Astron. Astrophys. 327, 1114-1122 (1997)

3. Results

3.1. Statistical analysis

3.1.1. Flare energies and accumulated flare frequency distribution

The 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 h coverage, we can expect the occurrence of only one flare with total energy release in excess of erg, as observed.

Fig. 1. Flare energy distributions

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 a and b are parameters that can be derived from the analysis of the accumulated frequency distribution. The accumulated frequency distribution of U-band flares is shown in Fig. 2, where the total flare energy release (E ) is plotted versus the mean occurrence rate ( ) of flares with energy larger than E (in ergs) detected during the time T (in seconds).

Fig. 2. The accumulated frequency distribution of U-band flares.

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 a and b were computed as and , with n the number of data points used in the linear fit.

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 nanoflare on the Sun), the average flare energy production of EV Lac in the U-band remains unchanged within the 2nd decimal digit, whilst a choice of only one order of magnitude greater yields a two times larger flare energy output.

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 b depends on the star age. Therefore, the flare characteristics, such the energy distribution, are linked to stellar rotation rates and, consequently, to magnetic activity levels.

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 n is the number of flares observed during the coverage time T in each season. In Fig. 3 the time behavior of the flare frequency parameter is shown for the subset of data acquired in the B-band with the 61-cm telescope, which is the most conspicuous data set we acquired with the same instrument in a single filter. The choice to inspect the time variability of flare frequency, separately for each filter and telescope, is to avoid effects linked to inhomogenities in the data. A slight modulation in the flare occurence is apparent from 1970 with a period close to 3 years. The observed time distribution can be tested against the null hypothesis, i.e. that no variation in flare frequency actually took place. We computed the expected number of flares for each season by multiplying the monitoring time by the average flare frequency that we found to be 0.17  (see Table 4).

Fig. 3. The time behavior of the yearly mean flare frequency.

Table 4. Mean flare activity level on yearly time-scale. The data considered are those acquired in the B-band with the 61-cm telescope. P is the probability that the observed flare frequency is different from the expected value only by chance.

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 h of coverage, has a high significance (0.01% probability that it was a chance result). Therefore, the apparent 3 years modulation in flare activity, that should have required a high level of flare occurence also in 1968, does not appear to indicate a permanent cycle.

3.1.3. Color-color energy correlations

The 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-scales

The distributions of the rise-times to the highest flare peak ( ) and of the descent times from flare maximum to quiescence (da ) are shown in Fig. 4 for U -band flares.

Fig. 4. Distribution of rise-times ( ) to the highest peak (top panel ) and descent times (da ) from the flare maximum to quiescence (bottom panel ), for the flares observed in U -band.

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 da ) are plotted versus flare energies ( ). Least square fits to the data give the following relationships:

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.

Fig. 5. Relationships between flare time-scales and flare energy.

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 indices

The 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.

Fig. 6. The colours of flaring plasmas at light maximum in the two-colour diagram (cf. text).

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_e =10,000 K and electron densities 10¹² and 10¹⁴ cm^-3, respectively; IV-V) optically thick plasmas at T_e =10,000 K and T_e =15,000 K, respectively; VI-VII-VIII) dwarf star upper layers heated by proton beams with threshold proton energy of 1, 2 and 5 MeV, respectively.

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 1-1.5 10⁴ K is apparent, but flare events compatible with proton beam are also observed.

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^d.45 0^d.01, with a confidence level of 99.6%). A similar period (P=4^d.44 0^d.02) results from the analysis of the B-band 1971 data, but with a confidence level of 96.9%. The U-band data acquired in the same year are more noisy than the B- and V- band data, therefore it was not possible to identify any significative periodicity. The 1971 V and B-V light curves are shown in Fig. 7, where phases were computed by adopting JD 2440793.5085 as initial epoch and P=4^d.45. The 1971 V light curve can be reproduced by a sinusoidal function with peak-to-peak amplitude of 0.11 mag and the light minimum at =0.18.

Fig. 7. EV Lac V and B-V light curves in 1971. Phases are computed using the ephemeris HJD0=2440793.5085, and P=4.45.

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.

Table 5. Mean seasonal magnitudes of EV Lac

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 h of U band monitoring with the 91-cm telescope (the best conditions to detect flares) were available. The total coverage (in hours), the total number of observed flares, the passband and the telescope aperture are given in the Fig. 8 for each curve. The phases have been computed by using JD 2440793.5085 as initial epoch and P=4^d.45 as done for the 1971 V-band light curve (cf. Fig. 7).

Fig. 8. Flare occurence rate versus photometric phase.

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.

Fig. 9. Top panel: V-band light curve in 1970. Middle panel: Normalized flare distribution in 1970. Bottom panel: V-band light curve in 1971.

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 (r) between (the number of observed flares/hour, binned in 0.1 phase intervals) and (the mean V value at the central phase of each bin). The result was r =0.77. The probability of determining such a correlation by chance from an uncorrelated population is 0.01 for N =10 data points, as in our case.

© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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