Photometric monitoring of 12 BL Lacertae objects

Abstract

We present the results of our monitoring the flux variability of 12 BL Lac objects, which have variabilities on time-scales ranging from hours to months. Individual sources are discussed in detail. Three of them, OY091, and , show significant rapid variation (hours). Two of them, 3C 66A and Mrk 501, exhibit significant variability on time-scales of months. We find that 3C 66A has a variability period of , supporting the 65-d period obtained by Lainela et al. The periodicity seems to be weak according to the Jurkevich test. We also find a correlation between colour index and B magnitude for Mrk 421. We discuss possible physical mechanisms on the basis of these observational phenomena.

galaxies: active, BL Lacertae objects: general, galaxies: jets, galaxies: nuclei, galaxies: photometry, gamma-rays: observations

Introduction

Variability at different wavelengths (UBVRI) has often been used to probe the central engine and physical process of active galactic nuclei (AGNs). In recent years, considerable progress has been made in studying the variability of blazars. The studies of long-term light curves have shown a variability periodicity of about 10 yr in some sources, such as OJ287 (Sillanpaa et al. 1988; Kidger et al. 1992), ON231 (Liu & Xie 1995) and 3C 345 (Schramm et al. 1993; Zhang et al. 1998). During the long-term monitoring , OJ287 was found to burst almost exactly at the predicted time (see e.g., Katajainen, Takalo & Sillanpaa 2000). Kranich et al. (1999) reported evidence of a 23-d variability of Mrk 501 in TeV and X-ray ranges. For another TeV γ-ray source, PKS , a flare with and a time-scale of 23-d occurring from JD 245 0321.193 to JD 245 0342.093 was observed (Xie et al. 1999). These results suggest that the variability of blazars on medium time-scales is also very interesting. The rapid variability for BL Lac objects is irregular (e.g. Xie et al. 1992). The periodicity of long-time-scale variability and the irregularity of rapid variability indicate that the origin of the variability on long time-scales is probably different from that for the variability on short time-scales.

The variability on short time-scales, which is useful for developing a new interpretation for the objects (Zamorani et al. 1984), has provided interesting results for some blazars (e.g. Urry et al. 1993, 1997; Wagner & Witzel 1995; Villata et al. 1997; Xie et al. 1999, and references therein). These studies have provided new strict constraints for models on the structure of the central engine and the physical process of blazars.

For explaining new observational results, the shocks-in-jet model seems to be more reasonable than other models (e.g. Wager & Witzel 1995; Hughes et al. 1998; Bloom et al. 2000; Peng et al. 2000; Romero et al. 2000). It is known that the optical photons in the beamed relativistic jet might be scattered into the γ-ray range by inverse Compton scattering. Optical monitoring simultaneous with the γ-ray satellite pointings presents an opportunity to test how these optical variations are correlated with variations in the γ-ray range. Intensive monitoring of a sample of BL Lac objects has been carried out since 1980 at the Yunnan Observatory in the B, V, R and I bands (Xie et al. 1988, 1992, 1996). We report here a brief description of 12 Lacertae objects as a sample of this monitoring programme.

Observations and Data Reduction

Most of the observations presented here were made by the 1.02-m RCC telescope equipped with an RCC CCD (1024 × 1024 pixels). The scale of the CCD is 0.38 per pixel. The telescope was used at the Cassegrain focus. The CCD observations of IES 2344+514 on 2000 November 28, 2000 December 18 and 19, and 2001 January 3, were carried out at the Cassegrain focus of the 60-cm telescope (made in Germany) at the Yunnan Observatory. The CCD was Tektronix TK512CF Grade I system of with dimensions, giving on this telescope a total field of with an image scale of 0.445 arcsec pixel⁻¹. The filters are as follows: B: GG385(2 mm)+BG12(1 mm)+BG18(1 mm), V: GG495(2 mm)+BG18(2 mm), R: RG610(3 mm)+66.2500(1 mm), I: RG710(3 mm) +60.5050(1 mm). The integration time is for most observations of the objects. Photometry was obtained by observing standard stars taken from the literature (Craine 1977; Smith et al. 1985; Fiorucci 1988;Villata et al.1997). These standard stars are listed with references in Table 1. Photometric measurements were made with the normal iraf routines. The results of our monitoring programme are listed in Table 2, where column 1 is the UT date, column 2 is the Julian Date, column 3 is the magnitude, column 4 is the rms error, and column 5 is the filter used.

Table 1

List of the sources.

Table 2

The observational data.

The data analysis techniques were detailed in our previous paper (Xie et al. 1994). The minimum variability time-scale, Δt_min, is defined as in Miller, Carini & Goodrich (1989). This time-scale corresponds to a minimal change of 30 per cent or more of the emitted intensity in the B, V, R and I bands (Urry & Padovani 1995). In addition, the amplitude of optical variability on this time-scale must be more than 5σ, where σ is the maximum total observational rms.

Results and Analysis

OY091 (PKS 2254+074)

OY091 is a BL Lac object. Its historical brightness variations were 1.6 mag. In 1981, a 1.8-mag outburst was detected (Pica et al. 1988). According to Pollock et al. (1979), Pica et al. (1988) reported 2.37-mag variations during 1.39 years, and after that flaring the brightness faded 1.3 mag in 18 days. We measured the faintest B-band magnitude for this object so far. It is (Xie et al. 1994). A rapid optical flare of 0.69 mag in 52 minutes in the B band was obtained on 1990 17 December (Xie et al. 1994).

We observed the object continuously for 4.5 hours in our monitoring programme on 1998 September 23. The source was in an active state. The maximum variations of about 0.64 mag in the B band within 91 minutes and 0.37 mag in the V band within 3.7 hours were observed. The results are shown in Table 2 and in Figs 1 and 2, respectively. In Figs 1 and 2 the upper panels display the light curves in the B and V bands for OY091, and the lower panels display the deviation from the mean of the magnitude difference between two comparison stars in the same field of the source, giving information on the reliability of the comparison stars for each frame. From Fig. 1, we can see that the source faded from to 18.38 mag in 11 minutes and then brightened from 18.38 mag to 17.74 mag in 80 minutes. The brightness variation is 0.64 mag in 80 minutes. The lower panel of Fig. 1 shows that the maximal deviation of the comparison star E is 0.06 mag in one night. Such large brightness variations are very rare. Some comparison of our results to variability time-scales observed by other authors is necessary. On 1995 November 3, Villata et al. (1997) observed a similar rapid increase 0.21 mag in 10 minutes. In addition, on 1995 July 24–25, a rapid variability with decreasing and in 23.97 hours had been observed for the source by Villata et al. (1997). Obviously, our result is consistent with the observations of Villata et al. (1997).

Figure 1

The light curve of OY091 in the B band on 1998 September 23 (the upper panel) and the deviation from the mean of the magnitude difference between the two comparison stars E and D (the lower panel).

Figure 2

The light curve of OY091 in the V band on 1998 September 23 (the upper panel) and the deviation from the mean of the magnitude difference between the two comparison stars E and C1 (the lower panel).

From Fig. 2, we also find that at the end of the observation on this night the source exhibited a rapid decrease in brightness of in 52 minutes. There is no correlation between the variabilities in the B and V bands.

The observations of the object on 1999 September 28 also show similar intraday variations of 0.50 and 0.43 mag in the B and V bands, respectively. The source faded from 17.71 to 17.97 in the B band, then brightened rapidly from 17.97 to 17.47 in 22 minutes. After that, the source faded slowly to 17.72 within 66 minutes. The results are shown in Table 2 and Figs 3 and 4. We have examined the CCD brightness measurements of the field standard star C. The maximal deviation of the field standard star C for one night is 0.06 mag (see the lower panel of Fig. 3). So we infer that the time evolution of the standard stars should not cause errors greater than 6 per cent.

Figure 3

The light curve of OY091 in the B band on 1998 September 28 (the upper panel) and the deviation from the mean of the magnitude difference between the two comparison stars C and B (the lower panel).

Figure 4

The light curve of OY091 in the V band on 1998 September 28 (the upper panel) and the deviation from the mean of the magnitude difference between the two comparison stars E and B (the lower panel).

The light curve in Fig. 4 shows that the source first gradually faded within 16 minutes, then gradually brightened within 13 minutes, after that, the source exhibited a slow decrease of in 2.1 hours. The maximal deviation of the comparison star E is 0.11 mag.

H0548ȁ22

H0548−322 is an X-ray-selected BL Lac object. Optical variability of more than 0.5 mag on a time-scale of 30 d has been found (Cruz-Gonzalez & Huchra 1984). H0548−322 has always been in our monitoring programme since 1992, and a flare of 0.58 mag in 55 minutes in the of V band was recorded on 1992 February 28 (Xie et al. 1996).

Our light curves for the R, V and B bands observed between 1996 December and 2000 January at the Yunnan observatory are shown in Fig. 5. They indicate that the source varied slowly during our monitoring period. The amplitudes of the observed variability are , , and . Also, the R, V and B-band light curves are well correlated with each other.

Figure 5

The light curve of in the B, V and R bands from 1996 December to 2000 January.

The results of our monitoring on 1999 January 17 and 23, and 2000 January 5, show that the source exhibited no significant variability on each night (see Table 2), while our monitoring on 1999 March 12, indicated significant variability, shown in Fig. 6. The upper panel of Fig. 6 displays the light curve in the R band, and the lower panel displays the deviation from the mean of the magnitude difference between two comparison stars B and A. From Fig. 6, we can see that, from 12.35 to 12.6 UT, the source brightened by 0.43 mag within 23 minutes, subsequently fading by 0.44 mag. The lower panel of Fig. 6 shows that the maximum deviation of the comparison star is 0.09 mag, indicating that the confidence level of this variability is not high.

Figure 6

The light curve of in the R band on 1999 March 12 (the upper panel) and the deviation from the mean of the magnitude difference between the two comparison stars A and B (the lower panel).

0048ȁ097

Repeated spectroscopic observations of 0048−097 failed to detect any emission or absorption lines in this object. The object appears to be stellar (Falomo et al. 1990). A lower redshift limit of can be derived from the absence of the extended nebulosity of the BL Lac host galaxy. The maximum magnitude variation over the considered period was , and the maximum optical polarization is 27.1 per cent (Stickel et al. 1993).

Our observation of the object on 2001 January 18, is the first in our monitoring programme. The results are displayed in Table 2 and Fig. 7. The upper panel of Fig. 7 the light curve of . The light curve shows that from 11:42 to 12:26 UT, the R brightness of the source increased by 0.38 mag in 45 minutes, then decreased by 0.32 mag in 20 minutes, and subsequently increased 0.38 mag in 17 minutes. The lower panel of Fig. 7 displays the deviation from the mean of the magnitude difference between two comparison stars 1 and 2. From Fig. 7, we can see that field comparisons are very stable. The maximal deviation of the differential magnitude between star 1 and star 2 in one night is only 0.05 mag. The results obviously show that the source was in an active state, and the confidence of the variability time-scale is high.

Figure 7

The light curve of in the R band on 2001 January 18 (the upper panel) and the deviation from the mean of the magnitude difference between the two comparison stars C1 and C2 (the lower panel).

IES244+514

IES2344+514 was only recently identified as a BL Lac object (Perlman 1996) based on its lack of optical emission lines and its Ca ii‘break strength’ being smaller than 25 per cent. It is the fourth closest known BL Lac object after Mrk 421, Mrk 501 and EXO (Padovani & Giommi 1995). Perlman et al. (1996) derived a 2-keV X-ray flux of 1.14 μJy, roughly one-third of the flux detected for Mrk 421 and Mrk 501, and measured an optical magnitude of with no galaxy subtraction. The Green Bank radio survey lists 5 GHz of which is about one-third and one-fourth of the flux of Mrk 421 and Mrk 501, respectively. IES2344+514 was discovered in Tev γ-rays (>350 Gev) by the Whipple Observatory telescope, and identified as a Tev γ-ray BL Lac object (Catanese et al. 1998).

This is the first observation of IES2344+514 in our optical monitoring programme. Our observations on 2000 January 5, November 28 and December 18 and 19, and 2001 January 1 and 17 show that it has no significant variation in a day (see Table 3). Its intraday maximum flickering amplitudes only are , . Fig. 8 shows a typical intraday light curve, which was observed on 2001 January 17. These results probably suggest that the short time-scale variation of Tev γ-ray blazars is a rare phenomenon in the optical band, because the synchrotron radiation peaks is in the UV/soft X-ray region in spectral energy distribution (SED), i.e., the flux variation (flare) is occurring mostly in X-ray, Gev and/or Tev γ-ray energy. A brightness decrease of 0.35 mag in 14 days had been observed in the V band on 2001 January 3–17. The upper panel of Fig. 9 shows the mean magnitude of IE2344+514 on each day, and the lower panel displays the mean of the magnitude difference between comparison stars 1 and 2. From the upper panel of Fig. 9, we can see that the mean brightness decreased by 0.23 mag in 14 days in the V band. From the lower panel of Fig. 9, we found that the maximal deviation of the deferential magnitude between stars 1 and 2 in 50 days is 0.03 mag. That is, the confidence level of this variability time-scale is high. These results imply that optical variability exists on a medium time-scale for this source.

Table 3

The optical variability time-scale.

Figure 8

The light curve of IES2344+514 in the V band on 2001 January 17 (upper panel) and the deviation from the mean of the magnitude difference between the two comparison stars C1 and C2 (lower panel).

Figure 9

The mean magnitude of IES2344+514 on each day (the upper panel) and the mean of the magnitude difference between comparison C1 and C2 (the lower panel).

1101+84 (Mrk 421)

Mrk 421 is the brightest BL Lac object at UV, X-ray and Tev γ-ray wavelengths. It was detected as the first extragalactic source of Tev γ-rays by the Whipple Observatory γ-ray telescope (Punch et al. 1992), and has been confirmed as a Tev γ-ray source by the HEGRA Collaboration (Petry et al. 1996). An extremely rapid burst of Tev photons from Mrk 421 was obtained, with the flux increasing by a factor of 20–25 on a time-scale about (Gaidos et al. 1996). A similar dramatic outburst in the optical band was obtained by Xie et al. (1998). Miller (1975) reported the light curve in the B band from 1899 to 1975 for Mrk 421. A rapid decline of 1.6 mag in brightness in 16 days was found in 1942 January, and the maximum magnitude variation over the considered period was (Miller 1975). The maximum magnitude variation detected by Villata et al. (1997) was about 0.4 mag in both R and B bands in about 60 days. The steepest variation so far is 0.19 mag in 1 day in the R band (Villata et al. 1997).

Since the BL Lac objects are violent variables, their colour indices should ideally be derived from simultaneous observations in the various colours. However, if the interval between the B and V observations is sufficiently small, these observations should be regarded as quasi-simultaneous. In our monitoring programme on 2000 May 1, the quasi-simultaneous observational results show that the intraday variabilities are , and , respectively.

The results are shown in Table 2. We notice that the maximal deviation of the differential magnitude between stars 1 and 2 is too large, so the confidence level of this variability time-scale is low.

From UT 14:58 (JD 245 1666.106) to UT 15:42 (JD 245 166.154) the interval between the B and V observations could be regarded as quasi-simultaneous. We find a strong correlation between the colour index and the magnitude in the B band, with a regression line  

(1)

with a linear correlation coefficient 0.99 and chance probability . It is evident that the colour index is also variable, and that it is greater when the source is fainter (see Fig. 10). For PKS 0735+178, OI 090.4, OJ–131 and ON+231, we have also observed similar correlations between colour index and B magnitude (Xie et al. 1992, 1991, 1990), while for Mrk 421, this is the first time we obtain this correlation. We should check the correlation in the future.

Figure 10

The colour index against the magnitude in the B band. The solid line is the regression line.

3C 66A

3C 66A was first identified optically by Wills & Wills (1974). It is a typical radio-selected BL Lac object, and it exhibited substantial variations in both brightness and polarization (Smith et al. 1987; Takalo et al. 1996, and references therein). An rapid decline of 0.52 mag in 42 min in the B band was observed (Xie et al. 1992).

Our new light curve observed between 1994 November and 1999 January at the Yunnan Observatory is shown in Fig. 11, where some data between 1995 and 1997 taken from the Tuorla Observatory (Katajainen et al. 2000) and the Vainu Bappu observatory (Ghosh et al. 2000) are represented by open circles and triangles, respectively, and our data by crosses.

Figure 11

The light curve of 3C 66A in the V band.

Fig. 10 shows that our light curve does not have any quiescent state at all. The brightness has fluctuated continuously, typically by on a time-scale of a few days. We analysed the light curve, and found a significant rapid variability in 1996 January. The maximum V-band brightness during our monitoring was 12.81 (JD 245 0320.260) and the minimum was 15.67 (JD 244 9660.188), which are so far the brightest and the faintest V-band values for this object, respectively. In our new monitoring programme, on 2001 January 19, 3C 66A showed a very stable state with . No significant short-time-scale variation was observed. For checking the medium-time-scale variability of about 65 d, which had been found by Lainela et al. (1999) and Katajainen et al. (2000), we analysed the light curve between JD 495 0300 and 495 0540 in Fig. 11 by using the powerful Jurkevich method (Jurkevich 1971). We consider the parameter ƒ, as did Kidger et al. (1992) and Liu & Xie (1995),  

(2)

where is the normalized value. In the normalized plot, a value of implies that , and hence there is no periodicity at all. A value of generally indicates that a strong periodicity exists in the data, whilst usually indicates that the periodicity, if genuine, is a weak one (Liu et al. 1997). The result of our analysis with for the period of is and . That is, . The value of shows that the periodicity of ∼63 d for 3C 66A is a weak one. Obviously, our results support the idea that there is medium-time-scale periodicity variability in the optical band for 3C 66A, which had already been obtained by Katajainen et al. (2000).

OI090.4 (PKS 0754+101)

The historical light curve of PKS 0754+101 had been study by Baumert (1980), who found 0.8-mag variability on a time-scale of a few days. Smith et al. (1987) obtained , while Katajainen et al. (2000) reported . Our monitoring programme lasted 7 years (1985–1991), and the range of variability of the object is in the V band (Xie et al. 1994). A rapid optical flare of 0.56 mag in 80 minutes in the B band was recorded on 1998 December 11, (Xie et al. 1991). In our monitoring programme on 1993 January 23 and 2001 January 18 the observations show that OI090.4 was in a stable state with on each night. The minimum, (JD 245 1285), was observed at the Yunnan Observatory on 1999 January 23. It is the faintest V-band measurement for this object so far (see Table 3).

Mrk 501 (1652+98)

Mrk 501 is a member of the 1-Jy (Kuehr et al. 1981) and the Einstein Slew Survey samples (Elvis et al. 1992), and is a well-known Tev γ-ray blazar (Aharonian et al. 1997;Bradbury et al. 1997). The multifrequency spectrum of this source is very similar to that of an high-energy peaked BL Lac objects (HBL) (Ghosh & Soundarararajaperumal 1995; Sambruna, Maraschi & Urry 1996). The source exhibits a strong flare at Tev γ-rays, indicating its short time-scale of at least 1 d (Sambruna et al. 2000). Mrk 501 had displayed optical variability on time-scales of weeks (Heidt & Wagner 1996). Optical observations in 1992–1993 showed that its intraday ‘flickening’ amplitude is smaller than 0.22 mag in the B and V bands (Xie et al. 1996). Ghosh et al. (2000) had searched the rapid variability of Mrk 501 on time-scales of minutes to hours and found a 0.13 mag variation within 12 minutes. During 1995 and 1996 the Tev γ-ray and optical emissions of this source were observed to be essentially constant by the Whipple γ-ray telescope (Quinn et al. 1997) and the 1-m telescope at the Yunnan Observatory (Xie et al. 2001). In our monitoring programme on 2000 January 3–4, the source was stable. The source exhibits oscillation of ΔR = 0.14 on each night (see Table 3).

OT081 (1749+096)

This source is strongly variable in both flux density and polarization at centimetre wavelengths, and its variability time-scale increases with wavelength (O'Dea et al. 1986). It outbursted on 1977 April 11 (Epstein, Landau & Rather 1980). Our observation on 1990 April 28 showed that a rapid fading of 0.48 mag within 27 minutes in the V band was followed by an outburst of 0.69 mag in 43 minutes, with a confidence level greater than 4σ but smaller than 5σ, where σ is the maximum total observational rms error. These results should be checked by more observations. We therefore carried out new observations over 6 days in 1999 May, but the source was not in an active state during these observations.

Due to the weather, only the limited number of data points have been recorded (see Table 2), and a more detailed analysis is impossible.

There are similar situations for the sources of Mrk 180, 0316+413 and 1028+511 (see Table 3).

Conlusions

We have presented some new results of our CCD photometry monitoring programme for the flux variability of 12 BL Lac objects from 1998 January to 2000 January. Since BL Lac objects have displayed variability on diverse time-scales, we chose only the objects with short and medium time-scale variabilities ranging from hours to months. The results are tabulated in Table 2.

Most of the objects in the sample were in the quiescent state during our monitoring. Only three of them, OY091, and , showed significant short time-scale variabilities. We also observed the RBL OY091 in an active state on 1998 September 23 and 28. The largest amplitude is about within 91 minutes and within 88 minutes, respectively. Observations of another RBL, , also showed that the source was in active state on 2001 January 18. Its largest amplitude is about within 65 minutes. During our monitoring, other RBLs (3C 66A, 0754+101, OT081, 0316+413 and Mrk 180) did not show rapid fluctuations in their brightness. Their optical emissions are substantially constant, although some small fluctuations are evident (see Table 2). Observations of the H0548–322 showed that this object was in active state too on 1999 March 12. The largest amplitude is within 32 minutes. During our observations on 1999 January 12 and 23, 2000 January 5, and 2001 January 19, H0548–322 did not show this kind of rapid brightness oscillations. Two of the objects, 3C 66A and Mrk 501, show significant medium-time-scale (months) variations. Using the Jurkevich test, we found that 3C 66A has variability with a period of , but this is weak. Our result is consistent with the 65-d period obtained by Lainela et al. in 1999. We also found a strong correlation between colour index and B magnitude for Mrk 421. We have discussed the possible physical mechanisms on the basis of these observational phenomena. On the other hand, for other XBLs, Mrk 501, IES2344+512, Mrk 421 and 1028+511, no significant short-time-scale variability was observed.

More recently, Kranich et al. (1999) reported that they had found evidence of a 23-d periodicity variation for a TeV source at TeV and X-ray energies. During our monitoring programme, the TeV γ-ray source IES2344+514 exhibited a decline of ∼0.35 mag within 14 days, and another TeV γ-ray source, 3C 66A, also exhibited a evidence of a weak periodicity variability, which is consistent with the results of Lainela et al. (1999). The multiband observed results mentioned above probably suggest that the medium time-scale variability is a common property for TeV γ-ray sources. Simultaneous observations are therefore very important for understanding the properties of TeV γ-ray source and BL Lac objects.

Discussion

The energy output of BL Lac objects is so enormous that only the accreting black hole model can explain the observed emission. In this model we have a supermassive black hole at the centre of BL Lac objects. Around the black hole, there is an accretion disc from which material falls into the central black hole. If one assumes that the microvariability detected for these objects is generated in the vicinity of a super massive black hole, one can estimates the size of the emitting region from , assuming no relativistic beaming. The time-scale identified from the present observations is the time during which the dramatic variability occurred (Miller et al. 1989). The duration of the event on 1998 September 23 is approximately 1.5 h for OY091. The duration of the event on 1999 March 12 for is approximately 32 min. The duration of the event on 2001 January 18, for 0048−097 is approximately 65 min. According to the definition of minimum variability time-scale presented in Section 2, all three minimum variability time-scales mentioned above are credible. If one assumes that the radiation is generated close to the black hole, then the mass of the supermassive black hole, M, is given by  

(3)

for the Schwarzschild black hole (Miller et al. 1989), or  

(4)

for the Kerr black hole (Abramowicz & Nobili 1982). If ΔL is the change in luminosity within the time interval Δt, then the efficiency for conversion of accreted matter into energy, η, can be written as (Fabian & Rees 1979)  

(5)

where ΔL is in erg s⁻¹ and Δt is in seconds. Using , we find and 0.56, respectively. These values of η suggest the presence of relativistic beaming in BL Lac objects. The observed variability is normally connected to changes in the accretion disc, changes in the accretion flow into the black hole, or shocks in the relativistic jet (Lainela et al. 1999). The observed rapid variability can be explained by variations of instability in the accretion disc.

According to present thin accretion disc theory, the long-term period optical variability can be explained by the following theories: (1) the binary black hole model, which was used by Sillanpaa et al. (1998) to explain the periodic variation of intensity of BL Lac object OJ287; (2) thermal and viscous instability of accretion disc of super-massive black hole model suggested by us (Liu, Xie & Bai 1995) to explain the period of 13.6 yr for ON231.

The medium-time-scale periodic variability can explained by the helical jet model (Marscher 1996; Lainela et al. 1999). Observational and theoretical studies of the medium-time-scale variability are very few and sporadic. The observations of 3C 66A in the optical range and of Mrk 501 at TeV γ-ray energy raise another question: How many BL Lac objects would show medium-time-scale periodic variability? Therefore more observations and more detailed theoretical analysis for medium-time-scale periodic variability of BL Lac objects are necessary.

Acknowledgments

We acknowledge support from the National Natural Science Foundation of China.