5 Rotation-activity relations

The rotation-activity connection has been extensively studied by Walter (1981),

Figure 8: XLF of single and binary stars of spectral type G, K and M. a) TTS, b) Pleiades, and c) Hyades. solid lines - single stars (s), dotted lines - binary stars assuming equal $L_{\rm x}$ from both components (b2), dashed lines - binary stars assuming only one X-ray emitting component (b1). See text for a more detailed description of these samples. All G type TTS in Taurus-Auriga are single stars, and therefore not displayed in this figure.

Walter & Bowyer (1981), Walter (1982), Bouvier (1990), Damiani et al. (1991), Grankin (1993), N95, Bouvier et al. (1997b), and Wichmann et al. (1998a).

In this section we study the subsample of the stars from Tables 2 to 7 with measured rotation periods $P_{\rm rot}$ or projected rotational velocity ${v \sin{i}}$. The choice of the best parameters describing the activity-rotation relation is not undisputed. We have, therefore, examined different parameter combinations. On the X-ray side we use the luminosity $L_{\rm x}$, the surface flux $F_{\rm s}$, and the ratio between X-ray and bolometric luminosity $L_{\rm x}/L_{\rm bol}$ to characterize the stars. Each star is represented by its mean X-ray luminosity or upper limit to $L_{\rm x}$ as described in Sect. 4. For binaries only one component is considered, because spectral types and rotation rates are in most cases known only for the primary. The stellar radii used to compute $F_{\rm s}$were determined from the Stefan-Boltzmann law. The rotation is described by the projected rotational velocity, ${v \sin{i}}$, or the rotation period, $P_{\rm rot}$. $P_{\rm rot}$ and ${v \sin{i}}$ of Pleiades and Hyades stars are listed in the Open Cluster Data Base. Values for the rotation rates of TTS are taken from N95, Bouvier et al. (1997b), and Wichmann et al. (1998a).

For the statistical analysis cTTS and wTTS have been combined to yield a larger sample, although generally wTTS are faster rotators than cTTS, and they are more X-ray luminous. A linear regression has been fitted to all pairs of rotation-activity combinations using the ASURV EM algorithm or the method by Schmitt (1985) for doubly censored data. In Table 11 we summarize the results of all correlation tests,

 

 

Table 11: Results of statistical tests with ASURV for the relation between X-ray emission and stellar rotation for TTS, Pleiads, and Hyads. The first two columns are the names of the two parameters to be compared. Next is the size of the sample, N, and in brackets the number of upper limits, $N_{\rm lim}$, to the rotation and X-ray parameter. Columns 5 and 6 give the probability that there is no correlation between the two parameters according to Kendall's and Spearman's test. The slope of a linear regression to the pair of parameters is given in Col. 7. For doubly censored data we have used the linear regression method of Schmitt (1985). All samples where $P_{\rm rot}$ is the rotation parameter have upper limits only in the X-ray parameters, and the EM algorithm is used.

Par 1	Par 2	N	$N_{\rm lim}$	Kendall	Spearman	slope
TTS
${\log{(v \sin{i})}}$	$\log{L_{\rm x}}$	65	(0/17)	0.0031	0.0047	$1.08 \pm 0.36$
${\log{(v \sin{i})}}$	$\log{F_{\rm s}}$	52	(0/14)	0.0009	0.0011	$1.57 \pm 0.46$
${\log{(v \sin{i})}}$	${\log{(L_{\rm x}/L_{\rm bol})}}$	52	(0/14)	0.0053	0.0040	$1.22 \pm 0.44$
$\log{P_{\rm rot}}$	$\log{L_{\rm x}}$	39	(0/7)	0.0000	0.0001	$-1.52 \pm 0.39$
$\log{P_{\rm rot}}$	$\log{F_{\rm s}}$	38	(0/6)	0.0000	0.0000	$-1.93 \pm 0.39$
$\log{P_{\rm rot}}$	${\log{(L_{\rm x}/L_{\rm bol})}}$	38	(0/6)	0.0001	0.0002	$-1.49 \pm 0.42$
Pleiades
${\log{(v \sin{i})}}$	$\log{L_{\rm x}}$	164	(6/53)	0.0000	0.0000	0.61
${\log{(v \sin{i})}}$	$\log{F_{\rm s}}$	164	(6/53)	0.0000	0.0000	0.67
${\log{(v \sin{i})}}$	${\log{(L_{\rm x}/L_{\rm bol})}}$	164	(6/53)	0.0000	0.0000	0.88
$\log{P_{\rm rot}}$	$\log{L_{\rm x}}$	46	(0/13)	0.0008	0.0005	$-0.42 \pm 0.11$
$\log{P_{\rm rot}}$	$\log{F_{\rm s}}$	46	(0/13)	0.0000	0.0001	$-0.52 \pm 0.11$
$\log{P_{\rm rot}}$	${\log{(L_{\rm x}/L_{\rm bol})}}$	46	(0/13)	0.0000	0.0000	$-0.66 \pm 0.12$
Hyades
${\log{(v \sin{i})}}$	$\log{L_{\rm x}}$	67	(41/2)	0.0008	0.0000	1.58
${\log{(v \sin{i})}}$	$\log{F_{\rm s}}$	67	(41/2)	0.0000	0.0001	1.56
${\log{(v \sin{i})}}$	${\log{(L_{\rm x}/L_{\rm bol})}}$	67	(41/2)	0.0003	0.0089	1.64
$\log{P_{\rm rot}}$	$\log{L_{\rm x}}$	21	(0/2)	0.0003	0.0004	$-1.13 \pm 0.32$
$\log{P_{\rm rot}}$	$\log{F_{\rm s}}$	21	(0/2)	0.0016	0.0016	$-0.94 \pm 0.28$
$\log{P_{\rm rot}}$	${\log{(L_{\rm x}/L_{\rm bol})}}$	21	(0/2)	0.3515	0.3489	$-1.51 \pm 0.32$

and also give the slopes of the linear regression. According to the statistical tests X-ray emission and rotation are clearly correlated for most of the examined stellar samples. For a given X-ray parameter the probability for a correlation with $P_{\rm rot}$ is in most cases larger than the probability for a correlation with ${v \sin{i}}$. This is probably due to the unknown inclination angle in ${v \sin{i}}$ whose arbitrary orientation tends to destroy any intrinsic correlation between the rotation and X-ray emission. Using $P_{\rm rot}$ should therefore be more meaningful. However, measurements of the actual periods (by spot modulation) are much sparser than spectroscopic observations of ${v \sin{i}}$, leading to a smaller data set.

We show correlations of all possible combinations of the above mentioned X-ray parameters with $P_{\rm rot}$ in Figs. 9 to 11 for TTS,

Figure 9: Relation between the rotation period and different X-ray parameters for TTS from the Taurus-Auriga region: top - X-ray luminosity, middle - X-ray surface flux, and bottom - Ratio between X-ray luminosity and bolometric luminosity. cTTS are represented by filled symbols and wTTS by open symbols. TTS of unknown nature are displayed as asterisks. Multiple stars are marked by boxes. The solid lines are linear regressions computed with the EM algorithm implemented in ASURV. The size of the errors bars varies a lot due to very different ROSAT exposure times, and they are sometimes smaller than the plotting symbol.

Figure 10: Same as Fig. 9 for the Pleiades.

Figure 11: Same as Fig. 9 for the Hyades.

Pleiads, and Hyads. Overlaid are the linear regressions corresponding to the power law relation from Table 11. The lowest significance is found in the Hyades. This may however be due to the limited range in rotation period (only two stars with $P_{\rm rot} < 4~{\rm d}$), and because the Hyades with known period have a small range of spectral types.