## 3. Results## 3.1. Statistical analysis## 3.1.1. Flare energies and accumulated flare frequency distributionThe fractional distributions of the number of flares versus their
total energy E are shown in the semi-log plots in Fig. 1
for U, B
and V bands. At low energies, the number of detected flares decreases
dramatically because of instrumental sensitivity limit. The range of
energies covers four decades in U and B distributions, and less than
three decades in the V distribution. The decrease of the number of
flares at high energies allows us to estimate the maximum energy
release through the flare mechanism operating on EV Lac. In fact, our
coverage time is sufficiently extended for our sample to be considered
complete consistently with known flare statistics (Shakhovskaya 1989).
Actually, according to the latter study, in a total observation time
comparable to our 1272
The rate of energy emitted as flares ( ), according to Lacy et al. (1976), is given by: where
and
correspond to the largest and the least
energetic flare which can be produced by the star, respectively, and
The dramatic decrease of the occurence rate at lower energies is due to instrumental detection limits (Gershberg 1972). Above the detection threshold the distribution is approximately linear and can be fitted by the relation: All flares observed in the U-band were included in the distribution: 95 of them were observed with the 91-cm telescope, 118 with the 61-cm telescope and 3 with the 30-cm telescope. Since the energy at which detection effects become apparent varies with the telescope used, we have excluded from the linear fitting those flares with energy lower than the onset of detection effects that was determined from observed 61-cm telescope observations (notice that the 3 flares observed with the 30-cm telescope are above this lower limit). The least-squares fit to the linear portion yields: where the estimated errors for the coefficients Taking and as the largest ( erg) and the smallest ( erg) flare energies observed in U-band, we derived: The ratio between the total energy released by flares and the energy emitted in the U-band by the quiet star (that is a parameter independent of the calibration of the quiescent flux from the star), thus results: a result only marginally different from that given by Lacy et al.
(1976) ( ). This confirms Lacy's at al. (1976)
conclusion that the flare energy spectrum of EV Lac is not affected by
time variablity as observed for other dMe sources. As derived by
numerical experiments,
is almost independent from the choice of
but is determined by the largest flare events.
In fact, even assuming
as low as
erg (a Cumulative frequency distributions of B- and V- band flares have also been computed. We found: Therefore: and The slope we found from the cumulative flare distributions ( for U, B, and V -band flares) is a typical
spectral index for flare stars in the vicinity of the Sun
(Shakhovskaya 1989). Gershberg (1989) showed that the spectral index
## 3.1.2. Flare occurence rate on time-scales of 1 .We have analyzed the time behavior of the flare occurence rate of
EV Lac on time-scales of 1
. The observed flare occurence rate is
, where
A
test comparing the observed and expected
numbers leads to the conclusion that the probability that the level of
activity in 1971, 1974 and 1977 by chance was higher than the mean
level is of order of 1.4%, 2.3%, and 0.6%, respectively. On the other
hand, the lack of flare detection in 1968, despite 88 ## 3.1.3. Color-color energy correlationsThe observations performed in more than one filter allow us to compare the flare energies in different colours. The analysis of 57 flares contemporarily observed in U and B -bands yields the following linear relation: where indicates the linear correlation coefficient, consistently with the relation found by Lacy et al. (1976) from the analysis of a more extended data sample, but concerning flares on eight stars. The analysis of 27 and 26 flares observed simultaneously in B and V -bands and in U and -V bands, respectively, leads to the following relations: The costant in relation ( 14) is quite different from the analogous one ( ) given by Lacy et al. (1976), and is not consistent with the extrapolation of the U-V relation that can be derived from relations ( 12) and ( 13). We believe that we underestimate the constant in relation ( 14) because of fewer data points are available to us and because of the more limited range of energy covered by our data (two decades) in comparison with about six decades covered by the Lacy et al. (1976) data. On the other hand, by using relation ( 13) to convert to V -band energies the flare energies measured in B -band, the data acquired contemporary in U and B can be also used to determining the U-V relation, thus increasing the number of data points to 60. The relation found in this way is closer to that obtained by Lacy et al. (1976): ## 3.1.4. Flare time-scalesThe distributions of the rise-times to the highest flare peak ( ) and of the descent times from flare maximum to
quiescence (
In early studies on stellar flares several authors (e.g. Haro &
Chavira 1955, Pettersen et al. 1984) inferred that flare durations are
correlated with spectral types, i.e. long duration flares more often
occur on the more luminous stars. However, Gershberg and Shakhovskaya
(1973) showed that the largest flares last longer than the smallest
flares. Therefore, since large flares preferably occur on luminous
stars while small flares dominate on faint stars because of contrast
effects, a spurious correlation between flare duration and spectral
type does result. We have also investigated the relationships between
flare time-scales and flare energies. In Fig. 5 the flare
time-scales ( and which confirm the existence of a general correlation between the time charactistics of a flare and its energy. Within each energy value the time-scales of individual flares span 1-2 order of magnitudes with the rise-time showing the largest scatter. We note that, for a given energy value, slow or long-duration flares could escape detection while fast or short-duration flares are easily detectable because they should have intense peak luminosities. Therefore, the bottom part of the trends shown in Fig. 5 does not suffer from detection limit.
Flares of equal energy output but different time-scales presumably reflect different physical characteristics of the flaring region such as size, strength of the magnetic field where the magnetic reconnection takes place, electron density and flaring plasma temperature. The slope of relation ( 16) is larger than in the analogous relation given by Pettersen (1989) but derived from the analysis of flares from a large sample of different type of stars - from the brightest dKe's to the faintest dMe stars. ## 3.1.5. Correlation between flare colour indicesThe colours of the most intense UBV flares at light maximum, computed according to Cristaldi & Rodonò (1975), are given in the two colour diagram in Fig. 6.
In the same figure the following models given by Gershberg et al.
(1991) are shown: I) blackbody emissions from 8,000 to 20,000 K;
II-III) hydrogen plasmas, optically thin in the Balmer continuum, with
T The EV Lac flare colour indices are spread over a large area in the
two colour diagram. However, a concentration close to the region of
optically thick plasma emissions at 1-1.5
10 ## 3.2. Light curves
Assuming that the light curves are due to rotationally modulated
visibility of surface inhomogenities, we have performed a Fourier
analysis of seasonal data series from 1969 to 1972 by periodogram
analysis for unequally spaced data (Scargle 1982, Horne & Baliunas
1986). The resulting periodograms reveal significant periodicity (with
confidence level greater than 99%) only for the data acquired in the
V-band in 1971 (4
In 1969, 1970, and 1972 the EV Lac magnitude was constant within 0.04 mag. Mean seasonal values of the EV Lac V magnitude, B-V and U-B colours are listed in Table 5, where we have not included the 1969 data because they were obtained in the instrumental system.
## 3.3. Phase distribution of flare occurence
One of the objectives of the present work was to investigate possible
spatial correlation between flares and photospheric spot regions. To
perform such an investigation, the behaviour of the flare occurrence
rate versus rotational phase was analyzed. For each data subset, we
computed the mean flare occurence rate, i.e. the ratio between the
number of flares and the flare coverage, in intervals of 0.1 phase
length. The resulting behaviour of flare occurence versus the
photometric phase are shown in Fig. 8 for the subsets 1968-1975.
The data acquired in 1976-1977 have not been considered for the
purpose of the present analysis because the total coverage obtained in
these years ( 32 h) is too short to get reliable conclusions.
To avoid effects due to inhomogeneties in the data (e.g. different
detection limits of the different instruments, etc.) the flare
occurrence rates have been separately computed for the data sets
acquired with the same telescope and passband. In most cases only the
data acquired in the U-band with the 61-cm telescope were used. This
is not the case for the 1969 data, because in this case only B-band
observations were available, and for the 1970 data, because about 98
A well defined behaviour of the flare occurrence rate versus phase is apparent only in the 1970 data. However, we believe that the lack of rotational modulation of the flare occurrence at the other epochs is not a conclusive result because of the higher threshold for flare detection with the 61-cm telescope than with the 91-cm telescope. To ascertain that the modulation of flare occurrence found in 1970 is not spuriously given by an anticorrelated modulation with the coverage time (), we have inspected the behaviour of coverage versus phase. The coverage appears slightly modulated in phase with the flare occurrence rate. Being at the denominator, the observed coverage behaviour cannot artificially enhance the flare occurence rate. For the sake of comparison, in Fig. 9 we have plotted the V-band light curves observed in 1970 and 1971 (top and bottom panels, respectively) and the flare occurrence behaviour in 1970 (middle panel). In 1970 the V-band light curve was flat, therefore the EV Lac photosphere was uniformely covered by spots or completely unspotted. On the contrary, in 1971 the V-band light curve had a minimum at phase 0.2, that implies a concentration of spots at that phase. The maximum of flare occurrence in 1970 lies in the phase interval 0.1-0.3. Therefore, the concentration of spots in 1971 was at almost the same stellar longitudes where flare activity was concentrated in the previous observational season.
To test the significance of this apparent correlation between the
site of preferred flare occurrence in 1970 and the site most covered
by spots in 1971, we have computed the linear correlation coefficient
( © European Southern Observatory (ESO) 1997 Online publication: April 6, 1998 |