High-frequency optical oscillation during the flare phase of the red
dwarf EV Lac
M.E. Contadakis1, S. Avgoloupis 2, J. Seiradakis 2
1
Department of Surveying and Geodesy,University of Thessaloniki, Greece
Department of Physics Section of Astronomy Astrophysics and Mechanics,University of
Thessaloniki, Greece
2
kodadaki@vergina.eng.auth.gr
Abstract
The observational support of the presence of high frequency low amplitude oscillations
reported by Zhillyaev et al. 2000 and Contadakis et al. 2004, is highly demanding and will
be done by the future observations and by carefully reanalysing the data from our files. In
this paper we present the results of the analysis of the B-light curve for a flare of
magnitude 1.01,which was observed on September,1993. Despite the low time resolution
(sampling interval 12s) we were able to detect transient low amplitude oscillations with
period ranging between 30s and 125s with a confidence level higher than 70%. This result
is in favour of (or does not contradict) the suggested explanation i.e the evolution of a fast
mode magneto-acoustic wave generated at the impulsive phase of the flare and travelling
through the magnetic loop.
1. Introduction
In order to investigate the fine structure of the flares light-curve in UBVRI, a program of
Many-sites Multi Channel simultaneous observations of flare stars by a network of
telescopes was initiated in 1998. Analysis of the B-light curve of three flares which were
observed at the Stephanion Observatory during the campaign of the year 1999,reveal the
presence of high frequency small amplitude oscillations with periods ranging between 6s
and 20s (Contadakis et al. 2004)in accordance with the results of the campaign of 1998
(Zhilyaev et al. 2000). In addition there is a tendency the higher frequencies to persist in
the late phases of the flare development. These observations are consistent with the
phenomenology of the evolution of a fast mode magneto-acoustic wave generated at the
impulsive phase of the flare and travelling through the magnetic loop. Roberts et al.
(1984) describe the evolution of an impulsively generated fast wave as follows: (1) Initial
periodic phase, beginning at a time ∝ 1/υAe after the initial impulse (Ae = external Alven
velocity) and characterized by an increase in amplitude and frequency. (2) A quasiperiodic phase, showing an increase in amplitude at a time ∝ 1/υA. (3) A periodic decay
phase, with a rapid decrease in amplitude (see also Williams et al. 2001, 2002) and
references therein.
The observational support of this interesting result is highly demanding and will be done
by the future observations and by carefully reanalysing data of our files. In this paper we
present the results of the analysis of the B-light curve for a flare of magnitude 1.01,which
was observed on September of 1993.
2.The Observations
The observations were carried out with the help of the 30 inch Cassegrain telescope
equipped with a single-channel UBVRI photometer (Mavridis et al. 1982) on September of
1993 and the data are been retrieved from our files. Since the today issue, i.e. high
frequency oscillations, was not in our concern at that time a rather low high integration
time was chosen in order to avoid the, so believed random and atmospheric noise. So the
time resolution of these data is 12 seconds. Table 1 displays the characteristics of the
flare and Figure 1 displays the observed light-curve.
Table 1. Characteristics of the flare
Date
Time max (U.T) tb(min)
13/9/1993
21h 12min
1.4
tb
ta
ta(min)
30.0
P(min)
7.08
∆b(mag)
1.01
: preflare duration
: after flare duration
P=∫ [(If -Io)/Io] dt : Emitted energy from the flare event
If
: light Intensity of the star during the flare phase
: Quiet star light Intensity
Io
∆b
: Increment of the flare b magnitude at maximum
σ
: standard error of the quiet state stellar intensity
Figure 1: Light-curve in the B color of the flare of EV-Lac on 13/9/1993
(If -Io)/Io
1.539
σ(mag)
0.0155
3. Data Analysis
Our data consist of flare to quiet star relative intensities (If -Io)/Io sampled every 12
second. From Table 1 it is seen that the standard error of our photometry is 0.0155
magnitude in the B color. This noise is of instrumental and atmospheric origin. Speaking in
terms of quiet state stellar light intensities, the induced noise of instrumental and
atmospheric origin is 0.144%. The Nyquist frequency is 0.0416 Hz. This means that we
can detect oscillations with smaller frequencies than this limit (i.e. periods larger than 24s).
We may assume that a certain part of the light curve is a continuous time series
representing an ergodic process. Then we may consider our data as a discrete time
series, which is one of the many samples potentially derived from the underlying
continuous time series, and estimate the Spectral Densities of the stellar light intensity (or
in our case relative flare to star intensity) applying Discrete Fourier Transform (DFT).
Frequencies of potentially intrinsic light fluctuations of the star correspond to these
frequencies which present a meaningful spectral density in the Periodogram. Here we
have to mention that in order to avoid the very small frequency intensity variations, i.e. the
main flare and gross characteristic of the light-curve, we treat the residuals from a moving
average smoothing, using a 5 sample values half width (i.e. 60s). In the subsequent
analysis in order to detect the presence of higher frequencies in specific parts of the light
curve we filter out the lower frequencies using a 2.5 sample values half width (i.e. 15s).
The DFT analysis of our data will provide an estimation of the Spectral Densities of the
underlying continuous light intensity (or in our case relative flare to star intensity)
variations of the star. The reliability of this estimation is given by the noise to signal ratio of
the estimated Spectral Density S and is σ/S = (BT)-1/2. Where the band width is
B=1/12s=0.0833Hz and the extent of the sample is T = N ×12s (Newland 1984). Since B
is constant the reliability of the Spectral Density estimation depends on the extent of the
sample and are σ/S=0.075 for the estimation of the Spectral Densities using the whole
light curve and σ/S=0.114 for the estimation of the Spectral Densities using part of the
light curve.
An approximate level of confidence of an estimation of a spectral density can be achieved
by assuming the spectral density distribution for a particular frequency. To this purpose we
assume that the measured spectral density S, which is a sum of products of values of our
sample xi (i=1...n) (i.e. relative intensity residuals) with Normal distribution N(0, σ), can be
approximated by a random variable with χn2 distribution of n statistical degrees of freedom
(Newland 1984)
S= χn2=x12 + x22 + ….+ xn2
(1)
The mean value and the variance of the spectral density are
ms = n σ2 and σs2 =2 nσ4
(2)
respectively, where σ,σs is the standard deviation of the sample and the spectral density
estimation respectively.
However not all of the well defined spectral densities correspond to an intrinsic stellar
intensity oscillation. Actually most of them are effect of the random and atmospheric noise.
In order to estimate the confidence level that a measured spectral density correspond to
an intensity oscillation, we get an estimation of the variance of the sample from the
relation (2) and we compare the variance of the sample σ2 with the variance of the quite
state flare intensity.
We set the hypothesis H0 according to which the estimated spectral density correspond to
the random noise if σ2= σq2 against the hypothesis H1, that the spectral density correspond
to an intensity oscillation if σ2> σq2 (σq2 is the variance of the stellar intensity in the quiet
state). It is well known that the random variable (n-1) σ2/ σq2 follow the chi-square
distribution (see for instance Fisz 1978).
Then the hypothesis H1 holds to a significant level α if (n-1) σ2/ σq2 > χα2.
Finally the estimated confidence level of a spectral density is the product of the above
calculated confidence level with the confidence level of the basic hypothesis which is 66%.
4. Results and Discussion
The results of the DFT analysis of the whole light-curve are being displayed in the
periodogram of Figure 2. From the Spectral Densities which are being displayed in this
periodogram, these with frequencies 0.00947, 0.01126 and 0.0123 Hz ( i.e. periods 105.6,
88.0 and 81.2 seconds) may be considered to correspond to intrinsic stellar oscillations
with a confidence level higher than 70% while those with frequencies 0.00805 and
0.01467 Hz do also with a confidence level ranging between 55% and 70% (Table 2).
Table 2. Potentially identified frequencies from the periodogram
of the whole light-curve
Frequency
(c/lag)
Frequency
(Hz)
Period
(sec)
Spect.Density
0.09659
0.00805
125.0
0.0414
0.11364
0.00947
105.6
0.05613
>70
0.13636
0.01126
88.0
0.05327
>70
0.14773
0.01230
81.2
0.05721
>70
0.17614
0.01467
68.1
0.04622
Noise/signal
Confidence level
%
70>P>55
70>P>55
0.075
In addition to the potentially identified frequencies there is a number of peaks, not reliably
identified according to our analysis, in the high-frequency site of the periodogram. In order
to have a close approach to this frequency range we filter out the lower frequencies by
treating the residuals from a moving average mean with a half width band of 30s and we
repeat the DFT analysis. The results are shown in the periodograms of Figure 3, which
concern the first 14 minutes of the light curve and Figure 4, which concern the last 20
minutes of the light curve. As it is obvious from the periodogram of the late stages of the
light-curve no reliable identification with intrinsic stellar intensity variation can be done
since the Spectral Densities are very weak (i.e. of the order of 10-3 ).
Figure 2: Periodogram for the whole light-curve of the flare
Table 3. Potentially identified frequencies from the periodogram of the first 14 minutes of
the light-curve
Frequency
Frequency
Period
Spec.Density
Conf.level
(c/lag)
(Hz)
(second)
(1/Hz)
(%)
0.14474
0.012067
82.9
0.03912
>70
0.17105
0.014254
70.15
0.05399
>70
0.19373
0.016144
61.94
0.05811
>70
0.32895
0.027412
36.45
0.03925
>70
0.39474
0.032895
30.40
0.02821
>70
Noise/signal
0.114
These frequencies correspond to the instrumental and atmospheric noise. On the contrary
from the periodogram of the early stage of the light curve five frequencies (i.e. 0.012067,
0.014254, 0.016144, 0.027412,0.032895 Hz which correspond to the periods 82.9,70.15,
61.94, 36.45 and 30.40 second) can be reliably identified to correspond to intrinsic
brightness fluctuations with a confidence level higher than 70%. Two of them are the
same (if we take into account the low frequency discrimination of this analysis) with the
last two of the first analysis. Table 3 displays all the characteristics of the identified
frequencies.
Figure 3: Periodogram of the first 14 minutes of the light-curve. Low frequencies are
being filtered out
Our analysis indicate that transient light oscillations with periods ranging from 30s up to
125s occur at the early stages of the flare. These oscillations can be reproduced from our
data assigning the respective phase to each point according to the period, and have an
amplitude of 0.08 magnitude. As an example the oscillation with period 1 minute at the
early stage of the flare is reproduced in the Figure 5. Error bars indicate the dispersion of
the points with the same phase.
The periods found in this analysis are larger than those reported by Zhilyaev et al. (2000)
and Contadakis et al. (2004) for flares on EV Lac and Rodono (1974) for a flare on HII
2411, and approach the order of magnitude of that reported for a flare on II Peg by
Mathioudakis et al. (2003).
Nevertheless we have to emphasise that light oscillations with similar periods at the early
stages of a flare are to be observed if we accept the explanation for the generation of such
high frequency transient light oscillations during the flare phase i.e the evolution of a fast
mode magneto-acoustic wave generated at the impulsive phase of the flare and travelling
through the magnetic loop.
Roberts et al. (1984) describe the evolution of an impulsively generated fast wave as
follows: (1) Initial periodic phase, beginning at a time ∝ 1/υAe after the initial impulse (Ae =
external Alven velocity) and characterized by an increase in amplitude and frequency. (2)
A quasi-periodic phase, showing an increase in amplitude at a time ∝ 1/υA. (3) A periodic
decay phase, with a rapid decrease in amplitude (see also Williams et al. 2002) and
references therein.
Figure 4: Periodogram of the last 20 minutes of the light-curve. Low frequencies are being
filtered out
Figure 5: Oscillation with period of one minute at the early stages of the flare
Our observational results are consistent with the phenomenology of the initial periodic and
quasi periodic phase. So the oscillations are weaker and of larger period than the reported
so far for EV Lac, which presumably correspond to a later phase of the development of
the magneto-acoustic wave.
5.Conclusions
Discrete Fourier Transform analysis of the flare of 13/9/1993 of the red dwarf EV Lac,
indicate that at the early stages of the flare small scale transient oscillations possibly
occur. The periods of these oscillations range between 30s and 125s sec. Their amplitude
is 0.08 magnitude. Our observational results agree with the up to day observational results
on high-frequency oscillations during the flare phase in the frame of the phenomenology of
a fast mode magneto-acoustic wave generated at the impulsive phase of the flare and
travelling through the magnetic loop. In particular the observed oscillations may be
considered that are being generated at the initial periodic and quasi-periodic phase of the
fast mode magneto-acoustic wave.
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