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Magnetic activity properties of M-type kepler stars
New Astronomy 66 (2019) 31–39
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Magnetic activity properties of M-type kepler stars Ahmad Mehrabi a b c
⁎,a,b
, Han He
T
c
Department of Physics, Bu-Ali Sina University, Hamedan 65178, 016016, Iran School of Astronomy, Institute for Research in Fundamental Sciences (IPM), 19395-5746 Tehran, Iran CAS Key Laboratory of Solar Activity, National Astronomical Observatories, Chinese Academy of Sciences, Beijing, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Stars: activity Stars: magnetic field Stars: rotation Stars: M-type
In this paper, we study the magnetic activity properties for a sample of M-type stars in Kepler field by using the two light-curve-based magnetic proxies, iAC and Reff, which describe the periodicity and magnitude of light-curve variations, respectively. It is found that (1) stars with short rotational period might have a relatively large value of R eff (mean of Reff) while stars with long period can’t have large R eff value; (2) there is a weak trend between the rotation period and the correlation of iAC and Reff, that is, the correlation of the two magnetic proxies tends to be positive for stars with shorter period and be zero and negative for stars with longer period; and finally (3) statistically speaking, the negative correlation stars can’t have a large value of R eff as positive correlation stars. We describe the similarities and differences of magnetic activities, base on current and previous analysis, for Mtype and G-type stars. The similarity of magnetic activity behaviors between M-type stars and G-type stars implies that the two kinds of stars obey the same magnetic dynamo mechanism, though their internal structures of the convective zone may be distinctly different.
1. Introduction
addition to this, one interesting objective is investigation of magnetic activity of stars in Kepler field. The magnetic activity is believed to initiate from convective zone of solar-like stars through dynamo process (Berdyugina, 2005), and two prominent phenomena could be seen in the light curves due to magnetic activity (He et al., 2015). (1) Inhomogeneous dark or bright magnetic features (e.g., dark starspots and bright faculae) might generate a gradual fluctuation by the process of rotational modulation (Debosscher et al., 2011). Such a rotational modulation can be used to estimate the rotational period of stars (Reinhold et al., 2013; Nielsen et al., 2013; McQuillan et al., 2013; 2014). (2) The magnetic activities produce the stellar flares that could be seen as sudden spikes in the light curves of the stars (Walkowicz et al., 2011; Maehara et al., 2012; Shibayama et al., 2013; Balona, 2015; Yun et al., 2016; 2017). Moreover, in Notsu et al. (2016) the properties of superflares on solar-type stars in Kepler field have been investigated and their energy have been compared with the flares of the Sun. The results of Maehara et al. (2012) and Shibayama et al. (2013) have shown that the superflares can only be discerned from the light curves for a small part of Kepler stars, whereas the fluctuation owing to the rotational modulation is common in Kepler light curves (Notsu et al., 2013; McQuillan et al., 2014). Since the physical cause of dark starspots and bright faculae is the magnetic field of stars and rotational modulation is due to these features (Berdyugina, 2005; Reiners, 2012), fluctuation information of
The Kepler mission is specifically designed to find the earth-size and smaller planet in or near habitable zone. To do this, Kepler observe around 115 square degree of sky in optical wavelength (Koch et al., 2010) to detect very tiny transit signal due to a earth-like planet. Kepler observes the light curves of around 1.5 × 105 stars continuously with very high precision to find the earth-like planet in a region of our Milky Way galaxy and determine the fraction of stars in our galaxy that might have such planets. Many methods and approaches have been proposed to characterize the complex flux variations of stellar light curves for studying transiting planets, such as wavelet transforms as used by the Kepler team (Stumpe et al., 2014), Fourier transforms (SanchisOjeda et al., 2014), Gaussian processes regression (Gibson et al., 2012), ARMA-type modeling (Chatfield, 2003), etc. Kepler needs to observe a large number of stars to get more chance to detect such a planet, and thus Kepler targets also provide a good sample to study the properties of stars. The asteroseismology of stars is one of the possibilities that can be investigated with Kepler targets. The asteroseismology of stars measures up to a powerful tool to study properties of stars in Kepler filed (Ceillier et al., 2016) as well as the extrasolar planets (Aguirre et al., 2015). A straightforward analysis of solar-like oscillation usually provides the surface gravity and mean density of the star as well as the astrosiesmic mass, radius and age. In ⁎
Corresponding author. E-mail addresses: [email protected] (A. Mehrabi), [email protected] (H. He).
https://doi.org/10.1016/j.newast.2018.07.007 Received 22 April 2018; Received in revised form 11 July 2018; Accepted 24 July 2018 Available online 25 July 2018 1384-1076/ © 2018 Elsevier B.V. All rights reserved.
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A. Mehrabi, H. He
maximum (minimum) for a full period (half period) lag time and this pattern repeats after each period. On the contrary, for a non periodic or random time series, the autocorrelation change almost randomly with a relatively small amplitude. To see examples regarding the autocorrelation functions for the periodic and random time series, reader can refer to He et al. (2015). The degree of periodicity of a time series as proposed by He et al. (2015), could be obtained by computing the average value of |ρ(h)| over the interval 0 < h < N/2:
rotational modulation in a light curve could be used to study the magnetic properties of stars (He et al., 2015). To examine how magnetic features affect light curve of stars, one can use the Sun as a good example. Investigations by Lean et al. (1998) and Fligge et al. (2000) have shown that both the sunspots and faculae play important roles in solar light curve modulation: the bright faculae regions increase and the dark sunspot regions decrease the flux of the Sun. The magnetic field of stars is not static and varies over time. Such a time variation leads to a complexity in the shape of the light curves which is confirmed for the Sun in Lanza et al. (2003) and He et al. (2015). Some careful studies of solar magnetic field have revealed that life-time of most sunspots is shorter than rotation period of the Sun while the bright faculae are more stable because of their larger special scale (Zirin, 1985; Solanki et al., 2000). According to above discussion, the degree of periodicity and amplitude of a stellar light curve could give us some information regarding to the activities of the magnetic features that dominate the rotational modulation (He et al., 2015). In He et al. (2015), two proxies of magnetic activity were defined to measure the degree of periodicity and amplitude of light curve fluctuations quantitatively and then their correlation was studied for the Sun and two solar-type stars observed with Kepler. These procedures have been done for a sample of G-type stars with different data processing by Mehrabi et al. (2017) and their results indicate that the two magnetic proxies are positively correlated for most of the G-type stars in the sample, while by increasing rotational period, the percentage of negative correlation stars, like the Sun, get larger. In addition, their study showed that negative correlation stars cannot have a large magnitude of fluctuation in their light curves. In this work, we are going to extend our study of stellar magnetic activity properties by using the two magnetic proxies for a sample of Mtype stars in the Kepler field. M-type stars are relatively smaller than Gtype stars and have different internal structures of convective zone. We are interested in the similarities and differences of magnetic activities between M-type and G-type stars. Since the rotational period distribution of M-type stars were different from the G-type stars, we should have considered a different data processing for M-type stars. In Section (2), we give a brief description of the two light-curve-based magnetic proxies and physical information of them. Section (3) explains the data processing and the criteria for the M-type Kepler targets selection step by step. In Section (4) we present the properties of the Mtype stars in our sample as well as results of the statistical analysis of magnetic activities. Finally in Section (5) we conclude and discuss about the important aspects of our results.
iAC =
t=N −1−h
(Xt + h − X )(Xt t=N −1 ∑t = 0 (Xt − X )2
− X)
N /2
ρ (h) dh.
(2)
(
h
)
roughly given by ρ (h) = 1 − L cos(ωh) which results iAC ≈ 0.48. A simple test reveals that the quantity iAC varies between (0,0.48) and can be used as a quantitative measure to measure the degree of periodicity of a given time series. The fluctuation amplitude is generally not uniform for a given light curve and varies over the time. So we consider an effective range of fluctuation Reff to describe the magnitude of light curve variation quantitatively. The distance between the crest and trough of the light curve, as used in Basri et al. (2011, 2013), considered here as fluctuation range of a light-curve. Since working with normalized light curve is more convenient, the light-curve data is normalized before defining the measure of fluctuation amplitude. A light curve is normalized to its median value as:
∼ Xt − X ∼ , (3) X ∼ where X is the median of the light curve. It is worth to notice that both Xt and xt yield the same values of ρ(h) and iAC. The effective fluctuation range for a typical light curve is given by xt =
R eff = 2( 2 x rms),
(4)
where xrms is the rms value of xt (see García et al., 2010; Chaplin et al., 2011),
x rms =
1 N
t=N −1
∑t=0
x t2 .
(5)
Consider a perfect sine wave with equation x = A sin ωt , The effective
(
A
)
flactuation range for this time series is R eff = 2 2 2 = 2A which is the distance between crest and trough. To see an illustrative example of this case, see He et al. (2015). As we mentioned above, the fluctuations in the light curve of a single star are due to existing magnetic features over the star. The effective range of fluctuation Reff is a proxy which reveals the size or spatial coverage of the magnetic features as emphasized in works (García et al., 2010; Basri et al., 2013; He et al., 2015). On the other hand, the iAC could be used to measure the stability of magnetic features. The correlation of these quantities will show how these two proxies related to each other and different correlation value reveals different magnetic behaviors. Similar analysis has been done for a sample of G-type stars, which are relatively bigger than M-type stars, in Mehrabi et al. (2017). In current work, we extend our study and compute the correlation for a sample of M-type Kepler targets. Notice that to make the analysis suitable for M-type stars, we make some modifications in data processing which will be described in the following section.
For a time series the autocorrelation function can be used as a measurement of periodicity of the time series (Chatfield, 2003). Since the periodic behavior in a typical light curve is due to steady rotational modulation of magnetic features on the star’s surface, this quantity can be used to find the stability of magnetic features over time. So the autocorrelation algorithm could be used to measure the periodicity of a light curve and reflect activity properties of magnetic features. That is, if the magnetic features producing the rotational modulation are stable, the periodicity of the light curve get stronger, and vice versa. Such a measurement has been introduced in He et al. (2015) which is reviewed in the following. For a typical N points light-curve as {Xt , t = 0, 1, 2, 3, …, N − 1}, the autocorrelation function (ACF) is defined via (Chatfield, 2003; McQuillan et al., 2013; He et al., 2015):
∑t = 0
∫0
The iAC gives a large value for a stable periodic time series and a small value for a random time series. To clarify it consider a perfect sine curve x = A sin(ωt ) with total length L, the autocorrelation function is
2. Two magnetic proxies
ρ (h) =
2 N
,
3. M-Type stars selection and Kepler data reduction
(1)
The Kepler targets observed by the photometry instrument in longcadence (LC) mode, i.e., one data point in every 29.4 minutes
where h is the time lag and X is the mean value of the time series. For a perfect periodic time series, the above function ρ(h) has 32
New Astronomy 66 (2019) 31–39
A. Mehrabi, H. He
(Jenkins et al., 2010) are used in this work. At first the raw data is passed to the Presearch Data Conditioning (PDC) module (Smith et al., 2012; Stumpe et al., 2012) of the Kepler data processing pipeline to remove any systematic uncertainties and after this step the data is ready to study the astrophysical signal. Like our previous work, we use the PDC flux data from Kepler Data Release 25 (Thompson et al., 2016) to study the magnetic activity of the M-type stars. The data files were downloaded from the MAST (Mikulski Archive for Space Telescopes) web site.1 During the primary Kepler mission, the photometry data of stars are collected and divided into 18 quarters (Q0–Q17). Each quarter (except Q0, Q1, and Q17) contains about three months of continuous data (Haas et al., 2010). Because the Kepler telescope rolls 90° about its axis between successive quarters, the data of a same star are obtained with different CCD modules (Koch et al., 2010), which leads to the discontinuities of Kepler light curves at the end of each quarter. In our previous work for G-type stars (Mehrabi et al., 2017), we obtained the iAC and Reff for each quarter, there was a limitation on the upper period of stars we considered (Recall that we did not consider a G-type star with period longer than 30 days in our previous work). To overcome this limitation, a different approach is adopted in this work, which will be described in the following. In fact, since a relatively large number of M-type stars in our sample have long rotational period compare to the length of the quarters, we change the data processing slightly to be suitable for a long rotational period star. The step by step procedures for Kepler targets selection and data processing for M-type stars are described as follows.
Fig. 1. An example of star with non-continues fluctuation amplitude in our sample. The blue line is the original Kepler light curve; the red line is the gradual component of the light curve after low pass filter (see step 7 in Section 3). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
dropped in our sample. 6. To fulfill the above criteria, we calculate the fluctuation amplitude (Eq. (4)) of the light curves in both sides of each gaps between R quarters and select stars under condition R eff1 < 2 . In fact, in each eff2 gap the Reff is calculated for both sides and larger one (smaller one) named Reff1(Reff2) then such procedure apply for all quarters and if for all of them the above condition are satisfied, the star is marked as selected. In this procedure, the fluctuation amplitude has been calculated for 20 days at each sides of each gaps. After this step, we left with 1248 targets which satisfied all our criterion and are depicted in H-R digram in Fig. 2. Note that here we use the relation
1. The rotational period of stars for a sample of Kepler targets in the catalog of McQuillan et al. (2014) are confirmed. Since the period of these stars are due to rotational modulation which is from magnetic activities, these stars have magnetic activity over their surfaces. The catalog consist around 34,000 main sequence stars and M-type stars are selected from this catalog for current work. Notice that the catalog of McQuillan et al. (2014) excludes known eclipsing binaries and Kepler Objects of Interest, so the magnetic activities of the binaries in the Kepler field are not considered in this work. In addition, our final targets have been checked with the catalog of Kirk et al. (2016) to confirm that not be an eclipsing binary. 2. To select the M-type stars from the catalog, at first, the B − V color for the stars are obtained via the relation B − V = 0.98(g − r ) + 0.22 (Jester et al., 2005), and then, by using appendix B of Gray (2005), those stars with B − V > 1.4 are selected as M-type stars. We found 3236 stars in this step. 3. In above selected M-type stars, those being continuously monitored are suitable for our analysis. For a star which has not been monitored in some quarters, the calculated correlation between the two magnetic proxies may not be reliable. So we select those stars which have data in all quarters from Q2 to Q16. Note that Q0, Q1, and Q17 are not full length (three months) quarters and the data in these three quarters are not adopted in this work. After this filter, we are left with 2469 stars. 4. The gaps in the light curves are filled with a linear interpolation algorithm and each quarter normalize to its own median then all quarters are sticked together to make a long and continues light curve. 5. After above step, we found that for some targets the fluctuation amplitude in two subsequence quarters are not continuous. An example of such stars is shown in Fig. 1. As it is clear from these light curves the fluctuation amplitude in first quarter is relatively larger than two subsequence quarters. We think that such thing is likely due to systematic uncertainty and these stars should be
( ) T
7.
8.
9.
10. 1
https://archive.stsci.edu/kepler/publiclightcurves.html. 33
4
L/ L⊙ = R2 Teff (Brown et al., 2011) to find the luminosity of the ⊙ stars. It is known that the Kepler light-curves consist of noises, outliers, flare spikes, and granulation driven flickers (Cranmer et al., 2014; Kallinger et al., 2014), so have potential to disturb the activity analysis via (iAC, Reff) pairs, they are filtered out using a low pass Sinc filter. After performing such low pass filter for a light curve, the gradual component which is suitable for our analysis remains (see Fig. 1 for an example; notice that the flares and instrumental outliers in the light curves are effectively filtered out). However the upper cutoff frequency of the filter has to be determined empirically. In this work, we adopt the value of 1 for the cutoff fre0.2P quency, where P is the rotation period of stars. (For more details, see our previous paper (Mehrabi et al., 2017).) Now to calculate the magnetic proxies, each light curve is divided to some segments in which each segment contain 10 rotational periods. After this step, the fluctuation amplitude (Reff) and stability measure (iAC) for each segment are calculated. Note that in this case we select 10 times of period to divide the whole light curve because this is close to the analysis for the Sun by He et al. (2015) in which the whole solar light curve is divided into one year segments and each segment covers more than 10 times of periods. In fact, in each segment, 10 times of rotational period of light curve provide sufficient fluctuation to calculate the two magnetic proxies. As we mentioned in our previous work, the gaps in light curves might make uncertainty in our results. So within each segment if there is a gap longer than 2 times of rotational period, the value of two proxies are set to null for the segment, and if null values repeat in 5 segments consecutively, the target is dropped out from the sample to avoid possible wrong results. Finally, we calculate the correlation of the two magnetic proxies for all stars in our sample. A discussion about uncertainty in quantities has been given by Mehrabi et al. (2017).
New Astronomy 66 (2019) 31–39
A. Mehrabi, H. He
Fig. 2. H-R diagram of the M-type stars selected from the catalog of McQuillan et al. (2014).
Fig. 4. The R eff versus period for stars in our sample.
Fig. 3. Distribution of mass and radius as a function of period for our sample of M-type stars.
of fluctuation amplitude over all segments of light curves for each star. In Fig. 4, the mean fluctuation amplitude versus rotation period is shown. This result shows that stars with short rotational period might have a relatively large value of fluctuation amplitude and hence larger spatial coverage of magnetic features. In contrast, stars with long rotational period, like the Sun, have tiny fluctuation amplitude. Actually, such behavior shows how different rotational periods affect magnetic activities of M-type stars. In this analysis, it is found that the star KIC=10063343 with period P=0.33 days has the maximum value of R eff among all stars. For this star, we found R eff = 0.1288 which indicates a relatively large coverage of magnetic features over its surface. If assuming the observed R eff value of a star is generated by a single circular dark spot on the stellar surface, then the area of the spot (relative to the stellar disk area) would be no
4. Results 4.1. Stellar parameters of the selected M-type stars Before study the evolution of two magnetic proxies calculated above, it is worth to show the distribution of mass and radius as a function of period for selected stars. The rotation period and mass of the stars are taken from the catalog of McQuillan et al. (2014) while the radius of stars are obtained from the Kepler Input Catalog (KIC) (Brown et al., 2011). The distributions of mass and radius of stars in our sample are shown in Fig. 3. It can be seen from Fig. 3 that the rotation period values of the selected stars are up to 50 days, so we have enough stars with small and large period to study the statistical properties. Since one of important factor in magnetic activity of stars is their rotational period and our sample contains stars in period range of 0.3–50 days, our results open a window to study the magnetic properties of M-type stars. In our previous work, for G-type stars we were not allowed to consider stars with period longer than 30 days due to statistical limitation, but in this work, we adopt a different approach to analyze the light curves as described in Section 3, and so we are able to consider longer periods. As it is expected, stars in our sample are smaller and less massive than the Sun. These stars have different internal structure rather than G-type stars and a comparison of magnetic activity results for M-type and G-type stars might be useful for stellar magnetic dynamo theories.
(
less than R eff 1 −
Fs (ν ) F0 (ν )
), where F (ν) and F (ν) indicate flux of spot and s
0
photosphere at a specific frequency. Assuming black-body radiation, the above factor can be given in terms of photosphere and spot temperatures T0 and Ts as
Fs (ν ) e hν / kT0 − 1 = hν / kT ≃ e hνΔT /(kT0 Ts). s − 1 F0 (ν ) e This factor is very small for G-type stars due to a large temperature contrast but for M-type stars ΔT ≃ 500K (Berdyugina, 2005), so this factor is not negligible at visual wavelengths.2 For the star KIC=10063343 (T0 = 4045K ) the radius of the spot (relative to the stellar radius) would be no less than 0.1288(1 − 0.3) ≈ 0.3. Note that for the Sun the quantity R eff ≃ 0.0005 which is smaller by a factor of 200 than the star we found in our sample.
4.2. Statistical properties of the magnetic proxies As we mentioned above, the fluctuation amplitude (Reff) provides information regarding the size or spatial coverage of magnetic features. Before presenting the results of correlation analysis between the two magnetic proxies in the following subsection, we investigate relation between the rotation periods and mean fluctuation amplitude of stars in our sample. The mean value of fluctuation amplitude R eff , is the mean
2
Assuming T0 = 4250, Ts = 3750, λ = 620nm we have
T0 = 4250, Ts = 3750, λ = 420nm we have 34
Fs (ν ) F0 (ν )
= 0.34
Fs (ν ) F0 (ν )
= 0.48 and for
New Astronomy 66 (2019) 31–39
A. Mehrabi, H. He
Fig. 5. The mean value of iAC as a function of fluctuation amplitude and period for all stars in our sample.
The star is peculiar in some sense and further investigation of its light curve might reveal extra information regarding magnetic activity and time evolution. In addition to above discussion, the statistical properties of iAC might be interesting. We measure the mean value of iAC over all quarters for each star and present it (iAC ) as a function of period and mean fluctuation amplitude in Fig. 5. The left panel shows the mean value of iAC versus period for each star. Long period stars not only have small fluctuation amplitudes (see Fig. 4) but also a small value of iAC which indicates a short life-time for small spots. On the other hand, for a fast rotating star, the fluctuation amplitude is relatively large as well as their life-times. The right panel of Fig. 5 present the mean value of iAC versus mean value of Reff for all stars in our sample and clearly confirms above discussion about the size and life-time of fluctuations on M-type stars.
Table 1 The correlation (Cor), mean values of iAC, mean value of Reff, period, mass, radius, and effective temperature of the stars presented in this work. This table is available in its entirety in a machine-readable form. KIC
Cor
iAC
R eff
P (days)
M (M⊙)
R (R⊙)
Teff (K)
892376 1162635 1569863 1570915 1572802 1577265
−0.03908 0.85164 0.47998 0.77010 0.52370 0.63419
0.27168 0.26048 0.32640 0.29389 0.32012 0.28707
0.00850 0.00638 0.02158 0.00831 0.05022 0.00791
1.532 15.678 13.321 12.315 0.374 15.153
0.4699 0.4497 0.3862 0.5646 0.495 0.5028
0.605 0.494 0.36 0.583 0.535 0.56
3973 3759 3591 4161 3878 3903
4.3. Correlation analysis for the two magnetic proxies Number of (iAC, Reff) pair for each star depend on the rotational period of the star. Dependency of these two proxies on each other could be quantified by correlation (denoted by Cor) of them. The correlation can be positive, zero, or negative, and might give us useful information about the magnetic activities of the stars and their evolutions (He et al., 2015; Mehrabi et al., 2017). iAC is a proxy that measure the stability (relative to the time scale of rotation period) of the magnetic features, and Reff shows the spatial coverage of the features and hence intensity level of magnetic activities. The evolution of these proxies in 3.8 years which is divided into 15 quarters (Q2-Q16), could be used to extract information about how the size and lifetime of the magnetic features change over the time (He et al., 2015; Mehrabi et al., 2017). However, a large value of iAC might be happened in a non-eclipsing (close) binaries. To avoid this possibility, we check 10 targets with most large value of iAC and did not see any regular pattern during all quarters. The correlation between the two quantities is given by
Fig. 6. Distribution of correlation between iAC and Reff.
presented. Our analysis indicates that most stars in our sample are positive correlated and only a few of them are negative correlated. In our analysis it is found that 9.5%, 5.8%, and 84.6% of the stars are in negative, zero, and positive correlation, respectively. So for most stars in our sample, the time variations of the two magnetics proxies are in same phase, that is, when the spatial coverage of the magnetic features that dominate the rotational modulation increases, the stability of the magnetic features also increase, and vice versa. It is worthwhile to mention that similar results have been found by several groups about size-life correlation in the sunspots (Petrovay and van Driel-Gesztelyi, 1997; Henwood et al., 2010). These groups have studied the size and decay rate of sunspots through different data and their results indicate positive correlation (similar to most M-type stars in our sample) between the size and lifetime of sunspots. Moreover, such analysis for a sample of G-type stars gives the results of 4.3%, 6.8% and 88.9% (Mehrabi et al., 2017), which are very close to results for the M-type stars. Statistically speaking, the above results show that the magnetic activity in both groups, G-type and M-type stars, act in a same way. In Fig. 7 the scatter plot of correlation (Cor) versus R eff (left panel) as well as correlation versus period (right panel) for all the stars in our
s=k
Cor =
s s − iAC)(Reff − R eff ) ∑s = 1 (iAC s=k
s=k
s s − iAC)2][ ∑s = 1 (Reff − R eff )2] [ ∑s = 1 (iAC
, (6)
where iAC and R eff indicate the mean values of iAC and Reff for all the 10period segments (see Section (3)) of a star, s is the serial number for the 10-period segments (from 1 to k), and k is the total number of the 10period segments. The correlation between two proxies are calculated for all the selected stars in our sample. A small part of our results is presented in Table 1 and the entire information of this table, in a machine-readable form, are available in the online journal. The correlation coefficient (Eq. (6)) are in the range (−1, 1) with Cor = −1 fully anti-correlated and Cor = 1 fully-correlated. Similar to our previous work, the correlation values are divided into 3 parts as − 1 < Cor < −0.1 (negative correlation), − 0.1 < Cor < 0.1 (zero correlation), and 0.1 < Cor < 1 (positive correlation). In Fig. 6, the distribution of correlation between iAC and Reff for all the selected stars is
35
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Fig. 7. Correlation (Cor) versus R eff (left panel) and versus period (right panel) for all the stars in the sample.
might be useful for stellar magnetic dynamo theory. From our results, it is found that: (1) Stars with short rotational period might have a relatively large value of fluctuation amplitude (quantified by R eff ) while stars with long period can’t have a large R eff value. This result indicate that the magnetic features on long period stars cover a relatively smaller spatial area than short period stars. (2) There is a weak trend between the rotation period and the correlation of iAC and Reff, that is, the correlation of the two magnetic proxies tends to be positive for stars with shorter period and be zero and negative for stars with longer period. In fact, almost all short period stars are positively correlated while for longer period stars the correlation has a more scattered distribution. We have found a few short period G-type stars with negative correlation in our previous work but for M-type stars there are almost no stars in the period interval of 0–10 days with negative correlation. Statistically speaking, the distribution of M-type stars in correlation versus period plane is similar as the G-type stars. (3) Statistically speaking, the negative correlation stars can’t have a large value of R eff as positive correlation stars. This means that the spatial coverage of magnetic features that dominate the rotational modulation can’t be large for a negative correlation star. Note that the Sun is a negative correlated star with very tiny fluctuation amplitude. In our sample we found a peculiar target (KIC 10063343) with both large R eff and negative correlation. We postpone a further analysis of its light curve to our future work. The similarity of magnetic activity behaviors between M-type stars and G-type stars implies that the two kinds of stars obey the same magnetic dynamo mechanism, though their internal structures of convective zone may be distinctly different. This conclusion is consistent with the result obtained by Wright and Drake (2016) using the X-ray emission as the proxy of stellar magnetic activity.
sample is depicted. The result shows that small values of R eff could happen for the negative, zero, and positive correlation stars, but statistically speaking, the negative correlation stars can’t have large R eff value. Note that we have detected the same character for G-type star (Mehrabi et al., 2017) and specifically the Sun is a negative correlation star with R eff = 0.00047 (He et al., 2015). In addition, it is found that the order of magnitude of R eff for positive, zero and negative correlation stars is the same as G-type stars. So our results suggest that the spatial coverage ratios (relative to the stellar disk area) of magnetic features over G-type and M-type stars, for each correlation group, may be roughly the same. One interesting point from our results is that, statistically speaking, fast rotators of M-type stars can not have negative correlation (see right panel of Fig. 7). There are almost no targets with negative correlation in the period interval of 0–10 days. In contrast, a few G-type stars have been found in this range with negative correlation (Mehrabi et al., 2017). On the other hand, for M-type stars with long period, we can find a relatively scattered distribution across different correlation groups, which is almost the same for G-type stars. Moreover, our results also demonstrate a weak trend between the rotation period and the correlation of iAC and Reff. That is, the correlation tends to be positive for stars with shorter period, and be zero and negative for stars with longer period. One possibility is that the correlation properties of stars depend on the their ages. Such a possibility will be considered in our future work. The similarity of magnetic activity between M-type and G-type stars have been reported in Wright and Drake (2016) using a completely different approach which supports our results. 5. Conclusion
Acknowledgment
In this paper, we study the magnetic activity properties for a sample of M-type stars in Kepler field by using the two light-curve-based magnetic proxies, iAC and Reff. The former indicates the stability of magnetic features and the latter reflects the spatial coverage of magnetic features. In our previous work, we analyze these quantities for a sample of G-type stars. In this work, we investigate how these magnetic proxies depend on the rotational period of stars and the correlation between them for a sample of M-type Kepler targets. Our results show similarities and differences between M-type and G-type stars which
This paper includes data collected by the Kepler mission. The data used in this paper were downloaded from the Mikulski Archive for Space Telescopes (MAST). Funding for the Kepler Mission is provided by the NASA Science Mission Directorate. The authors of this paper gratefully acknowledge the entire Kepler team and all people who contribute to the Kepler mission, which provides us the opportunity to conduct this study.
Appendix In this part, we introduce 10 targets with a large value of iAC in our sample. These targets provide a good sample to study the properties of magnetic activities over M-type stars. Further analysis over these targets might reveal more information regarding the size and location of spots and faculae over these stars. In this part we only introduce these stars and are not going to study properties of them. These targets are presented in Table 2 . We also plot the light curves of these targets and show two of them in Fig. 8. The light curves of all targets are provided as online material.
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Table 2 The KIC and other properties of 10 targets with maximum values of iAC. KIC
Cor
iAC
R eff
P (days)
M (M⊙)
R (R⊙)
Teff (K)
6370174 8572066 9473985 6871896 9032388 5954552 7350067 2581444 9472906 3439126
0.42153 0.60117 0.08841 0.82403 0.902493 0.48006 0.08887 0.68210 0.90317 −0.03430
0.40857 0.40772 0.40770 0.39265 0.391517 0.39099 0.38661 0.38656 0.38275 0.380914
0.03520 0.03702 0.016873 0.023255 0.02034 0.019946 0.038538 0.01459 0.02362 0.05238
14.99 9.339 20.931 14.243 19.825 14.922 3.514 11.976 33.085 7.9
0.314 0.513 0.388 0.530 0.583 0.497 0.261 0.580 0.408 0.577
0.327 0.55 0.453 0.625 0.646 0.524 0.2 0.6 0.451 0.664
3386 3928 3595 4053 4297 3887 3227 4220 3659 4236
Fig. 8. The light curves for two M-type stars with a large value of iAC. In both figures, top left panel is for quarter 2 and last panel is for quarter 16. 37
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Supplementary material Supplementary material associated with this article can be found, in the online version, at 10.1016/j.newast.2018.07.007 .
oscillation as seen by the kepler mission. AAP 570, A41. https://doi.org/10.1051/ 0004-6361/201424313. arXiv:1408.0817. Kirk, B., Conroy, K., Prša, A., Abdul-Masih, M., Kochoska, A., Matijevič, G., Hambleton, K., Barclay, T., Bloemen, S., Boyajian, T., Doyle, L.R., Fulton, B.J., Hoekstra, A.J., Jek, K., Kane, S.R., Kostov, V., Latham, D., Mazeh, T., Orosz, J.A., Pepper, J., Quarles, B., Ragozzine, D., Shporer, A., Southworth, J., Stassun, K., Thompson, S.E., Welsh, W.F., Agol, E., Derekas, A., Devor, J., Fischer, D., Green, G., Gropp, J., Jacobs, T., Johnston, C., LaCourse, D.M., Saetre, K., Schwengeler, H., Toczyski, J., Werner, G., Garrett, M., Gore, J., Martinez, A.O., Spitzer, I., Stevick, J., Thomadis, P.C., Vrijmoet, E.H., Yenawine, M., Batalha, N., Borucki, W., 2016. Kepler eclipsing binary stars. VII. the catalog of eclipsing binaries found in the entire kepler data set. AJ 151, 68. https:// doi.org/10.3847/0004-6256/151/3/68. arXiv:1512.08830. 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Nielsen, M.B., Gizon, L., Schunker, H., Karoff, C., 2013. Rotation periods of 12 000 mainsequence kepler stars: dependence on stellar spectral type and comparison with v sin i observations. AAP 557, L10. https://doi.org/10.1051/0004-6361/201321912. arXiv:1305.5721. Notsu, Y., Maehara, H., Shibayama, T., Honda, S., Notsu, S., Namekata, K., Nogami, D., Shibata, K., 2016. Statistical properties of superflares on solar-type stars with kepler data. Proceedings of the 19th Cambridge Workshop on Cool Stars, Stellar Systems, and the Sun. CS19. pp. 119. https://doi.org/10.5281/zenodo.59138. arXiv:1608. 00186. Notsu, Y., Shibayama, T., Maehara, H., Notsu, S., Nagao, T., Honda, S., Ishii, T.T., Nogami, D., Shibata, K., 2013. Superflares on solar-type stars observed with kepler II. photometric variability of superflare-generating stars: a signature of stellar rotation and starspots. APJ 771, 127. https://doi.org/10.1088/0004-637X/771/2/127. arXiv:1304.7361. 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New Astronomy journal homepage: www.elsevier.com/locate/newast
Magnetic activity properties of M-type kepler stars Ahmad Mehrabi a b c
⁎,a,b
, Han He
T
c
Department of Physics, Bu-Ali Sina University, Hamedan 65178, 016016, Iran School of Astronomy, Institute for Research in Fundamental Sciences (IPM), 19395-5746 Tehran, Iran CAS Key Laboratory of Solar Activity, National Astronomical Observatories, Chinese Academy of Sciences, Beijing, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Stars: activity Stars: magnetic field Stars: rotation Stars: M-type
In this paper, we study the magnetic activity properties for a sample of M-type stars in Kepler field by using the two light-curve-based magnetic proxies, iAC and Reff, which describe the periodicity and magnitude of light-curve variations, respectively. It is found that (1) stars with short rotational period might have a relatively large value of R eff (mean of Reff) while stars with long period can’t have large R eff value; (2) there is a weak trend between the rotation period and the correlation of iAC and Reff, that is, the correlation of the two magnetic proxies tends to be positive for stars with shorter period and be zero and negative for stars with longer period; and finally (3) statistically speaking, the negative correlation stars can’t have a large value of R eff as positive correlation stars. We describe the similarities and differences of magnetic activities, base on current and previous analysis, for Mtype and G-type stars. The similarity of magnetic activity behaviors between M-type stars and G-type stars implies that the two kinds of stars obey the same magnetic dynamo mechanism, though their internal structures of the convective zone may be distinctly different.
1. Introduction
addition to this, one interesting objective is investigation of magnetic activity of stars in Kepler field. The magnetic activity is believed to initiate from convective zone of solar-like stars through dynamo process (Berdyugina, 2005), and two prominent phenomena could be seen in the light curves due to magnetic activity (He et al., 2015). (1) Inhomogeneous dark or bright magnetic features (e.g., dark starspots and bright faculae) might generate a gradual fluctuation by the process of rotational modulation (Debosscher et al., 2011). Such a rotational modulation can be used to estimate the rotational period of stars (Reinhold et al., 2013; Nielsen et al., 2013; McQuillan et al., 2013; 2014). (2) The magnetic activities produce the stellar flares that could be seen as sudden spikes in the light curves of the stars (Walkowicz et al., 2011; Maehara et al., 2012; Shibayama et al., 2013; Balona, 2015; Yun et al., 2016; 2017). Moreover, in Notsu et al. (2016) the properties of superflares on solar-type stars in Kepler field have been investigated and their energy have been compared with the flares of the Sun. The results of Maehara et al. (2012) and Shibayama et al. (2013) have shown that the superflares can only be discerned from the light curves for a small part of Kepler stars, whereas the fluctuation owing to the rotational modulation is common in Kepler light curves (Notsu et al., 2013; McQuillan et al., 2014). Since the physical cause of dark starspots and bright faculae is the magnetic field of stars and rotational modulation is due to these features (Berdyugina, 2005; Reiners, 2012), fluctuation information of
The Kepler mission is specifically designed to find the earth-size and smaller planet in or near habitable zone. To do this, Kepler observe around 115 square degree of sky in optical wavelength (Koch et al., 2010) to detect very tiny transit signal due to a earth-like planet. Kepler observes the light curves of around 1.5 × 105 stars continuously with very high precision to find the earth-like planet in a region of our Milky Way galaxy and determine the fraction of stars in our galaxy that might have such planets. Many methods and approaches have been proposed to characterize the complex flux variations of stellar light curves for studying transiting planets, such as wavelet transforms as used by the Kepler team (Stumpe et al., 2014), Fourier transforms (SanchisOjeda et al., 2014), Gaussian processes regression (Gibson et al., 2012), ARMA-type modeling (Chatfield, 2003), etc. Kepler needs to observe a large number of stars to get more chance to detect such a planet, and thus Kepler targets also provide a good sample to study the properties of stars. The asteroseismology of stars is one of the possibilities that can be investigated with Kepler targets. The asteroseismology of stars measures up to a powerful tool to study properties of stars in Kepler filed (Ceillier et al., 2016) as well as the extrasolar planets (Aguirre et al., 2015). A straightforward analysis of solar-like oscillation usually provides the surface gravity and mean density of the star as well as the astrosiesmic mass, radius and age. In ⁎
Corresponding author. E-mail addresses: [email protected] (A. Mehrabi), [email protected] (H. He).
https://doi.org/10.1016/j.newast.2018.07.007 Received 22 April 2018; Received in revised form 11 July 2018; Accepted 24 July 2018 Available online 25 July 2018 1384-1076/ © 2018 Elsevier B.V. All rights reserved.
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maximum (minimum) for a full period (half period) lag time and this pattern repeats after each period. On the contrary, for a non periodic or random time series, the autocorrelation change almost randomly with a relatively small amplitude. To see examples regarding the autocorrelation functions for the periodic and random time series, reader can refer to He et al. (2015). The degree of periodicity of a time series as proposed by He et al. (2015), could be obtained by computing the average value of |ρ(h)| over the interval 0 < h < N/2:
rotational modulation in a light curve could be used to study the magnetic properties of stars (He et al., 2015). To examine how magnetic features affect light curve of stars, one can use the Sun as a good example. Investigations by Lean et al. (1998) and Fligge et al. (2000) have shown that both the sunspots and faculae play important roles in solar light curve modulation: the bright faculae regions increase and the dark sunspot regions decrease the flux of the Sun. The magnetic field of stars is not static and varies over time. Such a time variation leads to a complexity in the shape of the light curves which is confirmed for the Sun in Lanza et al. (2003) and He et al. (2015). Some careful studies of solar magnetic field have revealed that life-time of most sunspots is shorter than rotation period of the Sun while the bright faculae are more stable because of their larger special scale (Zirin, 1985; Solanki et al., 2000). According to above discussion, the degree of periodicity and amplitude of a stellar light curve could give us some information regarding to the activities of the magnetic features that dominate the rotational modulation (He et al., 2015). In He et al. (2015), two proxies of magnetic activity were defined to measure the degree of periodicity and amplitude of light curve fluctuations quantitatively and then their correlation was studied for the Sun and two solar-type stars observed with Kepler. These procedures have been done for a sample of G-type stars with different data processing by Mehrabi et al. (2017) and their results indicate that the two magnetic proxies are positively correlated for most of the G-type stars in the sample, while by increasing rotational period, the percentage of negative correlation stars, like the Sun, get larger. In addition, their study showed that negative correlation stars cannot have a large magnitude of fluctuation in their light curves. In this work, we are going to extend our study of stellar magnetic activity properties by using the two magnetic proxies for a sample of Mtype stars in the Kepler field. M-type stars are relatively smaller than Gtype stars and have different internal structures of convective zone. We are interested in the similarities and differences of magnetic activities between M-type and G-type stars. Since the rotational period distribution of M-type stars were different from the G-type stars, we should have considered a different data processing for M-type stars. In Section (2), we give a brief description of the two light-curve-based magnetic proxies and physical information of them. Section (3) explains the data processing and the criteria for the M-type Kepler targets selection step by step. In Section (4) we present the properties of the Mtype stars in our sample as well as results of the statistical analysis of magnetic activities. Finally in Section (5) we conclude and discuss about the important aspects of our results.
iAC =
t=N −1−h
(Xt + h − X )(Xt t=N −1 ∑t = 0 (Xt − X )2
− X)
N /2
ρ (h) dh.
(2)
(
h
)
roughly given by ρ (h) = 1 − L cos(ωh) which results iAC ≈ 0.48. A simple test reveals that the quantity iAC varies between (0,0.48) and can be used as a quantitative measure to measure the degree of periodicity of a given time series. The fluctuation amplitude is generally not uniform for a given light curve and varies over the time. So we consider an effective range of fluctuation Reff to describe the magnitude of light curve variation quantitatively. The distance between the crest and trough of the light curve, as used in Basri et al. (2011, 2013), considered here as fluctuation range of a light-curve. Since working with normalized light curve is more convenient, the light-curve data is normalized before defining the measure of fluctuation amplitude. A light curve is normalized to its median value as:
∼ Xt − X ∼ , (3) X ∼ where X is the median of the light curve. It is worth to notice that both Xt and xt yield the same values of ρ(h) and iAC. The effective fluctuation range for a typical light curve is given by xt =
R eff = 2( 2 x rms),
(4)
where xrms is the rms value of xt (see García et al., 2010; Chaplin et al., 2011),
x rms =
1 N
t=N −1
∑t=0
x t2 .
(5)
Consider a perfect sine wave with equation x = A sin ωt , The effective
(
A
)
flactuation range for this time series is R eff = 2 2 2 = 2A which is the distance between crest and trough. To see an illustrative example of this case, see He et al. (2015). As we mentioned above, the fluctuations in the light curve of a single star are due to existing magnetic features over the star. The effective range of fluctuation Reff is a proxy which reveals the size or spatial coverage of the magnetic features as emphasized in works (García et al., 2010; Basri et al., 2013; He et al., 2015). On the other hand, the iAC could be used to measure the stability of magnetic features. The correlation of these quantities will show how these two proxies related to each other and different correlation value reveals different magnetic behaviors. Similar analysis has been done for a sample of G-type stars, which are relatively bigger than M-type stars, in Mehrabi et al. (2017). In current work, we extend our study and compute the correlation for a sample of M-type Kepler targets. Notice that to make the analysis suitable for M-type stars, we make some modifications in data processing which will be described in the following section.
For a time series the autocorrelation function can be used as a measurement of periodicity of the time series (Chatfield, 2003). Since the periodic behavior in a typical light curve is due to steady rotational modulation of magnetic features on the star’s surface, this quantity can be used to find the stability of magnetic features over time. So the autocorrelation algorithm could be used to measure the periodicity of a light curve and reflect activity properties of magnetic features. That is, if the magnetic features producing the rotational modulation are stable, the periodicity of the light curve get stronger, and vice versa. Such a measurement has been introduced in He et al. (2015) which is reviewed in the following. For a typical N points light-curve as {Xt , t = 0, 1, 2, 3, …, N − 1}, the autocorrelation function (ACF) is defined via (Chatfield, 2003; McQuillan et al., 2013; He et al., 2015):
∑t = 0
∫0
The iAC gives a large value for a stable periodic time series and a small value for a random time series. To clarify it consider a perfect sine curve x = A sin(ωt ) with total length L, the autocorrelation function is
2. Two magnetic proxies
ρ (h) =
2 N
,
3. M-Type stars selection and Kepler data reduction
(1)
The Kepler targets observed by the photometry instrument in longcadence (LC) mode, i.e., one data point in every 29.4 minutes
where h is the time lag and X is the mean value of the time series. For a perfect periodic time series, the above function ρ(h) has 32
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(Jenkins et al., 2010) are used in this work. At first the raw data is passed to the Presearch Data Conditioning (PDC) module (Smith et al., 2012; Stumpe et al., 2012) of the Kepler data processing pipeline to remove any systematic uncertainties and after this step the data is ready to study the astrophysical signal. Like our previous work, we use the PDC flux data from Kepler Data Release 25 (Thompson et al., 2016) to study the magnetic activity of the M-type stars. The data files were downloaded from the MAST (Mikulski Archive for Space Telescopes) web site.1 During the primary Kepler mission, the photometry data of stars are collected and divided into 18 quarters (Q0–Q17). Each quarter (except Q0, Q1, and Q17) contains about three months of continuous data (Haas et al., 2010). Because the Kepler telescope rolls 90° about its axis between successive quarters, the data of a same star are obtained with different CCD modules (Koch et al., 2010), which leads to the discontinuities of Kepler light curves at the end of each quarter. In our previous work for G-type stars (Mehrabi et al., 2017), we obtained the iAC and Reff for each quarter, there was a limitation on the upper period of stars we considered (Recall that we did not consider a G-type star with period longer than 30 days in our previous work). To overcome this limitation, a different approach is adopted in this work, which will be described in the following. In fact, since a relatively large number of M-type stars in our sample have long rotational period compare to the length of the quarters, we change the data processing slightly to be suitable for a long rotational period star. The step by step procedures for Kepler targets selection and data processing for M-type stars are described as follows.
Fig. 1. An example of star with non-continues fluctuation amplitude in our sample. The blue line is the original Kepler light curve; the red line is the gradual component of the light curve after low pass filter (see step 7 in Section 3). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
dropped in our sample. 6. To fulfill the above criteria, we calculate the fluctuation amplitude (Eq. (4)) of the light curves in both sides of each gaps between R quarters and select stars under condition R eff1 < 2 . In fact, in each eff2 gap the Reff is calculated for both sides and larger one (smaller one) named Reff1(Reff2) then such procedure apply for all quarters and if for all of them the above condition are satisfied, the star is marked as selected. In this procedure, the fluctuation amplitude has been calculated for 20 days at each sides of each gaps. After this step, we left with 1248 targets which satisfied all our criterion and are depicted in H-R digram in Fig. 2. Note that here we use the relation
1. The rotational period of stars for a sample of Kepler targets in the catalog of McQuillan et al. (2014) are confirmed. Since the period of these stars are due to rotational modulation which is from magnetic activities, these stars have magnetic activity over their surfaces. The catalog consist around 34,000 main sequence stars and M-type stars are selected from this catalog for current work. Notice that the catalog of McQuillan et al. (2014) excludes known eclipsing binaries and Kepler Objects of Interest, so the magnetic activities of the binaries in the Kepler field are not considered in this work. In addition, our final targets have been checked with the catalog of Kirk et al. (2016) to confirm that not be an eclipsing binary. 2. To select the M-type stars from the catalog, at first, the B − V color for the stars are obtained via the relation B − V = 0.98(g − r ) + 0.22 (Jester et al., 2005), and then, by using appendix B of Gray (2005), those stars with B − V > 1.4 are selected as M-type stars. We found 3236 stars in this step. 3. In above selected M-type stars, those being continuously monitored are suitable for our analysis. For a star which has not been monitored in some quarters, the calculated correlation between the two magnetic proxies may not be reliable. So we select those stars which have data in all quarters from Q2 to Q16. Note that Q0, Q1, and Q17 are not full length (three months) quarters and the data in these three quarters are not adopted in this work. After this filter, we are left with 2469 stars. 4. The gaps in the light curves are filled with a linear interpolation algorithm and each quarter normalize to its own median then all quarters are sticked together to make a long and continues light curve. 5. After above step, we found that for some targets the fluctuation amplitude in two subsequence quarters are not continuous. An example of such stars is shown in Fig. 1. As it is clear from these light curves the fluctuation amplitude in first quarter is relatively larger than two subsequence quarters. We think that such thing is likely due to systematic uncertainty and these stars should be
( ) T
7.
8.
9.
10. 1
https://archive.stsci.edu/kepler/publiclightcurves.html. 33
4
L/ L⊙ = R2 Teff (Brown et al., 2011) to find the luminosity of the ⊙ stars. It is known that the Kepler light-curves consist of noises, outliers, flare spikes, and granulation driven flickers (Cranmer et al., 2014; Kallinger et al., 2014), so have potential to disturb the activity analysis via (iAC, Reff) pairs, they are filtered out using a low pass Sinc filter. After performing such low pass filter for a light curve, the gradual component which is suitable for our analysis remains (see Fig. 1 for an example; notice that the flares and instrumental outliers in the light curves are effectively filtered out). However the upper cutoff frequency of the filter has to be determined empirically. In this work, we adopt the value of 1 for the cutoff fre0.2P quency, where P is the rotation period of stars. (For more details, see our previous paper (Mehrabi et al., 2017).) Now to calculate the magnetic proxies, each light curve is divided to some segments in which each segment contain 10 rotational periods. After this step, the fluctuation amplitude (Reff) and stability measure (iAC) for each segment are calculated. Note that in this case we select 10 times of period to divide the whole light curve because this is close to the analysis for the Sun by He et al. (2015) in which the whole solar light curve is divided into one year segments and each segment covers more than 10 times of periods. In fact, in each segment, 10 times of rotational period of light curve provide sufficient fluctuation to calculate the two magnetic proxies. As we mentioned in our previous work, the gaps in light curves might make uncertainty in our results. So within each segment if there is a gap longer than 2 times of rotational period, the value of two proxies are set to null for the segment, and if null values repeat in 5 segments consecutively, the target is dropped out from the sample to avoid possible wrong results. Finally, we calculate the correlation of the two magnetic proxies for all stars in our sample. A discussion about uncertainty in quantities has been given by Mehrabi et al. (2017).
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Fig. 2. H-R diagram of the M-type stars selected from the catalog of McQuillan et al. (2014).
Fig. 4. The R eff versus period for stars in our sample.
Fig. 3. Distribution of mass and radius as a function of period for our sample of M-type stars.
of fluctuation amplitude over all segments of light curves for each star. In Fig. 4, the mean fluctuation amplitude versus rotation period is shown. This result shows that stars with short rotational period might have a relatively large value of fluctuation amplitude and hence larger spatial coverage of magnetic features. In contrast, stars with long rotational period, like the Sun, have tiny fluctuation amplitude. Actually, such behavior shows how different rotational periods affect magnetic activities of M-type stars. In this analysis, it is found that the star KIC=10063343 with period P=0.33 days has the maximum value of R eff among all stars. For this star, we found R eff = 0.1288 which indicates a relatively large coverage of magnetic features over its surface. If assuming the observed R eff value of a star is generated by a single circular dark spot on the stellar surface, then the area of the spot (relative to the stellar disk area) would be no
4. Results 4.1. Stellar parameters of the selected M-type stars Before study the evolution of two magnetic proxies calculated above, it is worth to show the distribution of mass and radius as a function of period for selected stars. The rotation period and mass of the stars are taken from the catalog of McQuillan et al. (2014) while the radius of stars are obtained from the Kepler Input Catalog (KIC) (Brown et al., 2011). The distributions of mass and radius of stars in our sample are shown in Fig. 3. It can be seen from Fig. 3 that the rotation period values of the selected stars are up to 50 days, so we have enough stars with small and large period to study the statistical properties. Since one of important factor in magnetic activity of stars is their rotational period and our sample contains stars in period range of 0.3–50 days, our results open a window to study the magnetic properties of M-type stars. In our previous work, for G-type stars we were not allowed to consider stars with period longer than 30 days due to statistical limitation, but in this work, we adopt a different approach to analyze the light curves as described in Section 3, and so we are able to consider longer periods. As it is expected, stars in our sample are smaller and less massive than the Sun. These stars have different internal structure rather than G-type stars and a comparison of magnetic activity results for M-type and G-type stars might be useful for stellar magnetic dynamo theories.
(
less than R eff 1 −
Fs (ν ) F0 (ν )
), where F (ν) and F (ν) indicate flux of spot and s
0
photosphere at a specific frequency. Assuming black-body radiation, the above factor can be given in terms of photosphere and spot temperatures T0 and Ts as
Fs (ν ) e hν / kT0 − 1 = hν / kT ≃ e hνΔT /(kT0 Ts). s − 1 F0 (ν ) e This factor is very small for G-type stars due to a large temperature contrast but for M-type stars ΔT ≃ 500K (Berdyugina, 2005), so this factor is not negligible at visual wavelengths.2 For the star KIC=10063343 (T0 = 4045K ) the radius of the spot (relative to the stellar radius) would be no less than 0.1288(1 − 0.3) ≈ 0.3. Note that for the Sun the quantity R eff ≃ 0.0005 which is smaller by a factor of 200 than the star we found in our sample.
4.2. Statistical properties of the magnetic proxies As we mentioned above, the fluctuation amplitude (Reff) provides information regarding the size or spatial coverage of magnetic features. Before presenting the results of correlation analysis between the two magnetic proxies in the following subsection, we investigate relation between the rotation periods and mean fluctuation amplitude of stars in our sample. The mean value of fluctuation amplitude R eff , is the mean
2
Assuming T0 = 4250, Ts = 3750, λ = 620nm we have
T0 = 4250, Ts = 3750, λ = 420nm we have 34
Fs (ν ) F0 (ν )
= 0.34
Fs (ν ) F0 (ν )
= 0.48 and for
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Fig. 5. The mean value of iAC as a function of fluctuation amplitude and period for all stars in our sample.
The star is peculiar in some sense and further investigation of its light curve might reveal extra information regarding magnetic activity and time evolution. In addition to above discussion, the statistical properties of iAC might be interesting. We measure the mean value of iAC over all quarters for each star and present it (iAC ) as a function of period and mean fluctuation amplitude in Fig. 5. The left panel shows the mean value of iAC versus period for each star. Long period stars not only have small fluctuation amplitudes (see Fig. 4) but also a small value of iAC which indicates a short life-time for small spots. On the other hand, for a fast rotating star, the fluctuation amplitude is relatively large as well as their life-times. The right panel of Fig. 5 present the mean value of iAC versus mean value of Reff for all stars in our sample and clearly confirms above discussion about the size and life-time of fluctuations on M-type stars.
Table 1 The correlation (Cor), mean values of iAC, mean value of Reff, period, mass, radius, and effective temperature of the stars presented in this work. This table is available in its entirety in a machine-readable form. KIC
Cor
iAC
R eff
P (days)
M (M⊙)
R (R⊙)
Teff (K)
892376 1162635 1569863 1570915 1572802 1577265
−0.03908 0.85164 0.47998 0.77010 0.52370 0.63419
0.27168 0.26048 0.32640 0.29389 0.32012 0.28707
0.00850 0.00638 0.02158 0.00831 0.05022 0.00791
1.532 15.678 13.321 12.315 0.374 15.153
0.4699 0.4497 0.3862 0.5646 0.495 0.5028
0.605 0.494 0.36 0.583 0.535 0.56
3973 3759 3591 4161 3878 3903
4.3. Correlation analysis for the two magnetic proxies Number of (iAC, Reff) pair for each star depend on the rotational period of the star. Dependency of these two proxies on each other could be quantified by correlation (denoted by Cor) of them. The correlation can be positive, zero, or negative, and might give us useful information about the magnetic activities of the stars and their evolutions (He et al., 2015; Mehrabi et al., 2017). iAC is a proxy that measure the stability (relative to the time scale of rotation period) of the magnetic features, and Reff shows the spatial coverage of the features and hence intensity level of magnetic activities. The evolution of these proxies in 3.8 years which is divided into 15 quarters (Q2-Q16), could be used to extract information about how the size and lifetime of the magnetic features change over the time (He et al., 2015; Mehrabi et al., 2017). However, a large value of iAC might be happened in a non-eclipsing (close) binaries. To avoid this possibility, we check 10 targets with most large value of iAC and did not see any regular pattern during all quarters. The correlation between the two quantities is given by
Fig. 6. Distribution of correlation between iAC and Reff.
presented. Our analysis indicates that most stars in our sample are positive correlated and only a few of them are negative correlated. In our analysis it is found that 9.5%, 5.8%, and 84.6% of the stars are in negative, zero, and positive correlation, respectively. So for most stars in our sample, the time variations of the two magnetics proxies are in same phase, that is, when the spatial coverage of the magnetic features that dominate the rotational modulation increases, the stability of the magnetic features also increase, and vice versa. It is worthwhile to mention that similar results have been found by several groups about size-life correlation in the sunspots (Petrovay and van Driel-Gesztelyi, 1997; Henwood et al., 2010). These groups have studied the size and decay rate of sunspots through different data and their results indicate positive correlation (similar to most M-type stars in our sample) between the size and lifetime of sunspots. Moreover, such analysis for a sample of G-type stars gives the results of 4.3%, 6.8% and 88.9% (Mehrabi et al., 2017), which are very close to results for the M-type stars. Statistically speaking, the above results show that the magnetic activity in both groups, G-type and M-type stars, act in a same way. In Fig. 7 the scatter plot of correlation (Cor) versus R eff (left panel) as well as correlation versus period (right panel) for all the stars in our
s=k
Cor =
s s − iAC)(Reff − R eff ) ∑s = 1 (iAC s=k
s=k
s s − iAC)2][ ∑s = 1 (Reff − R eff )2] [ ∑s = 1 (iAC
, (6)
where iAC and R eff indicate the mean values of iAC and Reff for all the 10period segments (see Section (3)) of a star, s is the serial number for the 10-period segments (from 1 to k), and k is the total number of the 10period segments. The correlation between two proxies are calculated for all the selected stars in our sample. A small part of our results is presented in Table 1 and the entire information of this table, in a machine-readable form, are available in the online journal. The correlation coefficient (Eq. (6)) are in the range (−1, 1) with Cor = −1 fully anti-correlated and Cor = 1 fully-correlated. Similar to our previous work, the correlation values are divided into 3 parts as − 1 < Cor < −0.1 (negative correlation), − 0.1 < Cor < 0.1 (zero correlation), and 0.1 < Cor < 1 (positive correlation). In Fig. 6, the distribution of correlation between iAC and Reff for all the selected stars is
35
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Fig. 7. Correlation (Cor) versus R eff (left panel) and versus period (right panel) for all the stars in the sample.
might be useful for stellar magnetic dynamo theory. From our results, it is found that: (1) Stars with short rotational period might have a relatively large value of fluctuation amplitude (quantified by R eff ) while stars with long period can’t have a large R eff value. This result indicate that the magnetic features on long period stars cover a relatively smaller spatial area than short period stars. (2) There is a weak trend between the rotation period and the correlation of iAC and Reff, that is, the correlation of the two magnetic proxies tends to be positive for stars with shorter period and be zero and negative for stars with longer period. In fact, almost all short period stars are positively correlated while for longer period stars the correlation has a more scattered distribution. We have found a few short period G-type stars with negative correlation in our previous work but for M-type stars there are almost no stars in the period interval of 0–10 days with negative correlation. Statistically speaking, the distribution of M-type stars in correlation versus period plane is similar as the G-type stars. (3) Statistically speaking, the negative correlation stars can’t have a large value of R eff as positive correlation stars. This means that the spatial coverage of magnetic features that dominate the rotational modulation can’t be large for a negative correlation star. Note that the Sun is a negative correlated star with very tiny fluctuation amplitude. In our sample we found a peculiar target (KIC 10063343) with both large R eff and negative correlation. We postpone a further analysis of its light curve to our future work. The similarity of magnetic activity behaviors between M-type stars and G-type stars implies that the two kinds of stars obey the same magnetic dynamo mechanism, though their internal structures of convective zone may be distinctly different. This conclusion is consistent with the result obtained by Wright and Drake (2016) using the X-ray emission as the proxy of stellar magnetic activity.
sample is depicted. The result shows that small values of R eff could happen for the negative, zero, and positive correlation stars, but statistically speaking, the negative correlation stars can’t have large R eff value. Note that we have detected the same character for G-type star (Mehrabi et al., 2017) and specifically the Sun is a negative correlation star with R eff = 0.00047 (He et al., 2015). In addition, it is found that the order of magnitude of R eff for positive, zero and negative correlation stars is the same as G-type stars. So our results suggest that the spatial coverage ratios (relative to the stellar disk area) of magnetic features over G-type and M-type stars, for each correlation group, may be roughly the same. One interesting point from our results is that, statistically speaking, fast rotators of M-type stars can not have negative correlation (see right panel of Fig. 7). There are almost no targets with negative correlation in the period interval of 0–10 days. In contrast, a few G-type stars have been found in this range with negative correlation (Mehrabi et al., 2017). On the other hand, for M-type stars with long period, we can find a relatively scattered distribution across different correlation groups, which is almost the same for G-type stars. Moreover, our results also demonstrate a weak trend between the rotation period and the correlation of iAC and Reff. That is, the correlation tends to be positive for stars with shorter period, and be zero and negative for stars with longer period. One possibility is that the correlation properties of stars depend on the their ages. Such a possibility will be considered in our future work. The similarity of magnetic activity between M-type and G-type stars have been reported in Wright and Drake (2016) using a completely different approach which supports our results. 5. Conclusion
Acknowledgment
In this paper, we study the magnetic activity properties for a sample of M-type stars in Kepler field by using the two light-curve-based magnetic proxies, iAC and Reff. The former indicates the stability of magnetic features and the latter reflects the spatial coverage of magnetic features. In our previous work, we analyze these quantities for a sample of G-type stars. In this work, we investigate how these magnetic proxies depend on the rotational period of stars and the correlation between them for a sample of M-type Kepler targets. Our results show similarities and differences between M-type and G-type stars which
This paper includes data collected by the Kepler mission. The data used in this paper were downloaded from the Mikulski Archive for Space Telescopes (MAST). Funding for the Kepler Mission is provided by the NASA Science Mission Directorate. The authors of this paper gratefully acknowledge the entire Kepler team and all people who contribute to the Kepler mission, which provides us the opportunity to conduct this study.
Appendix In this part, we introduce 10 targets with a large value of iAC in our sample. These targets provide a good sample to study the properties of magnetic activities over M-type stars. Further analysis over these targets might reveal more information regarding the size and location of spots and faculae over these stars. In this part we only introduce these stars and are not going to study properties of them. These targets are presented in Table 2 . We also plot the light curves of these targets and show two of them in Fig. 8. The light curves of all targets are provided as online material.
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Table 2 The KIC and other properties of 10 targets with maximum values of iAC. KIC
Cor
iAC
R eff
P (days)
M (M⊙)
R (R⊙)
Teff (K)
6370174 8572066 9473985 6871896 9032388 5954552 7350067 2581444 9472906 3439126
0.42153 0.60117 0.08841 0.82403 0.902493 0.48006 0.08887 0.68210 0.90317 −0.03430
0.40857 0.40772 0.40770 0.39265 0.391517 0.39099 0.38661 0.38656 0.38275 0.380914
0.03520 0.03702 0.016873 0.023255 0.02034 0.019946 0.038538 0.01459 0.02362 0.05238
14.99 9.339 20.931 14.243 19.825 14.922 3.514 11.976 33.085 7.9
0.314 0.513 0.388 0.530 0.583 0.497 0.261 0.580 0.408 0.577
0.327 0.55 0.453 0.625 0.646 0.524 0.2 0.6 0.451 0.664
3386 3928 3595 4053 4297 3887 3227 4220 3659 4236
Fig. 8. The light curves for two M-type stars with a large value of iAC. In both figures, top left panel is for quarter 2 and last panel is for quarter 16. 37
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Supplementary material Supplementary material associated with this article can be found, in the online version, at 10.1016/j.newast.2018.07.007 .
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