4 X-ray emission vs. Rossby number

The historical importance of the Rossby number is strictly related to the dynamics of the magnetic fields embedded and frozen in the plasma. As already anticipated in Sect. 1 the Rossby number $R_{\rm o}=P_{\rm rot}/\tau_{\rm conv}$ is usually considered a good indicator of the efficiency of the dynamo mechanism in the generation and amplification of the stellar magnetic fields, and consequently of the stellar magnetic activity. However, while the rotation period can be directly measured, $\tau_{\rm conv}$ is usually empirically determined (Noyes et al. 1984; Stepien 1994) or derived from theoretical models (Gilliland 1985; Kim & Demarque 1996; Pizzolato et al. 2001).

The study of the relationship between X-ray emission and rotation in different mass ranges (or spectral classes), described in the previous section, allowed us to clearly separate saturated stars from non-saturated stars and, therefore, it made possible a new determination of an empirical X-ray-based Rossby number. In order to evaluate this parameter for all the stars in our sample we have determined an empirical X-ray-derived function of the stellar mass, $\tau_{\rm e}(M)$, and we have used this function as an empirical time scale in place of  $\tau_{\rm c}$, the convective turnover time. In this section we show how the ratio $R_{\rm e} = P_{\rm rot}/\tau_{\rm e}$ effectively provides us with a unique, mass-independent prediction of the X-ray emission level, for both saturated and non-saturated stars, a property which makes $R_{\rm e}$ possibly preferable to  $P_{\rm rot}$.

   
4.1 Empirical determination of the Rossby number from X-ray data

Following the original idea of Noyes et al. (1984), and a working approach similar to that adopted by Stepien (1994), we have explored the possibility to obtain an improvement in the correlation between activity and rotation by using the Rossby number $R_{\rm e}=P_{\rm rot}/\tau_{\rm e}(M)$ instead of the rotation period, where $\tau_{\rm e}(M)$ is a mass-dependent function derived from the observational data available.

Our work is characterized by two improvements with respect to the Stepien (1994) work: (i) a larger sample, including more stars in all rotation ranges and more low-mass stars, and (ii) a quantitative separation of the two X-ray emission regimes, a crucial point for the determination of the mass dependent function  $\tau_{\rm e}(M)$.

We have first considered the family of power laws:

$$L_{\rm x}/L_{\rm bol}=A_k P_{\rm rot}^{-2}$$ (1)

which describe the behavior of the non-saturated stars in Fig. 8 (right panel), with A_k dependent on the mass range considered. We stress that Stepien (1994) made no distinction between saturated and non-saturated stars in his activity-rotation relationships.

A single relationship for all the stars in our sample can be obtained by scaling:

$$P_{\rm rot} \Rightarrow \frac{P_{\rm rot}}{\tau_{\rm e}(M)}$$ (2)

where the mass-dependent function $\tau_{\rm e}(M)$ can be obtained as (A_k/C)^1/2, with C= const., so that

$$L_{\rm x}/L_{\rm bol}= C \left(\frac{P_{\rm rot}}{\tau_{\rm e}}\right)^{-2} \cdot$$ (3)

The new variable $P_{\rm rot}/\tau_{\rm e}$ can be dubbed "X-ray empirical Rossby number" and denoted by $R_{\rm e}$. An important point is that, by using this empirically-derived Rossby number, we may effectively describe the activity level, represented by $L_{\rm x}/L_{\rm bol}$, as a single mass-independent function of the parameter $R_{\rm e}$, taking into account both the rotation and any effect linked to differences in internal stellar structure for stars of different masses (Fig. 9).

Figure 9: X-ray to bolometric luminosity ratio vs. empirical Rossby number for all the stars in our sample. The meaning of the symbols is the same as in Fig. 3.

It is to be mentioned, however, that the adopted procedure does not permit to determine absolute values of  $\tau _{\rm e}$, but only the functional dependence of such an empirical time scale on the stellar mass; the function  $\tau _{\rm e}$ must be properly scaled in order to be compared with other empirical or model-derived convective turnover times. The value of  $\tau _{\rm e}$ listed in Table 3 for each mass range was obtained by applying a constant scaling factor such that the value of  $\tau _{\rm e}$ for solar-mass stars matches the Noyes's semi-empirical prediction of the convective turnover time of the Sun. The relationship between this  $\tau _{\rm e}$ and the stellar mass has been used to calculate the values of $R_{\rm e}$ plotted in Fig. 9.

In Fig. 10 we show a comparison between the function  $\tau_{\rm e}(M)$ and the theoretical convective turnover time, $\tau_{\rm c}$, derived from two stellar structure models, the model by Kim & Demarque (1996) and the more recent model by Ventura et al. (1998). The latter was employed for the computation of the characteristic turnover time also for stars with $M<0.5~M_{\odot}$, including fully-convective stars with $M/M_{\odot}=0.3$ and $M/M_{\odot}=0.2$. Both models give a global estimate of this time scale by integrating over the whole convective region. For ease of comparison the function $\tau_{\rm e}(M)$ in Fig. 10 is scaled in such a way that our empirical time scale for a solar-mass star coincides with the theoretical convective time predicted by Ventura et al. (1998). We find that the empirically X-ray-derived function  $\tau _{\rm e}$ follows  $\tau_{\rm c}$ for stars in the mass range 0.6-1.2; for lower-mass stars, the empirical timescale is still in agreement with the model convective time, even if the paucity of stars with $P_{\rm rot}>10$ days makes the comparison particularly critical.

Figure 10: Comparison between our empirically-determined  $\tau _{\rm e}$(asterisks), and theoretical predictions by Kim & Demarque (dash-dotted line) and by Ventura et al. (1998) (dashed line). Horizontal lines cover the mass ranges considered, while the asterisks are placed at the median of the masses of the corresponding bin.

In order to compare our empirical time scale with the values computed with the Noyes et al. (1984) formula, we have completed our analysis by deriving  $\tau _{\rm e}$ also as a function of the B-V color, using the results reported in Sect. 3.3. In Fig. 11 we have plotted the Noyes function and our empirical $\tau_{\rm e}(B-V)$, properly scaled as in Table 3. The two formulations are very similar for 0.5<B-V<1.0, and our data confirm the Noyes's prediction also in the B-V range 1.0-1.4, where the Noyes study was based on the data of 5 stars only (dashed line in Fig. 11). For B-V>1.4 we find an indication of increasing  $\tau _{\rm e}$as already seen in Fig. 10 for stars with $M<0.5~M_{\odot}$.

Figure 11: Comparison between empirically-determined  $\tau _{\rm e}$(asterisks), scaled $L_{\rm bol}^{-1/2}$ (squares), and the Noyes et al. (1984) semi-empirical formulation (thin solid and dashed line).

4.2 Alternative interpretation of the empirical Rossby number

In Sect. 3.2 we have already demonstrated that a single power-law provides a good mass-independent description of the $L_{\rm x}$ vs. $P_{\rm rot}$ relationship, for non-saturated stars. Does the $L_{\rm x}/L_{\rm bol}$ vs. $R_{\rm e}$ relationship represent a real improvement?

The scaling:

$$L_{\rm x} \Rightarrow \frac{L_{\rm x}}{L_{\rm bol}}$$ (4)

introduces a color-dependent vertical shift of the power-laws, which can be compensated by the horizontal shift provided by the scaling:

$$P_{\rm rot} \Rightarrow \frac{P_{\rm rot}}{L_{\rm bol}^{-1/2}}\cdot$$ (5)

In fact, we show in Fig. 11 that the empirically-determined color function $\tau_{\rm e}(B-V)$ is equivalent to $L_{\rm bol}^{-1/2}$, except for a constant scaling factor, where $L_{\rm bol}$ was computed as the mean value of the bolometric luminosities of the stars in each color bin. The above argument suggests that for stars in the non-saturated regime the two relationships $L_{\rm x}$ vs. $P_{\rm rot}$and $L_{\rm x}/L_{\rm bol}$ vs. $R_{\rm e}$ are equivalent.

On the other hand, for saturated stars, the X-ray luminosity is related only to  $L_{\rm bol}$. From this point of view, we can argue that there is no need to invoke a Rossby number: $P_{\rm rot}$ is the only parameter required to described the behavior of the non-saturated stars, while $L_{\rm bol}$ is the crucial parameter in the saturated regime.

An alternative interpretation can be sustained by noting that the $L_{\rm x}/L_{\rm bol}$ vs. $R_{\rm e}$ relationship describes adequately both saturated and non-saturated stars. From this second point of view, we are left with one open issue: if we accept $R_{\rm e}$ as a fundamental dimensionless parameter of the dynamics of plasma flows in the stellar interiors, determining also the efficiency of the dynamo action, then the stellar activity level is best represented by the ratio $L_{\rm x}/L_{\rm bol}$, but why the X-ray emission level should be determined by the bolometric luminosity?

In any case, the observed agreement between the functional forms of $L_{\rm bol}^{-1/2}$ and $\tau_{\rm e}(B-V)$ seems to indicate a possible physical link between the stellar bolometric luminosity and the "effective" time scale relevant for the dynamo action. Future investigations on fully-convective late M-type dwarfs or stars in evolutionary phases across the Hertzsprung gap, will allow us to test in more detail whether stellar activity in these stars can be described in the same framework as in main sequence stars.

A final point is illustrated in Fig. 12 which shows an interesting almost linear relationship between our empirical convective turnover time and the rotation period at which the saturated X-ray emission is reached. This plot suggests that the saturation is triggered when a critical value of the ratio between these two time scales is reached, and this value is almost independent from the stellar mass, at least for main-sequence stars. Moreover, since  $\tau _{\rm e}$ scales as  $L_{\rm bol}^{-1/2}$, also $P_{\rm rot}^{\rm sat}$ scales in the same way; to a good approximation:

$$P_{\rm rot}^{\rm sat}\approx 1.2\left(\frac{L_{\rm bol}}{L_{\odot}}\right) ^{-1/2}$$ (6)

which is accurate within a factor two.

Figure 12: Relationship between our empirical time scale and the rotation period at which the saturated X-ray emission is reached.