Far Ultraviolet Data
The C IV 1550 š A line fluxes were taken from several
sources, these included Simon & Fekel (1987), Rutten et
al. (1989, 1991) and Robinson et al. (1994). For some ob­
jects, in particular the fainter sources, the original IUE
SWP images were de­archived and re­analysed using the
Starlink IUEDR package. Some differences were appar­
ent when compared to previous reductions although these
were not major and should be considered more as repre­
sentative of the observational errors. The above references
also included the factor to convert the observed flux to
surface flux, the bolometric flux and the rotation periods.
Then to derive the total radiative power output from the
hot outer layers, the C IV 1550 š A relationship devised by
Bruner & McWhirter (1988) was applied. The various de­

2 J.G. Doyle: Letter to the EditorMechanical heating levels in ultra­fast rotating dwarfs
Table 1. Data summary .. fC IV = C IV 1550 š A doublet observed flux, f bol = bolometric flux at the Earth, C = conversion
factor for the surface flux, F tot = the total surface radiative output as a result of the mechanical heating, F tot =F bol = ratio of
total radiative output to bolometric flux, P rotation period in days and P=Ü = ratio of the rotation period to the turnover time
in the surface convection zone
Star Sp Ty. B­V fC IV f bol Log C Log FC IV Log F tot Log F tot =F bol Log P Log P=Ü
HD 1835 G2V 0.66 8:50 \Gamma14 0:76 \Gamma7 17.88 4.81 7.17 \Gamma3.29 +0.89 \Gamma0.21
HD 8358 G5V/G5V 0.72 2:95 \Gamma13 0:15 \Gamma7 18.58 6.05 8.51 \Gamma2.25 \Gamma0.28 \Gamma1.48
HD 16673 F6V 0.52 7:00 \Gamma14 1:30 \Gamma7 17.82 4.67 7.02 \Gamma3.92 +0.76 +0.06
HD 16895 F8V 0.49 1:30 \Gamma13 5:80 \Gamma7 17.11 4.30 6.62 \Gamma4.33 +0.86 +0.28
HD 17925 K2V 0.87 2:93 \Gamma13 1:30 \Gamma7 17.46 4.93 7.30 \Gamma3.28 +0.82 \Gamma0.44
HD 19373 G0V 0.59 1:20 \Gamma13 6:50 \Gamma7 16.96 4.04 6.34 \Gamma4.43 +1.26 +0.33
HD 20630 G5V 0.68 4:49 \Gamma13 3:30 \Gamma7 17.21 4.86 7.22 \Gamma3.51 +0.96 \Gamma0.18
HD 22049 K2V 0.88 8:34 \Gamma13 1:00 \Gamma6 16.51 4.43 6.76 \Gamma3.75 +1.05 \Gamma0.28
HD 22403 G2V 0.70 8:30 \Gamma13 0:16 \Gamma7 18.59 6.51 9.01 \Gamma1.79 +0.28 \Gamma0.89
HD 25680 G1V 0.62 9:70 \Gamma14 1:20 \Gamma7 17.72 4.71 7.06 \Gamma3.74 +0.96 \Gamma0.05
HD 25998 F5V 0.46 3:75 \Gamma13 1:70 \Gamma7 17.63 5.21 7.60 \Gamma3.27 +0.41 \Gamma0.03
HD 26354 K1V 1.00 1:70 \Gamma13 1:47 \Gamma8 18.48 5.71 8.15 \Gamma2.50 +0.41 \Gamma0.95
HD 27836 G1V 0.60 7:60 \Gamma14 0:24 \Gamma7 18.41 5.28 7.68 \Gamma3.10 +0.82 \Gamma0.14
HD 27859 G1V 0.60 4:00 \Gamma14 0:21 \Gamma7 18.49 5.08 7.38 \Gamma3.42 +0.82 \Gamma0.14
HD 27991 F7V 0.49 1:40 \Gamma13 0:68 \Gamma7 18.08 5.23 7.62 \Gamma3.30 +0.58 +0.002
HD 30495 G1V 0.63 1:87 \Gamma13 1:70 \Gamma7 17.52 4.79 7.15 \Gamma3.60 +0.88 \Gamma0.13
HD 34411 G2V 0.63 ! 1:60 \Gamma13 3:50 \Gamma7 17.23 !4.43 !6.76 \Gamma4.01 +1.38 +0.34
HD 35296 F8V 0.53 5:20 \Gamma13 2:60 \Gamma7 17.45 5.30 7.70 \Gamma3.30 +0.53 \Gamma0.21
HD 36705 K1V 0.83 1:19 \Gamma12 6:48 \Gamma8 17.81 5.89 8.34 \Gamma2.29 \Gamma0.28 \Gamma1.60
HD 39587 G1V 0.59 4:92 \Gamma13 4:60 \Gamma7 17.15 4.84 7.20 \Gamma3.61 +0.72 \Gamma0.21
HD 64096 G1V 0.60 ! 0:75 \Gamma13 2:30 \Gamma7 17.43 !4.30 !6.62 \Gamma4.17 +1.23 +0.27
HD 71071 G5V 0.95 0:20 \Gamma13 2:65 \Gamma8 17.96 4.26 6.58 \Gamma3.80 +1.22 \Gamma0.14
HD 72905 G2V 0.62 2:60 \Gamma13 1:50 \Gamma7 17.62 4.97 7.34 \Gamma3.39 +0.52 \Gamma0.49
HD 78366 F9V 0.60 ! 3:60 \Gamma14 1:20 \Gamma7 17.76 !4.32 !6.51 \Gamma4.33 +1.03 +0.07
HD 82558 K0V 0.92 3:60 \Gamma13 2:80 \Gamma8 18.17 5.73 8.17 \Gamma2.45 +0.20 \Gamma1.15
HD 82885 G8V 0.78 0:80 \Gamma13 2:27 \Gamma7 17.39 4.29 6.62 \Gamma4.12 +1.26 +0.03
HD 90089 F2V 0.37 3:50 \Gamma13 2:00 \Gamma7 17.74 5.41 7.82 \Gamma3.35 \Gamma0.20 \Gamma0.14
HD 91480 F1V 0.34 2:10 \Gamma13 2:10 \Gamma7 17.76 5.15 7.54 \Gamma3.61 \Gamma0.31 \Gamma0.04
HD 97334 G0V 0.61 9:28 \Gamma14 0:83 \Gamma7 17.89 4.86 7.22 \Gamma3.59 +0.88 \Gamma0.10
HD 101501 G8V 0.72 ! 2:54 \Gamma13 2:23 \Gamma7 17.35 !4.76 !7.12 \Gamma3.58 +1.23 +0.03
HD 109358 G0V 0.59 ! 2:11 \Gamma14 5:40 \Gamma7 17.04 !3.43 !5.68 \Gamma5.16 +1.25 +0.32
HD 110897 G0V 0.55 ! 1:70 \Gamma14 1:20 \Gamma7 17.71 !4.08 !6.38 \Gamma4.55 +1.11 +0.30
HD 111456 F5V 0.46 1:60 \Gamma13 1:20 \Gamma7 17.84 5.30 7.70 \Gamma3.48 +0.23 \Gamma0.21
HD 113139 F2V 0.36 6:40 \Gamma13 2:70 \Gamma7 17.59 5.40 7.81 \Gamma3.22 \Gamma0.36 \Gamma0.23
HD 114710 G0V 0.57 1:85 \Gamma13 5:20 \Gamma7 17.12 4.39 6.72 \Gamma4.12 +1.09 +0.22
HD 115383 G0V 0.59 4:10 \Gamma13 2:20 \Gamma7 17.49 5.11 7.49 \Gamma3.35 +0.70 \Gamma0.23
HD 131156 G8V 0.72 6:67 \Gamma13 4:00 \Gamma7 17.05 4.87 7.23 \Gamma3.42 +0.79 \Gamma0.41
HD 141004 G0V 0.60 7:74 \Gamma14 4:68 \Gamma7 17.15 4.04 6.34 \Gamma4.48 +1.26 +0.30
HD 142373 F8V 0.56 ! 6:60 \Gamma14 3:70 \Gamma7 17.27 !4.30 !6.62 \Gamma4.43 +1.23 +0.39
HD 144284 F8V/IV 0.53 1:30 \Gamma12 6:40 \Gamma7 17.08 5.19 7.58 \Gamma3.30 +0.49 \Gamma0.25
HD 146361 F6V/F8V 0.51 3:10 \Gamma12 1:40 \Gamma7 17.78 6.27 8.75 \Gamma2.17 +0.06 \Gamma0.60
HD 149661 K2V 0.82 1:02 \Gamma13 1:50 \Gamma7 17.51 4.52 6.86 \Gamma3.83 +1.33 +0.03
HD 154417 F9V 0.58 1:27 \Gamma13 1:00 \Gamma7 17.84 4.94 7.31 \Gamma3.53 +0.88 \Gamma0.02
HD 155886 K0V/K1V 0.86 3:30 \Gamma13 6:00 \Gamma7 16.79 4.61 6.95 \Gamma3.92 +1.33 +0.005
HD 160346 K3V 0.96 4:13 \Gamma14 0:87 \Gamma7 17.53 4.15 6.46 \Gamma4.01 +1.53 +0.17
HD 165341 K0V 0.86 5:53 \Gamma13 6:90 \Gamma7 16.75 4.49 6.82 \Gamma3.77 +1.29 \Gamma0.03
HD 166181 G5V 0.74 5:70 \Gamma13 0:24 \Gamma7 18.34 6.10 8.56 \Gamma2.16 +0.26 \Gamma0.97
HD 174429 K0V 0.79 1:80 \Gamma13 2:51 \Gamma8 18.60 5.86 8.31 \Gamma2.69 \Gamma0.03 \Gamma1.30
HD 175742 K2V 0.91 2:00 \Gamma13 0:20 \Gamma7 18.11 5.62 8.04 \Gamma2.58 +0.46 \Gamma0.88
HD 178450 G8V 0.73 1:90 \Gamma13 0:23 \Gamma7 18.23 5.57 7.99 \Gamma2.66 +0.33 \Gamma0.88
HD 190007 K4V 1.14 1:17 \Gamma14 0:45 \Gamma7 17.70 3.77 6.05 \Gamma4.31 +1.47 +0.09
HD 197890 K0V 0.88 1:60 \Gamma13 0:65 \Gamma8 18.92 6.12 8.59 \Gamma2.14 \Gamma0.54 \Gamma1.82
HD 197890 K0V 0.88 1:60 \Gamma13 0:65 \Gamma8 18.92 6.12 8.59 \Gamma2.14 \Gamma0.54 \Gamma1.82
HD 200391 G0V/G5V 0.65 6:40 \Gamma13 0:32 \Gamma7 18.32 6.13 8.60 \Gamma2.23 \Gamma0.16 \Gamma1.24

J.G. Doyle: Letter to the EditorMechanical heating levels in ultra­fast rotating dwarfs 3
Table 1 .. cont
Star Sp Ty. B­V fC IV f bol Log C Log FC IV Log F tot Log F tot =F bol Log P Log P=Ü
HD 201091 K5V 1.18 1:24 \Gamma13 3:90 \Gamma7 16.58 3.67 5.94 \Gamma4.23 +1.58 +0.18
HD 201092 K7V 1.37 ! 8:00 \Gamma14 2:20 \Gamma7 16.66 !3.56 !5.82 \Gamma4.18 +1.68 +0.27
HD 206860 G0V 0.59 1:81 \Gamma13 1:10 \Gamma7 17.83 4.71 7.06 \Gamma3.43 +0.66 \Gamma0.26
BD 26 730 K5V 1.12 4:70 \Gamma13 0:19 \Gamma7 17.91 5.58 8.00 \Gamma2.19 +0.26 \Gamma1.12
Gl 103 K7V 1.39 5:18 \Gamma13 0:20 \Gamma7 17.45 5.21 7.60 \Gamma2.20 +0.19 \Gamma1.22
Gl 182 M1V 1.37 1:02 \Gamma13 0:85 \Gamma8 18.03 5.04 7.42 \Gamma2.54 +0.66 \Gamma0.75
Gl 278C M0V 1.49 4:43 \Gamma13 0:12 \Gamma7 17.83 5.48 7.89 \Gamma2.02 \Gamma0.09 \Gamma1.52
Gl 285 M4V 1.61 2:80 \Gamma13 0:94 \Gamma8 17.76 5.21 7.60 \Gamma2.14 +0.44 \Gamma1.01
Gl 388 M4V 1.54 5:50 \Gamma13 0:29 \Gamma7 17.44 5.18 7.57 \Gamma2.33 +0.43 \Gamma1.01
Gl 411 M2V 1.51 ! 1:55 \Gamma14 0:49 \Gamma7 17.19 !3.38 !5.63 \Gamma4.25 +1.68 +0.25
Gl 517 K5V 1.18 1:30 \Gamma13 0:10 \Gamma7 18.35 5.46 7.87 \Gamma2.48 +0.60 \Gamma0.79
Gl 630.1 M4V 1.60 3:60 \Gamma14 0:17 \Gamma8 18.29 5.85 8.29 \Gamma2.23 +0.10 \Gamma1.34
Gl 719 M0V 1.19 4:83 \Gamma13 0:16 \Gamma7 17.81 5.49 7.90 \Gamma2.11 +0.58 \Gamma0.81
Gl 803 M1V 1.43 4:68 \Gamma13 0:26 \Gamma7 17.54 5.21 7.60 \Gamma2.35 +0.69 \Gamma0.73
Gl 867A M1V/M1V 1.50 3:20 \Gamma13 1:70 \Gamma8 17.57 5.08 7.46 \Gamma2.35 +0.61 \Gamma0.82
Gl 873 M4V 1.60 3:49 \Gamma13 0:17 \Gamma7 17.56 5.10 7.48 \Gamma2.31 +0.64 \Gamma0.81
Gl 890 M2V 1.42 3:50 \Gamma13 0:44 \Gamma8 18.51 6.05 8.51 \Gamma1.64 \Gamma0.37 \Gamma1.79
rived observational parameters are summarized in Table
1.
Table 2. Ratio of the radiative output due to hydrogen as a
function of the bolometric flux for the chromospheric models of
Houdebine et al. (1996) ... column 2 assumes Tmin = 2660K
while column 3 has Tmin = 3000K
log M log FH =F bol log FH =F bol
\Gamma3:00 \Gamma1.37 \Gamma0:83
\Gamma3:50 \Gamma2.22 \Gamma1:44
\Gamma4:00 \Gamma3.06 \Gamma1:97
\Gamma4:15 \Gamma3.24 \Gamma2:07
\Gamma4:30 \Gamma3.34 \Gamma2:14
\Gamma4:50 \Gamma3.62 \Gamma2:24
3. Results
In Fig. 1a we present a plot of the total radiative output
as a result of the mechanical heating, F tot , versus the rota­
tion period. The objects in Fig. 1a have a range of rotation
periods from less than 0.3 days to 50 days, despite this fact
there is little or no evidence for a leveling­off in the radia­
tive output in the faster rotators although an increased
scatter is apparent. As found by Burner & McWhirter
(1988) for the Sun and Doyle (1996) for several stellar ob­
jects, the error in the derived output based on the C IV
relationship and that based on a proper emission measure
technique is better than a factor of two. Thus some of the
scatter amongst the faster rotators probably reflects the
intrinsic variability of these objects. This scatter increases
slightly if one plots the ratio of the radiative output to the
bolometric flux (Fig. 1b). The straight line fits through the
data points in both the above figures are given on each di­
agram. Based on this figure the total energy output as a
result of this mechanical heating amounts to a maximum
of ¸ 1% of the bolometric flux, similar to that estimated
by Vilhu & Walter (1987).
However, when the above ratio is plotted against the
Rossby number, log P=Ü , with Ü as defined by Noyes et al.
(1984), this scatter is much reduced and a clear `levelling­
off' is observed starting at log P=Ü ! \Gamma1 which corre­
sponds to rotation periods of ¸ 2 days. In this figure we
give two straight lines fits, the solid line is the fit including
all the data points while the dashed line only includes data
with log P=Ü ? \Gamma1. For the fastest rotators, the difference
between the observed ratio and either the solid or dashed
lines is a factor of 5--10. As noted in the Introduction,
there will been several papers dealing with this `levelling­
off' of the line flux in the faster rotators. The question
then remains, why? Is it because the surface of these fast
rotators are completely covered with active regions (Vilhu
1987), or is it due to a change in the dynamo in the fast
rotators (Knobloch et al. 1981). The first suggestion has
being discussed by several authors, while the latter has re­
ceived rather little attention. Since the `levelling­off' only
becomes apparent when the convective turnover time is
introduced, this is an area which requires further inves­
tigation. Here we ask an alternative; are we missing an
important source of radiative losses in these objects?
As mentioned by Burner & McWhirter (1989), their
formulation did not include losses not to hydrogen. In a

4 J.G. Doyle: Letter to the EditorMechanical heating levels in ultra­fast rotating dwarfs
recent paper on chromospheric modelling, Houdebine et
al. (1996) complied a series of chromospheric models cal­
culating the radiative output from hydrogen as a func­
tion of the hydrogen column mass in the lower transition
zone/chromosphere (see their paper for a proper descrip­
tion of the models). Their model atmospheres range from a
column mass of log M = \Gamma3:0 to \Gamma8:0 at T = 8000K. In
a previous paper, Houdebine & Doyle (1994) constructed
a model atmosphere for the active star AU Mic (P = 4
days), based on H ff & H fi. They found a column mass
at 8000K of log M = \Gamma4:1. Using a more recent determi­
nation of the interstellar medium attenuation to AU Mic's
Ly ff flux (Doyle et al. 1996) and an observation of AU Mic
in the Pa fi spectral region, Doyle & Andretta (1996) de­
termined a column mass in excellent agreement with the
previous value.
In Table 2 we present a summary of Houdebine et al.'s
(1996) findings for the ratio of the total output due to
hydrogen compared to the bolometric flux. These model
atmospheres apply to an M1 star similar to AU Mic and
thus the derived values can not be used to infer an abso­
lute flux for objects of different spectral types. However,
the general run of increasing hydrogen contribution with
increasing hydrogen column mass is applicable. For AU
Mic, which has a rotation period of 4 days, the output
due to hydrogen as a result of the mechanical heating
for the Tmin = 3000K model is a factor of 2--3 greater
than that due to the hotter chromospheric/coronal layers.
For the Tmin = 2660K, the losses due to hydrogen are
less than that from the hot outer layers. In relation to
the present study, the most interesting point is that for
log M ? \Gamma4:0, the hydrogen contribution increase dra­
matically, e.g. there is a factor of 4--8 more losses due to
hydrogen at log M = \Gamma3:5 than at \Gamma4:15 in both models.
This would therefore suggest that there need not be a
change in the dynamo mechanism in the ultra­fast rota­
tors, nor are these objects required to be entirely saturated
with active regions. Instead, this suggests that a domi­
nant potion of the released mechanical energy is in the
form of an enhanced continuum plus hydrogen line emis­
sion. In fact, evidence of an enhanced continuum has been
noted previously by Simon et al. (1985). Further model
atmospheres calculations are however required for K &
G dwarfs in order to explore the hydrogen contribution
properly, plus an expansion of the M dwarf calculation in­
cluding new molecular line opacities (Doyle & Andretta
1996). As shown in Table 2, the amount of losses due
to hydrogen depends on (i) these fast rotators having a
high density chromosphere, for the M dwarfs this implies
log M ? \Gamma4:0 at the base of the transition zone and (ii) a
high temperature minimum of the order of 3000K. Refer­
ence to Fig. 1c would thus suggest that the total mechan­
ical flux input in these fast rotators would be ¸ 0:1F bol as
suggested by Linsky (1991) and not the more frequently
quoted 10 \Gamma2 F bol .
Fig. 1. (a) The total radiative output from the chromospheric
& coronal region versus the rotation period, (b) ratio of the
above radiative output to the bolometric flux versus rotation
period and (c) ratio of the radiative output to the bolometric
flux versus the Rossby number, the parameters of the straight
line fit refer to the solid line, i.e. a fit to all the data points,
while the dashed line only includes data with log P=Ü ? \Gamma1

J.G. Doyle: Letter to the EditorMechanical heating levels in ultra­fast rotating dwarfs 5
Acknowledgements. Research at Armagh Observatory is grant­
aided by the Dept. of Education for N. Ireland while partial
support for software and hardware is provided by the STAR­
LINK Project which is funded by the UK PPARC. I would also
like to thank Chris Lloyd for his help in de­archiving some of
the IUE data used in this work from STADAT at the Ruther­
ford Appleton Laboratory. Partial support for this project was
provided via the PPARC grant GR/K04613.
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