DEPENDENCE OF THE MEAN LUMINOSITY OF FLARE STARS ON THE AGE OF THE SYSTEM

Astrophysics, Vol. 38, No. 2, 1995 DEPENDENCE OF THE MEAN LUMINOSITY OF FLARE STARS ON THE AGE OF THE SYSTEM A. A. Akopyan We study the luminosity of the brightest flare star and the mean luminosity of flare stars in a system as functions of the age of the system. We obtain estimates for the mean value and variance of the initial luminosity function of flare stars. 1. Introduction. Haro [1] was the first to conjecture that the flare-star stage in the life of red dwarf stars is regular and evolutionary and follows the stage of T Tau stars. The regularity of this evolutionary stage was confirmed by Ambartsumyan [2] by estimating the total number of flare stars in the Pleiades system, which turned out to be of the same order as the total number of dwarf stars in that system. The dependence of the luminosity of the brightest flare star on the age of the system containing these stars was pointed out by Haro and Chavira [3]: there is a relation between the earliest spectral class of flare stars and the age of the system. Subsequently Kunkel [4], using data on flare stars of two systems (Pleiades and Hyades), confirmed the existence of this dependence of the luminosity of the brightest flare star of a system on its age. Later the dependence of the mean luminosity of flare stars of a given system on its age was established by Mirzoyan and Brutyan [5] and then confirmed [6] on the basis of observational data of the nearest systems. It was shown that the observed luminosity function of flare stars is shifted toward lower luminosities as the system ages. The only system not satisfying this law turned out to be the stellar association Mon OB1 NGC 2264). However, the present author [7], using the method of estimating the age of young associations from the known relations of the number of flare stars and stars of T Tau type in them, obtained estimates of the ages of several star systems, including NGC 2264, showing that this dependence holds for all the systems studied in [6]. Figure 1 gives the dependence of the luminosity function of the brightest flare star and the mean luminosity of flare stars on the age of the system based on the data of [6]. The relations given in Fig. 1 differ slightly from the analogous relations given in [4] and [6]. First of all, we have used the values for the ages of the youngest systems obtained in [7], which differ from those used in [6]. Second, in order to take account of bias in the observational data due to the different distances of the systems studied, only flare stars with absolute photographic luminosity m < 11 m were used in [6] (these stars are accessible to observation for all the systems studied). The introduction of this restriction was necessary in order to prove that this dependence is real, rather than the result of observational bias; however the restriction itself is also selective. In particular, the mean luminosity of flare stars is significantly overestimated in older systems in comparison with young systems, since flare stars having luminosity m > 11 m form the major part of them due to the shift of the luminosity function. Since old systems are closest to us, the result is a decreased slope of the dependence (~., log t), where t is the age of the system (d~./d log t ~- 0.8). Despite the presence of bias due to distance, in the present paper we have used all the available data without any restrictions on luminosity. By removing the restriction m < 11 m we hope to obtain values for the mean luminosity of flare stars that are closer to the true values and to obtain an upper bound on the slope of the relation (~, log t). Indeed, not taking account of the selective effect of distance leads to a relative overestimate of the mean luminosities of flare stars of distant and younger systems, which leads to an elevated value of the slope of the relation: d-~./d log t -~ 2.2. Hence for the true slope of this relation we have 0.8 < d~./dlogt < 2.2. Translated from Astrofizika, Vol. 38, No. 2, 1995. Original article submitted July 16, 1995. 0571-7256/95/3802-0157512.50 1996 Plenum Publishing Corporation 157

12. Pleiades ~ -~' 1:4GC2254 x x /x t I! 6 7 Ig t 8 9 Fig.1. Curve 1 and the crosses give the dependence of the luminosity of the brightest flare star of a system on the age of the system. Curve 2 and the dots give the dependence of the mean luminosity of flare stars on the age of the system. Curve 3 gives the theoretical dependence of the mean luminosity of flare stars on the age of the system in the case of a normal initial distribution of luminosity of flare stars with distribution parameters ~a = 10.m 5 and ~ -- 3 m. 2. A possible interpretation of the dependence of the mean luminosity of flare stars on the age of the system. According to [8] the dependence of the mean luminosity of the flare stars of a system on the age of the system is a direct corollary of the dependence of the rate of evolution of stars on their masses, as a result of which the more massive stars, i.e., the brighter stars, pass through the flare star stage of evolution faster than the less massive stars of low luminosities. In favor of this interpretation is the fact that the relative number of flare stars among red dwarfs increases sharply as we pass from stars of high lunainosity to those of lower luminosity (cf. [9]). We assume that the initial distribution function of the luminosities of flare stars is the same for all systems. This makes it possible to regard given systems as realizations of the same system at different times. Comparison of the distribution functions of the luminosities of bright stars of the two closest and best-studied systems, the Pleiades and Praesepe [10, 11] shows that this assumption is valid at least for these systems. In this case the emergence of bright stars in the process of evolution from the set of flare stars naturally leads to a shift in the luminosity of the brightest flare star and in the mean luminosity of flare stars of a system toward lower luminosities. Here the dependence of the luminosity of the brightest flare star of the system on the age of the system, especially for large values of the age, can be interpreted as a dependence between the luminosity of stars and the duration of the stage of flare activity. Indeed, if the age of the system is much larger than the characteristic time of star formation, one can assume that all the flare stars of the given system have the same age, approximately equal to that of the system. In this case flare activity of a star indicates that the duration of the stage of flare activity for it is larger than the age of the system. For the brightest flare stars of the system this means that the duration of the stage of flare activity equals the age of the system, since the slightly brighter stars have already left that stage. 3. An estimate of the parameters of the initial luminosity function of flare stars. The mean luminosity of flare stars can be represented as 158 if mdf =, (i) o(t)

where ~.(t) is the mean luminosity of flare stars, too(t) is the luminosity of the brightest flare star, F is the initial luminosity function of the flare stars, and t is the age of the system. Breaking the integral in the numerator of (1) into two parts, we can write m df - m df = oo Jmoo, (2) ff o(t) where moo is the luminosity of the brightest flare star of the initial luminosity function of flare stars, i.e., " oo = too(0). By successively taking the quantity m outside the integral obtain the following equality from (2). where ff~ = E O0 f rao ( t ) too(t) m df as m = moo and m = too, we J moo fr~o ~n -mo df ~, - moo df.' ~oo < ~.( t ) < oo,, df /) 0 /70 m. df is the mean luminosity of flare stars of the initial luminosity function of the flare stars, i.e., ~. = ~.(0). According to the sense of the distribution function F(moo) = 0, F(oc) = 1. Taking account of this, we obtain from (3) moof(mo) + m - ~F(mo) < ~n < roof(too) + ~ - ~F(rno). (4) The quantities mo and ~ can be determined immediately from the observed luminosity functions [6]. Since moo = m0(0), the quantity moo can be determined by extrapolating the observed dependence (too,log t) (Fig. 1) to reasonably small values of log t, from which it follows that moo = 2m-3 TM" The analogous procedure for determining the quantity ~ is not justified due to the much lower precision in the determination of the (~, log t) curve, and may lead to large errors. The quantity moo can also be estimated on the basis of other considerations. It is to be expected that the luminosity of the brightest flare star of the initial luminosity function will have a value close to the luminosities of the brightest flare stars of those young systems where active star formation is occurring (Orion, NGC 7000, NGC 2264). This consideration also leads to the estimate ~- 3 TM (see Fig. 1). The quantity F(mo) has the following physical meaning. It is the portion of flare stars that, according to this interpretation, have left the stage of flare activity (m. < m0), i.e., F(m0) is the portion of normal stars in the total number of dwarf stars: df (3) g. (5) - U. + Nf' where N~ is the number of normal dwarf stars and Ny the number of flare stars. At present for the majority of systems considered the determination of the quantities N~ and IVy is fraught with significant errors. In this respect the best-studied system is the Pleiades, which due to its relative nearness and the significant number of observations of its flare stars over a considerable time is less subject to various sources of bias. The number N~ for the Pleiades system can be estimated from the known luminosity functions of the bright stars given in the papers of Jones [12] and Stauffer et al. [10]. It is of the order 150-200. The number of flare stars in the Pleiades can be estimated by Ambartsumyan's method [2] using observational data. According to [13] it is of the order of 1000. However, in estimating the number of flare 159

stars flares have been used from flare stars of the field and false flares, which are errors of observation and analysis. Taking account of flare stars of the field may lead to a 10% reduction in the number of flare stars of the Pleiades system now recorded [14]. The higher estimates given in [10, 15] are mostly based on a determination of the number of flare stars of the field by studying the proper motions of the stars. However it has been shown in [16] that flare activity of a star is a more reliable indicator of its membership in a system than the criterion based on proper motions. This conclusion follows from the fact that all flare stars of the Pleiades system, independently of their probability of belonging to the system computed on the basis of proper motions [10], have a marked tendency to concentrate about the center of the system (cf. Fig 1 in [16]). According to [15], the proportion of pseudoflare stars can be significant among stars with a single recorded flare when the photographic amplitude of the flare satisfies Am < 1 m, which by our crude estimates can lead to a lowering of the number of known flare stars by at most 10%. For that reason we assume that as a result of taking account of these factors the number of known flare stars can be lowered by 20%. The method of Ambartsumyan in turn may lower the estimate of the total number of flare stars by 30%. However it must be kept in mind that the Ambartsumyan method itself gives only a lower bound on the number of flare stars [17]. Taking account of this circumstance may completely compensate for the given lowering and even more. Therefore we take the estimate N S = 1000 [13] as the lower estimate for the total number of flare stars. From (5) we obtain F(mo) = 0.13-0.16. For the value F(mo) = 0.15 of (4) we obtain an estimate of the mean value of the initial luminosity function of flare stars 10~. 15 < ~Z < 10m. 75. The variation of the mean luminosity of flare stars A~ = ~, - ~ over the lifetime of the system can be estimated from the expression F(mo)(~, - too) < A~, < F(mo)(~,, - moo), which follows immediately from (4). For the Pleiades system A~ is of order 1 (0.66 < A~ < 1.26). Similarly one can estimate the variance of the initial luminosity function of flare stars. For the observed variance we have 0.2._~ o /?, " (m - ~)~ df or, breaking the integral in the numerator into two parts, 0 df (m - ~)2dF - (m- ~)2dF 0.2.= E oo E oo, (6) 0 Taking the quantity (m -~)2 outside the integral successively as (m- ~)2 = (too -~)2 and (m- ~)2 = /2 O0 O0 (moo- ~)2 and taking account of the fact that (m-~)2df = ~2+ i~2, where -~2 = (m- ~)2dF is the variance of the initial luminosity function for flare stars, we obtain from (6) 160 a2 > 0.2 + F(mo)[(~, - too)" - 2] _ Am2, "/ a2 < ~2 + F(mo)[(m - moo) 2 - ~2] _ Am2. j, (7)

The observed value of the variance of the luminosity of flare stars of the Pleiades system is a 2 = 3. Substituting this value into (7) we obtain the following estimate of the variance of the initial luminosity function of flare stars: 4.5 < 32 < 12.1 or 2.1 < 3 < 3.5. 4. The normal distribution approximation. To determine the form of the initial luminosity function of flare stars it is necessary to have the relations given in Fig. 1 for (m0,1ogt) and (~,logt) with much greater precision and for a wider time interval. However for a significant part of the flare stars the duration of the stage of flare activity is an order of magnitude or more higher than the time necessary for complete or partial decay of the physical system they belong to. In other words a significant number of stars leave the mother system and become field stars without ceasing to be flare stars [14]. As a result of partial or complete decay of the system the process of transition of flare stars into normal stars is not Completed within the physical system; therefore one can hardly expect that it will be possible to enlarge the time interval significantly by the possible discovery of old systems of flare stars and determine precisely the form of the initial luminosity function of flare stars. However consideration of the forms of observed luminosity functions of flare stars given in [6] gives some grounds for taking the normal distribution as a first approximation to the initial luminosity of flare stars. Assume that the initial luminosity function is normal: df - v'~ exp - ~ cr ' (8) where N. is the mean and ~2 the variance of the luminosities of the initial luminosity function of flare stars. Substituting (8) into (1), we obtain a relation between the mean luminosity of flare stars and the luminosity of the brightest flare star: ~. = ~, +, (9) where R is the Mills ratio known from mathematical statistics (cf., for example, [t8]). Using the observed relation (rn0,1ogt) (Fig. 1) from relation (9) one can construct the expected dependence of the mean luminosity of flare stars on the age of the system in the case of a normal initial distribution. In constructing the relation given in Fig. 1 we used the parameter values N = 10~.5 and ~ = 3 TM in accordance with the estimates obtained above. As can be seen from Fig. 1 the agreement with the observed dependence is good for the closest and oldest systems (Pleiades, Praesepe). In the approximation using a normal initial distribution of luminosities of flare stars one can also estimate the mean duration of the stage of flare activity. It was pointed out above that the dependence of the luminosity of the brightest flare star on the age of the system (m0, log t) can be interpreted as a relation between the luminosity and the duration of the stage of flare activity, i.e., as (m, lnt), where T is the duration of the stage of flare activity. As cah be seen from Fig. 1, this dependence is well represented in linear form m = alnt+ b, (10) where a = 1.1 and b = -11.8 are determined from the relation (m0, log t) (Fig. 1). Substituting (10) into (8), one can obtain a logarithmically normal distribution function for the duration of the stage of flare activity: qo(t)dt= -------zl. exp[ - 1 (lnt - lnt) 2] with mean value InT - r3. - b and variance ~. The values of the coefficients a and b determine the a mean duration of the stage of flare activity: lnt = 20 or 5-108 yrs. 161

i i I I I 120 Z 50 I 4 8 12 15 Fig. 2. Observed distribution of luminosities of flare stars of the Orion system (histogram) and a comparison with the normal distribution with parameters ~ = 10.m 5 and ~ = 3.m m Among the young distant systems the best studied in terms of flare-star statistics is the system of flare stars of the Orion association. However due to the comparatively large distance of that system flares have been recorded up to the present time only for stars with m < 12m-13 m. Here for the flare stars of lowest luminosities the minimal amplitude of a recorded flare is of the order 4m--5 m and higher. Naturally in this case a significant number of flare stars of lower luminosities with lower-amplitude flares are lost. By comparing the observed luminosity function of flare stars with the normal distribution one can take account of the contribution of possible bias in the estimate of the total number of flare stars. To this end Fig. 2 gives the observed luminosity function of flare stars of the Orion system (from [6]) in comparison with the normal distribution, which was studied above as a first approximation to the initial luminosity function. The distributions given in Fig. 2 are normalized so as to obtain good agreement of the left tails of the distributions (corresponding to the bright stars) free of bias due to distance. As can be seen from Fig. 2, taking account of the bias may significantly increase, at least by a factor of 2, the number of flare stars, which at present is estimated to be 2000 [19]. 5. The possible role of decreased luminosity of stars. Flare activity of a star is a manifestation or consequence of a nonequilibrium physical state of the star, which gives grounds for supposing that during the flare stage the luminosity of the star may vary significantly in proportion to its approximation to the equilibrium state. Naturally a decrease in luminosity of stars introduces certain variations into these relations. In particular it leads to a shift not only in the mean luminosity of flare stars and the luminosity of the brightest flare star, but also to a shift in the maximum of the observed luminosity function of flare stars. Such a shift appears, for example, in comparing the luminosity function of flare stars of the TOT system and the Pleiades/Praesepe system. This shift (~ 2 m) is difficult to explain as the influence of bias, since the given systems lie at nearly the same distance. Starting from this one may conjecture that the decrease in luminosities of flare stars makes a certain contribution to the dependence of the mean luminosity of flare stars on the age of the system. Preliminary analysis shows that taking account of decreased luminosities of flare stars leads to a lowered estimate of the quantities ~, ~, and NI, the number of flare stars in Orion obtained in Sects. 3 and 4 above. 6. Conclusion. We have studied the dependence of the luminosity of the brightest flare star of a system and the mean luminosity of flare stars in it on the age of the system. Within the context of the interpretation of these relations proposed in [8], in the present paper we have estimated the parameters (mean and variance) of the initial luminosity function of flare stars. In the normal distribution approximation we have also estimated the mean duration of the stage of flare activity and the influence of bias due to distance. We have proposed an alternative explanation of the observed dependences in which we take account of the fact that the luminosities of individual flare stars may decrease over time. The observational data available at present do not allow an unambiguous determination of the relative contribution of this factor to the formation of the data of these relations. 162

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