2.1. Data Sources

The rotational periods of field and open cluster stars contained in 28 publications were collected in order to investigate the evolution of angular momentum during the main sequence. Twenty three of these papers are dedicated to 17 open star clusters in the Milky Way. The open cluster sample was put together with the intention of covering the largest possible range of ages after the termination of the disk locking phase. With an estimated age of 13 and 3500 Myr, h Persei and M 67 are the youngest and oldest open clusters inspected in this work. The central positions (epoch 2000.0), angular radii (Θ), parallaxes, ages, color excesses (E(B-V)) and metallicities ([F_e/H]) of these clusters are laid out in Table 1 (references in Appendix A). Some of these numbers were used in data selection and extinction correction, §2.2 and §2.3. The cluster age, metallicity and assumed initial helium abundance (Y₀) are the in-put parameters for the evolutionary tracks used to derive the physical properties of cluster stars from their visual and infrared colors, §2.4.

Rotational periods of field stars were computed by several research groups using observations from the CoRoT satellite (^{Affer et al. 2012}), the HATNet survey for transiting extrasolar planets (^{Hartman et al. 2011}), and the Kepler satellite project (^{Nielsen et al. 2013}; ^{Reinhold et al. 2013}; ^{McQuillan, Mazeh & Aigrain 2014}).

To homogenize the data set, the Vizier¹ facility was used to collect the spectral type, visual magnitude and classication of each star as reported by SIMBAD², their G, BP and RP band magnitudes and parallax from Gaia DR2³, and their J, H and K magnitudes from the Two Micron All Sky Survey⁴ (herein 2M ASS).

This information is collected in a number of tables containing stellar coordinates (degrees, epoch 2000; precise within 1 arcsec), spectral types (‘*?’ when unknown), visual magnitude as reported in SIMBAD (as a reference), Johnson-Cousins visual magnitudes calculated from the Gaia DR2 G, BP and RP band magnitudes and the transformation laws found by ^{Evans et al. (2018)}, J, H and K 2M ASS magnitudes, rotational periods (days) and Gaia DR2 parallaxes (miliarcsec). When there is no information for any of these quantities, the corresponding field value is 999.9.

These tables include all the stars reported in each rotational data source. Rotational periods in the Kepler field and the Pleiades, M 34, Hyades and Praesepe open clusters were searched for by more than one research group. Additional tables excluding duplicate stars were produced for these clusters and the field. Mean periods are reported for stars with multiple measurements, with the period error being equal to half the difference between the shortest and longest period.

Several filters were applied to select stars with good photometric and astrometric data quality and no possible caveats regarding contamination from other motives of variability besides single star rotation. As explained in the following subsection, distinctive flags are added when these filters are activated.

2.2. Data Filters

A ag following the visual magnitude derived from the Gaia data is set to ‘XX’ under column GP , when it could not be calculated because at least one of the Gaia DR2 G, BP and RP magnitudes and their errors is missing, or the photometric flux over error in any of these bands is smaller than 5. Otherwise, the Gaia ag GP is set to ‘-’.

A ag following 2M ASS magnitudes is set to ‘XX’ under column 2P if either one of the J, H and K magnitudes and their errors is absent, the photometric flux over error in any band is smaller than 5, the 2M ASS quality ag is not set to ‘AAA’ and the star may be contaminated by an extended source (their X flg is larger than zero). Otherwise, the 2M ASS ag 2P attached to the J, H and K magnitudes is set to ‘-’.

Databases were used to identify possible non main sequence stars, object types that may indicate that the angular momentum evolution of a single star may have been affected by an external agent (such as stellar companions and planetary systems) or light curve modulations produced by other sources of variability besides single star rotation. Namely, the following types (in parenthesis, condensed name for object classification recommended by SIMBAD):

When present, these name tags are reported under column label Comment. This Comment may also include the following name tags

The final comment is set to ‘-’ when the star has not been associated to any of the preceding types and the stellar parallax and period have been adequately measured. There are electronic versions of tables for primary data from field and open cluster stars (http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=J/other/RMxAA/56.1, http://ftp://cdsarc.u-strasbg.fr/pub/cats/J/other/RMxAA/56.1). As an example, see Table 2.

The number of stars with measured rotational periods in each cluster and field is given under Column Per in Table 3. The number of stars that were selected after passing all data filters is under Column OK in the same table. Notice that there is a large fraction of rejected stars in most open clusters. There are many reasons for this to be so, but the most frequent is insufficiently accurate parallax (94, 60 and 47% in h Persei, M 37 and NGC 6819), generally in clusters that are farther away. Stellar parallax is also poor in a large fraction of stars associated to the two open clusters that are closer to us, Hyades and Coma Berenices (43 and 21%). The next two main reasons for rejection are poor or non-existent 2M ASS photometry and association to a multiple system, with very high levels of incidence in h Persei, IC 4665 and NGC 6819. Parallax was too inaccurate in 59% of the HAT N et field stars, and 42% of the CoRoT field stars are not in the main sequence. Fortunately, all these problems affected a much smaller fraction of Kepler field stars.

In total, the rotational period has been measured for 3769 members of the 17 clusters listed in Table 1, and for 45235 field stars observed by the CoRoT , HAT N et and Kepler projects. Out of these, the data filters approved 1876 cluster and 32641 field stars.

2.3. Extinction Correction

Gaia DR2 provides an estimate of line-of-sight extinction, A_G, and reddening, E(BP - RP ) for each star. After a careful analysis of the Gaia DR2 database, ^{Andrae et al. (2018)} concluded that these line-of-sight extinction estimates are not accurate at the single star level and, concurrently, that there are no clearly defined transformations between A_G and A_V , and E(BP - RP ) and E(B - V ). Consequently, absolute magnitudes of open cluster stars were determined assuming that the color excess is the same for all stars and is as given in Table 1. This is a particularly weak assumption in young clusters, where dust distribution can be very inhomogeneous at the intra-cluster and individual level. For eld stars, the extinction model is a smooth vertically-exponential dust disk, and the extinction coe cient at galactic latitude b and distance d is

where C(0) is the mid-plane extinction coefficient and h is the model scale-height, which is taken as equal to 0.14 kpc (Koppen & Vergely 1998; ^{Marshal et al. 2006}; ^{Brown et al. 2011}). The assumed mid-plane extinction coefficients are C_V = 1.0, C_J = 0.35, C_H = 0.25 and C_K = 0.15 magnitudes per kiloparsec (^{Indebetouw et al. 2005}). Implicitly, line of sight variations are supposed to be a minor correction.

2.4. Stellar Parameters

Stellar parameters were inferred matching (within error bars) the observed absolute magnitudes in the Johnson-Cousins V band, and the Bessel & Brett J, H and K bands, with those deduced from the latest Yale-Potsdam stellar isochrones (^{Spada et al. 2013}; ^{Spada et al. 2017}, herein YaPSI) and the semi-empirical color-temperature calibration of ^{Worthey & Lee (2011)}. The 2M ASS magnitudes were con-verted to the Bessel & Brett system using formulae derived by ^{Carpenter (2001)}. The parameter space of YAPSI models is dense enough to carry out de-tailed interpolations over a broad spectrum of possibilities: from the birth line to the onset of helium ignition in the core, 0.15 to 5.0 M_ʘ , [Fe/H] = -1:5 to 0.3 and Y₀ = 0.25 to 0.37. Notice that these are iron and helium abundances at the birth line, not the larger values they have at a posterior time. Besides the usual parameters (mass, gravity, luminosity, temperature, etc.), the YaPSI archives also include the depth of the convective envelope, the convective overtime timescale and the moments of inertia of the entire star and its convective envelope. These quantities are essential in the understanding of the evolution of angular momentum, magnetic activity and mass loss.

The physical parameters of open cluster stars are assumed to be equal to the mean values of models having 27 different age, [Fe/H] and Y0 combinations, with each of these parameters being equal to the minimum, mean and maximum values given in Table 1. For instance, the YaPSI models used for the Pleiades are 27 combinations of three age estimates (100, 120 and 140 Myr), three initial iron abundances (-0.05, 0.00 and 0.05) and three initial helium abundances (0.27, 0.28 and 0.29). No attempt is made to estimate and use the initial abundances of iron and helium, but these are probably close to the minimum values. In Table 3, the number of stars with a model accounting for their visual and infrared magnitudes is under Column VIR, and in Column IR when only the infrared magnitudes were matched. To exclude multiple observations of individual stars, data les from the Pleiades, M 34, Hyades and Praesepe clusters were merged into one (name-All).

The visual and infrared magnitudes of a substantial fraction of stars in the Hyades (65%), Pleiades (57%) and NGC 2547 (29%) open clusters could not be reproduced with any of the corresponding YaPSI models, suggesting that there may be larger than explored differences in the chemical makeup of stars that are thought to be part of these open clusters and/or that extinction corrections were inadequate in a large number of them. In this respect, model matching including visual magnitudes was substantially more ineffective in IC 4665, NGC 2547, Pleiades, M 37, M 35, Praesepe and Hyades, which may imply that uniform extinction is an inadequate assumption for these relatively young associations. Notice that this does not hold in every young cluster (for instance, h Persei, M 48, NGC 2301 and Blanco 1) and certainly not in those that are older than 1000 Myr. Fluctuations in the predicted values of all physical parameters were less than 10% in nearly all open cluster stars. The most notable exception is NGC 2301, where all predicted stellar parameters fluctuate more than 30% in every star.

Model results were equally discouraging in 1 out of 4 stars from the Hyades and 3 out of 7 in NGC 6819. Unfortunately, being more massive than 1.35 M_ʘ, the few remaining stars from the relatively and conveniently old NGC 6819 open cluster are probably not subject to rotational braking. To use a sufficiently large amount of data, the impending analysis will include cluster stars where the visual magnitude could not be matched.

The physical parameters of field stars were extracted after inspecting a set of models with an initial close to solar-like metallicity and helium abundance ([Fe/H] = 0.0 and Y ₀ = 0.28) and 37 possible ages, from 1 to 13000 Myr. The observed period was used as a roughly confined mass dependent age restriction. This is imperative, since there are important changes in the moment of inertia at the beginning (first ≈ 100 Myr) and during the second half of the main sequence, as can be appreciated in Figure 1.

Fig. 1 Ratio of the moment of inertia at time t, I(t), with respect to the moment of inertia at t = 1000 Myr. Dotted lines are for M=M_ʘ = 0.5 (green), 0.6 (black), 0.7 (red) and 0.8 (blue). Continuous lines are for M=M_ʘ 0.9 (green), 1.0 (black), 1.1 (red) and 1.2 (blue). The color figure can be viewed online. 

The visual and infrared magnitudes of a very small fraction of HAT N et field stars with high quality data and no other possible sources of variability could be reproduced with a YaPSI model, and nearly half of these models were theoretically inaccurate at the 30% level or more. A stellar model matching visual and infrared magnitudes could be found for the majority of CoRoT field stars that survived the data filters, but more than half of them were also inexact to a very high level. To exclude multiple observations, Kepler field data les were merged into Kepler-All. At least one stellar model was able to account for the visual and/or infrared magnitudes of 90% of main sequence Kepler field stars. The predicted mass, temperature, radius and gravity (its logarithm) of the vast majority of these stars (at least 8 out of 10) is theoretically accurate at the 90% level or better, and less than 66% certain in roughly 1 out of 40 stars. The moment of inertia is precise at the 90% level in 8 out of 10 stars, but very un-certain in 1 out of 10. Results are slightly worse for the stellar luminosity. Since many more Kepler field stars were observed with a higher quality and success rate, CoRoT and HAT N et field stars will no longer be considered. Being more reliable, the ensuing analysis will only deal with Kepler field stars for which the visual and infrared magnitudes are reproduced.

At least one model matched the infrared magnitudes of 1311 and 30115 open cluster and field stars, and the visual and infrared magnitudes of 566 and 28364 of these stars. Electronic versions of tables with physical parameters of stars in each cluster and field are available upon request (http://vizier.u-strasbg.fr/viz-bin/VizieR?-source= J/other/RMxAA/56.1, ftp://cdsarc.u-strasbg.fr/pub/cats/J/other/RMxAA/56.1). As an example, see Table 4.

3.1. Rotational Period as a Function of Mass and Time. First Round.

With their data, ^{McQuillan, Mazeh & Aigrain (2014)} produced a figure for the logarithm of the rotational period as a function of mass for Kepler field stars (their Figure 1). Almost all periods are within rather well defined lower and upper mass dependent limits. Referring to empirical gyrochronology models, they mention that the upper envelope period distribution is broadly consistent with an age of 4500 Myr. The rotational period lower boundary was not discussed and, consequently, no age estimate was provided for this border.

^{McQuillan, Mazeh & Aigrain (2014)} also point out that their figure exhibits a bimodal period distribution for stellar masses less than ≈ 0:6 M_ʘ . The existence of large dispersions and bimodality in the rotation rates of young open cluster G and K main sequence stars has been noticed for some time (^{Soderblom, Jones & Walker, 1983}). Tracing fast and moderate-to-slow rotation sequences in several young clusters, ^{Meibom et al. (2011}a) found that this distribution tends to vanish in older clusters, with more massive stars being more likely to con-verge on a single rotational sequence. The bimodal period distribution of late K and M dwarf field stars is in line with this observation.

The rotational period as a function of mass for the Kepler field stars that passed all data filters and had their visual and infrared colors reproduced by a YaPSI model (as described above), is shown in the upper panel of Figure 2. The Sun, α Cen A and B (4850 ± 500 Myr; ^{Bazot et al. 2007}; ^{Dewarf, Datin & Guinan 2010}) and 16 Cygnus A and B (6800 ± 400 Myr; ^{Davies et al. 2015}) are included in this figure. ^{McQuillan, Mazeh & Aigrain (2014)} introduced a limit of T_eff < 6500K in their analysis, since they were only interested in stars with convective envelopes. When this limit is not introduced, an unanticipated result is a large number of slow rotators that are more massive than ≈1:3 M_ʘ . A smaller number of such stars are also found in the ^{Reinhold et al. (2013)} analysis of the Kepler database. If on the main sequence, these stars are predicted to be rotating more rapidly, since they have a very shallow convective envelope (if at all) and therefore no enhanced magnetic field to lessen their angular momentum. On the other hand, longer periods and lower temperatures are expected if they are turning away from main sequence. This combination is found in a number of cases. As can be seen in Figure 2, very few stars less massive than ≈0:5 M_ʘ were left after the data filters were applied and this may be the reason why the bimodal period distribution of low mass field stars is no longer apparent. Besides these two questions, there are no other overall differences between this figure and Figure 1 in ^{McQuillan, Mazeh & Aigrain (2014)}.

Fig. 2 Rotational period (P, days) as a function of mass (solar units). Upper and lower panel: Kepler eld stars (black). Lower panel: stars from the Pleiades (red, 120 Myr), M 35 (magenta, 155 Myr), Praesepe (green, 750 Myr), NGC 6811 (yellow, 1000 Myr), NGC 752 (yel-low, 1350 Myr) and M 67 (cyan, 3500 Myr) open clusters. Only open cluster stars with errors less than 10% in their observational parameters are shown in this figure. Open stars and filled triangles are for Cen A and B (yellow) and 16 Cygnus A and B (red). The data for these stars are in Table 6. The color figure can be viewed online. 

Most of the lower boundary of the cone defined by the Kepler field stars draws the rotational period of stars once they have stabilized in the main sequence, sometime after reaching a steady rotational shear and moment of inertia. An age estimate of the lower boundary for the period distribution can be established plotting the period as a function of mass in stars from young open clusters with known age. The lower panel of Figure 2 also contains the rotational periods of stars from several open clusters. To avoid cluttering, only open cluster stars with errors less than 10% in their observational parameters are shown in this figure. The same conclusions are obtained when all the stars used in the numerical analysis (errors less than 20%) are included in this plot. It is quite remarkable that most of the lower boundary coincides with the position occupied by slowly rotating Pleiades stars, setting the location of a zero age rotational main sequence for late type stars at approximately 120 Myr after birth. The location of slow rotators in the M 35 cluster runs in parallel but with slightly shorter periods, suggesting that this cluster is not 155 Myr old, but younger than the Pleiades.

There is a single, well de fined, mass dependent period distribution in the Pleiades, M 35 and Praesepe open clusters, as long as the stellar mass is larger than ≈ 0:8 M_ʘ . Early convergence to a single period distribution in the high mass range is sup-ported by the fact that no fast 0.7 - 1.2 M_ʘ rotators were found in M 37 (450 Myr) and M 48 (500 Myr). On the other hand, M 34 is a 210 Myr old open cluster supporting the notion that convergence to a single distribution in this mass range may take more than 120 Myr; the rotational periods of most Pleiades and M 34 stars more massive than ≈ 0.8 M_ʘ are comparable, but quite a few stars in M 34 are fast rotators with significantly lower periods than Pleiades and M 35 stars with a similar mass. Unfortunately, in NGC 2301 (180 Myr) there are no rotational periods for stars more massive than ≈0:8 M_ʘ . In conclusion, high quality data indicate that after ≈120 Myr, most ≈0:8 1:2 main sequence stars converge to the rotational sequence defined by the Pleiades. This is earlier than previously thought (around 600 Myr). One of the possible reasons for this discrepancy is that past estimates used a larger mass bin (at least 0.2 M_ʘ in ^{Gallet & Bouvier 2013} and 2015) and combined data from clusters with different ages (^{Johnstone et al. 2015}a). Notice that the dispersion - and the impression that there is no convergence - will increase when these operations are carried out. Finally, there are signs of a bimodal period distribution in Pleiades, M 35 and Praesepe (750 Myr) stars less massive than ≈0:8 M_ʘ , but no indication of such a thing in older clusters. Notice that there are relatively few fast rotators in Praesepe and, at least down to ≈0:5 M_ʘ , apparently none in clusters older than 750 Myr, such as NGC 6811 (1000 Myr) and NGC 752 (1350 Myr). Thus, nearly all 0.5 - 0.8 M_ʘ stars seem to merge into a single mass dependent rotational evolutionary sequence sometime after 750 Myr.

The relative position of the rotational period distributions of stars from the Pleiades, M 35, Prae-sepe, NGC 6811 (1000 Myr), NGC 752 (1350 Myr) and M 67 (3500 Myr) open clusters leaves no doubt that there is an evolutionary sequence. Stars from the oldest open cluster depicted in this figure, M 67, do not draw a similarly clear continuous line as the Pleiades stars, and the upper boundary (long periods) drawn by Kepler field stars is not as well defined. An interesting point is that α Cen A and B, which are ≈1000 older than M 67, are close to the site occupied by stars from this open cluster. This indicates that once they merge into a single rotational sequence, angular momentum evolution of single solar type stars (mass between 0.9 and 1.1 M_ʘ) is weakly dependent on binarity when the orbital period is sufficiently long (79 years in α Cen AB). Additionally, it implies that rotational braking of solar type stars is increasingly ineficient as they grow old. The second conclusion is strongly supported by the fact that there is hardly any difference between the rotational period of these stars and 16 Cygnus A and B, which are ≈3000 Myr older than the M 67 cluster. It is worth noting that the long-term rotational evolution of the Sun as a single star does not seem to have been affected in a major way by our planetary system. It remains to be seen which conditions are necessary for this to happen.

3.2. Rigid Body Angular Momentum as a Function of Mass and Time

Since angular momentum and rotational energy are the physical quantities that change during stellar evolution, we can not expect a clearly defined signature for the mass and time dependence of the mean rotational period at the stellar surface. A comprehensive and definite knowledge of these quantities - angular momentum and energy - is still beyond reach, since little is known on the radial dependence of the angular velocity as a function of stellar mass and age. Assumptions have to be made, and these are greatly influenced by what we know about the Sun, which is well advanced into its rotational history (as evinced in Figure 2).

Helioseisimology has shown that the radiative core of the Sun rotates rigidly at a rate of 431 nHz (period close to 27 days) at least down to ≈0:15 R_ʘ (^{Kozennik & E -Darwich, 2012}; ^{E -Darwich & Korzennik, 2013}). Taking a radial average, the convective envelope rotates at a similar rate (or slightly higher according to ^{Tomczyk, Schou & Thompson 1995}). Since the outer layers are slowed down by magnetic braking, this signifies that angular momentum transport from the radiative core to the convective envelope has been equally efficient for some time.

But not much is known on the internal rotation of other stars, and ^{Lund, Miesch & Christensen-Dalsgard (2014)} argued that asteroseismological measurements of Sun-like stars cannot result in clean-cut inferences on the radial pro le of the rotational period. Even so, ^{Nielsen et al. (2017)} concluded that the interior and envelope rotation rates cannot differ by more than ≈30% in 5 solar-like stars (insofar as mass and age are concerned), and ^{Collier Cameron et al. (2009)} claim that F, G and K main sequence stars rotate as rigid bodies after ≈600 Myr. Fully convective main and pre-main sequence stars are expected to rotate as rigid bodies, and ^{Charbonnel et al. (2013)} showed that the rotational frequency of a solar mass ZAMS star peaks at ≈0:25 R_ʘ and is approximately constant beyond ≈0:5 R_ʘ , if angular momentum is redistributed by internal gravity waves.

Modeling the rate at which angular momentum is lost at the stellar surface, and examining the constraints on internal angular momentum transport which can be inferred from the 0.2 M_ʘ mass bin period distributions of several clusters, ^{Denissenkov et al. (2010)} concluded that 0.4 ≤ M/M_ʘ ≤ 1.2 stars with an initial rotation period of less than ≈2 days, will rotate as rigid bodies during most of their main sequence evolutionary stage. But if the initial period is between 2 and 4 days, the core-envelope coupling timescale will be 50 ± 25 Myr when M = 1.0 ± 0.1 M_ʘ , or 175 ± 25 Myr when M = 0.8 ± 0.1 M_ʘ . More ex-tended periods lead to longer coupling times. Notice that by the time they reach the zero age rotational main sequence, ≈120 Myr after birth, the rotational period is close to 10 days when M/M_ʘ ≈ 0.7 and less than 5 days if M/M_ʘ > 1 (see Figure 2). Obviously, the initial rotational periods had to be smaller.

From the preceding paragraphs, it seems that a rigid rotation model may be not too far-o from reality. It may even be an almost precise description of stars that are older than a few hundred Myr, but not too close to the end of their main sequence life-time. Needless to say, rigid body rotation is also the simplest way to carry out an inspection of an-gular momentum evolution. The rest of this paper will discuss this subject, under the hypothesis that late-type main sequence stars rotate as rigid bodies. Arguing that there seems to be moderate differential rotation between core and envelope in stars older than 100 Myr, ^{Johnstone et al. (2015}a) introduced the rigid body assumption into the magnetic braking torque formula derived by Watt et al. (2012), to estimate the evolution of rotation and winds. In this work, the solid body assumption is used to determine the mass and time dependence of the angular momentum and, thereon, develop very simple equations for the evolution of the Rossby number (a surrogate for magnetic activity), the torque, and the mass loss rate.

The product J = I Ω= 2π I/P - where J is the angular momentum if the star is indeed rotating as a rigid body, I is the moment of inertia of the whole star as given by the YaPSI model,Ω is the observed angular frequency and P is the rotational period - is plotted as a function of stellar mass in Figure 3. For the same reason as in Figure 2, only open cluster stars with errors less than 10% in their observational parameters are shown in this figure. Take notice on the following:

Empirical relations between rigid body angular momentum and age for a specific mass can be found using either one or a combination of the Pleiades, Praesepe and terminal main sequence isochrones.

The simplest and more often used relation is an inverse square root law (a Skumanich-like relation). Choosing the Praesepe isochrone as the anchor line,

with J_pre as given in Equation 3 and t in Myr. Notice that this solution assumes a time invariant shape for the angular momentum distribution as a function of mass, in this case de fined by the Praesepe open cluster stars. Comparing Equations 2 and 3, we can see that this assumption is not verified in the Pleiades star cluster, and we can expect poor results when Equation 6 is applied to stars that are somewhat younger than the Praesepe cluster. On the contrary, the similar shape of the Praesepe and terminal (for M/M_ʘ ≥ 0.9) angular momentum distributions, implies that satisfactory results can be anticipated when Equation 6 is applied to older solar type stars.

Obviously, better adjustments valid for slowly rotating stars older than 120 Myr, can be obtained from three parameter solutions involving the three isochrones, such as the following exponential law,

where t is in Myr, and A; B and C are mass dependent constants. These are given in Table 5 for a few selected masses. The table includes two e-folding times for the angular momentum, Τ₀ and Τ₁₂₀ (in Myr), i.e., the time it takes to reduce it by a factor equal to 1/e after the birth line, t = 0, and after the age of the Pleiades, t =120. The e-folding time after the birth line may be deceptive, since there is no reason why we should expect that the power law approximation can be extrapolated to stellar ages that are less than 120 Myr. The e-folding time after the Pleiades rotational isochrone shows that the torque is increasingly vigorous in more massive stars. An interesting result is that after spending ≈13 to 19% of their main sequence lifetime (≈ 15% after discounting the Pleiades age), all 0.5 to 1.2 M main sequence stars loose ≈85% of the angular momentum they had when they were 120 Myr old.

Since a simpler exponential relation, J = A exp(Bt), can be summoned for fast rotators with a saturated magnetic field (^{Chaboyer et al. 1995}) and a fixed moment of inertia (^{Dennisenkov 2010}; ^{Amard et al. 2016}), it is worth noticing that such a relation is completely at odds with the data when it is calibrated with the Pleiades and Praesepe isochrones. Thus, the stars defining these isochrones are not in the saturated regime and, obviously, are not fast rotators.

The mean and minimum stellar angular momenta found in the open clusters listed in Table 1 are displayed in Figures 4 and 5. By reason of insufficient data, stars less massive than 0.65 M_ʘ are not considered. Since there is a limited amount of information for clusters older than ≈1000 Myr, main sequence stars with ages derived from asteroseismological data modeling are included. Listed in Table 6, some of these stars are binaries or host planetary systems (as the Sun, which is also included). The inverse square root and exponential relations between angular momentum and age have been added to these figures (continuous line, equation 6; dotted line, equation 7).

Fig. 4 Rigid body angular momentum (J47, 10⁴⁷ gr cm² s ¹) as a function of time for stellar masses between 0.6 and 1.2 M in 0.1 M intervals. Mean and minimum angular momentum values for the open cluster stars listed in Table 1 are shown in black and red. Single stars from a cluster or from the group of stars listed in Table 6 are shown in blue. The Sun is represented with a red circle and the green triangles in the right hand side figures stand for the angular momentum of the terminal rotational main sequence. The continuous and dotted lines are the inverse square root (green) and exponential fits (magenta) to the angular momentum of the lower and upper mass limits written at the top right hand side of these figures. The color figure can be viewed online. 

Fig. 5 Same caption as in Figure 4. The color figure can be viewed online. 

These figures show that most of the angular momentum of all late type main sequence slow rotators, is lost during their first ≈500 Myr. This implies that the torque is exceptionally strong during this period. Thereon, the torque must be much more moderate, since angular momentum is lost in a very sedate fashion. As expected, the inverse square root relation de-fined by the Praesepe isochrone is inconsistent with the mean and minimum values of the angular momentum in clusters that are younger than 450 Myr. For these clusters, the exponential t is more often than not compatible with the minimum value of the angular momentum. When M/M_ʘ ≥ 0:9 and the cluster age is larger than ≈450 Myr, both relations are consistent with the mean value of the angular momentum, and their difference is usually smaller than the observational uncertainties. This statement is also valid when stars are older than 750 Myr, and their mass is somewhat larger than 0.8 M_ʘ . With a couple of exceptions, these observations can be applied to stars with asteroseismological data.

According to equations 6 and 7, the rigid body angular momentum of a 4500 Myr solar mass star should be 2.1 and 2.2 10⁴⁸ gr cm² s¹, only 10% larger than the rigid body angular momentum of the Sun, 1.9 10⁴⁸ gr cm² s¹. Since scaling is implicitly included, the similarity with the power law t is not surprising. On the other hand, the semblance with the inverse square root relation is suggestive, since no scaling is built into it. These equations can also be used to produce an age estimate using the rigid body angular momentum; in the solar case the result is 5500 and 5600 Myr.

Thus, the rigid body angular momentum (as well as the rotational period) gives an indication of the evolutionary stage of any main sequence star, but the observational database shows that it can not be used to determine with any degree of precision the age of F and G main sequence stars younger than ≈450 Myr or K type stars younger than ≈750 Myr. This limit is likely to be substantially higher for later spectral types. On the other hand, the exponential relation supported by the Pleiades, Praesepe and terminal isochrones for the rigid body angular momentum, or the simple inverse square root relation based on the Praesepe isochrone, may provide acceptable but not too precise estimates for the ages of older stars.

3.3. Rotational Period as a Function of Mass and Time. Second Round

The rigid body angular momentum depends on the detected rotational period and the unobserved moment of inertia (J = 2πI/P ). Since the moment of inertia is mass, metallicity and time dependent, the relation between rotational period and age should be more complicated.

The metallicity dependence of the moment of inertia is weak, but not insignificant. For other-wise identical conditions, higher metallicities pro-duce larger moments of inertia, with an up to ≈10% difference between the highest ([F e=H] = 0.3, Y₀ = 0:31) and lowest ([F e=H] = -0.5, Y₀ = 0:25) values considered by YaPSI, the gap being smaller for lower mass stars.

As mentioned before, the time dependence of the moment of inertia can be substantially more important. The evolution of the moment of inertia of 0.5 - 1.2 M_ʘ stars with a solar composition was depicted in Figure It shows that it declines very rapidly at the beginning of the main sequence. After 100 Myr or so it settles at a relatively stable value that lasts until roughly half the main sequence lifetime. Dur-ing this stage, the moment of inertia of ≈0:4 to 1.2 M_ʘ stars is approximately equal to its 1000 Myr old value. YaPSI models for 1000 Myr old stars with [F e=H] = 0.0 and Y₀ = 0.28, lead to the following t to the moment of inertia for a stellar mass range between 0.5 and 1.2 M_ʘ ,

The regression coefficient and the square root of the mean square error are equal to 0.98 and 0.09. The metallicity effect may add an additional uncertainty of ≈5% in the high mass limit, less than this in lower mass stars. For a one solar mass star, the moment of inertia predicted by this Equation is 6:45x10⁵³ gr cm², somewhat less than the precise value, 6:71 x 10⁵³ gr cm², and appreciably smaller than the moment of inertia of a 4500 Myr old one solar mass star, 6:95 x 10⁵³ gr cm².

Equation 8 and the inverse square root relation between angular momentum and age lead to the following direct connection between age and rotational period during the time span where the moment of inertia is approximately constant,

This equation returns a disappointing age of 6600 Myr for the Sun. The exponential t leads to a more intricate relationship, but gives the same age for our star, i.e., ≈1000 Myr more than the age estimate based on the rigid body angular momentum.

In the second half of their main sequence lifetime, stars build an increasingly larger moment of inertia and the mean rotational period will increase even in the absence of rotational braking. Notice that the consequence on surface rotation may be more pronounced, since differential rotation may result as the core contracts and spins up and the envelope expands and slows down. These effects are still insignificant in lighter stars, but gain in importance in 1.2, 1.1, 1.0 and 0.9 M_ʘ stars as soon as they are ≈ 1400, 3600, 4500 and 7900 Myr old. For instance, without rotational braking, the solar period will be around 32, 38 and 68 days when our star is 8000, 9000 and 10000 Myr old. The rotational period would be equal to 38, 47 and 90 days if the evolution of angular momentum resembles the inverse square root relation (Equation 6) or 39, 49 and 95 days if it is described by the exponential law (Equation 7). Since the longest rotational period of the one solar mass stars included in this paper is 63 days, these numbers suggest that other drivers of rotational braking (mass loss, magnetic field intensity, wind acceleration) are secondary agents in the evolution of stellar periods during the second half of the main sequence. Actually, there may be some evidence for inefficient magnetic braking in the Sun, where the high latitude wind was found to be super- Alfvenic close to the solar surface (^{McComas et al. 2000}).

3.4. Magnetic Activity as a Function of Mass and Time

Stellar dynamos sustain poloidal-toroidal magnetic fields (the α-Ω mechanism) as long as Coriolis forces dominate inertial forces within the convective region (^{Durney & Latour 1978}). The relation between these forces is quantified with the Rossby number, which can be defined as

where τ_c is the convective turnover time. A strong argument for the conjecture that the magnetic field of main sequence stars with a convective envelope is produced by the α-Ω dynamo mechanism is based on the observation that the X-ray and chromospheric to bolometric luminosity ratios are proportional to Ro-2 o in the non-saturated regime (^{Noyes et al. 1984}; ^{Pizzolato et al. 2003}). This thesis has been disputed by ^{Reiner, Schüssler & Passegger (2014)}, arguing that these relations hold if the convective overturn time scales as Lbol-1/2. Since both quantities are roughly constant during most of main sequence, this proportionality should be true for any star during this relatively stable time period. On the other hand, the proportionality constant may depend on the stellar mass. Interestingly, during this stable phase Lbol1/2×τc≃30(L_bol in solar units, τ_c in days) if 0.4 ≤ M/M_ʘ ≤ 1, close to 20 if M/M_ʘ = 1.1 and around 7 when M/M_ = 1.2. Admittedly, the convective turnover time provided by YAPSI is associated to the tachocline, not to the position where the dynamo is more effective. Even so, these numbers suggest that the Parker dynamo mechanism has an effect on magnetic braking, at least in non-fully convective stars less than, or as massive as, the Sun. Whichever the case, ^{Reiner et al. (2014)} rightly emphasize the need to explore and consider a wider range of mechanisms for the generation of magnetic fields in low mass main sequence stars.

Thus, assuming that the α-Ω mechanism is operative, magnetic activity and rotational braking will tend to be suppressed when the Rossby number is large. It has been argued that this explains the existence of anomalously rapid rotation in some old field stars (^{van Saders et al. 2016}), and it may be the reason why KIC 4914923, KIC 5184732 and KIC 10963065 have such an atypically large angular momentum (see Table 6).

The value of the convective turnover time depends on the convection parameter (usually taken as equal to the solar value, α = 1.875) and the assumed depth of the region where the dynamo is being generated. Under the same circumstances, lengthier convection times and smaller Rossby numbers are obviously associated to dynamos running closer to the tachocline but, except for the scale difference, the mass and time dependence of the convective turnover time does not seem to be affected by this supposition (^{Landin, Mendes & Vaz 2010}).

According to YaPSI models with solar-like composition ([Fe/H] = 0.0 and Y₀ = 0.28), the convective turnover time of all late type main sequence stars is equal to a few hundred days during the first 10 to 100 Myr. Later on, it remains roughly constant, almost up to the end of main sequence. Consequently, as stars spin down during their main sequence evolution, the Rossby number will increase, the α - Ω mechanism will weaken and magnetic braking will become increasingly inefficient. Since the convective turnover time during this stage is ≥ 100 days when M ≤ 0:6 M_ʘ, and ≈70, 55, 40, 30, 15 and 5 days when M = 0.7, 0.8, 0.9, 1.0, 1.1 and 1.2 M_ʘ, the demise of the dynamo effect may begin significantly earlier in stars with shallow convective regions.

The history of the Rossby number for 0.5 to 1.2 M_ʘ stars was determined using the exponential and inverse square root approximations for the evolution of the angular momentum (similar results are obtained). In these calculations the Rossby number of a 4500 Myr old one solar mass star is equal to 0.87. This is significantly less than an often quoted value (2.16 in ^{van Saders et al. 2016}) but, as discussed above, part of this difference is probably related to the assumed depth of the dynamo producing region. Thus, depending on the precise definition of the convective turnover time, either one of these numbers (both close to one) can be taken as the Rossby number where magnetic activity and rotational braking may begin to decline.

The evolution of the Rossby number is shown in Figure 6. Notice that R_o ≥ 1 for 1.2, 1.1, 1.0 and 0.9 M_ʘ stars once they are older than ≈500, 2500, 5500, and 9500 Myr. In 1.2 and 1.1 M_ʘ stars, the cessation of the dynamo mechanism may occur be-fore there is a significant change in their moment of inertia. With the un-anticipated exception of 1.2 M stars, the possible disruption of the magnetic dynamo occurs much later than the T₁₂₀ e-folding times for the angular momentum.

Fig. 6 Evolution of the Rossby number during the main sequence stage, assuming that the angular momentum is as given in equation 6. Dotted lines stand for M/M_ʘ = 0:5 (green), 0.6 (black), 0.7 (red) and 0.8 (blue). Continuous lines stand for M/M_ʘ = 0:9 (green), 1.0 (black), 1.1 (red) and 1.2 (blue). The color figure can be viewed online. 

^{Kitchainov & Nepomnyashchikh (2017)} had noticed that the interruption of large scale dynamos may be the reason why gyrochronology fails to predict the age of older stars. If this is correct, their surviving magnetic fossil field may be the main rotational braking source during the time interval be-tween the fading α-Ω mechanism and the swelling moment of inertia. Later on, the transformation of the moment of inertia will eventually determine the evolution of the rotational period. In less massive stars, dynamo activity will decline at the same time as the moment of inertia escalates, and their continuously changing relative importance will impress the history of the rotational period during the second half of the main sequence stage.

3.5. Torque as a Function of Mass and Time

Differences in angular momentum between pairs of isochrones with respect to the youngest of the two, are displayed on the left hand side of Figure 7 (associated formulas are shown in this figure). The corresponding mean loss rates per Gyr are shown on the right hand side.

Fig. 7 On the left, differences in the angular momentum between the Pleiades, Praesepe and terminal isochrones, relative to the angular momentum of the younger one (Δ = Δ). On the right, logarithm of the mean relative angular momentum loss rate per Gyr, log(Δ J//J Δ t), for the time interval between the Pleiades and terminal isochrone (continuous line), the Praesepe and terminal isochrones (dotted line) and the Praesepe and Pleiades isochrones (short-dashed line). 

Close to 60% and 25% of the initial rigid body angular momentum of 1.2 and 0.5 M_ʘ slowly rotating stars is lost during the few hundred million years separating the Pleiades and Praesepe isochrones. At this stage, the mean loss rate is around 90% and a bit more than 40% per Gyr for 1.2 and 0.5 M stars. Loss rates are much smaller after ≈750 Myr. Near the end of their main sequence lifetime, 88% and 86% of the initial angular momentum of 1.2 and 0.9 M_ʘ stars has been lost. At this point in time, the oldest 0.7 and 0.5 M stars have lost 80% and 73% of the angular momentum they had when landing on the Pleiades isochrone. From these two figures, it is once again clear that the relative and absolute rigid body angular momentum loss rates are always appreciably larger in shorter-lived, more massive, stars.

The inverse square root and exponential ts to angular momentum evolution (Equations 6 and 7), lead to the following expressions for the angular momentum loss rates (i.e., the torque τ = dJ/dt) acting on slowly rotating late type main sequence stars older than approximately 120 Myr,

with t in Myr, and constants A, B and C as given in Table 5. Notice that in both cases the torque can be re-written in the following way

with K = 1=2 and γ = -1 in the inverse square root relation, and K = BC and γ = C-1 in the exponential approximation.

These torques are usually smaller than those where the radiative core is spinning faster than the envelope, since there is less angular momentum when the entire star rotates at the surface rate. This can be visualized in Figure 8. Beyond 100 Myr, both of them are up to a factor of 5 smaller than the torques produced by a couple of models for the slowly rotating branch of one solar mass stars, where solid body rotation is not assumed and angular momentum is transferred from the radiative core to the convective envelope in two different ways (^{Amard et al. 2016}).

Fig. 8 Absolute value of the specific torque per year, τ_s, as a function of time (Myr), for a stellar mass equal to 1.0 M_ʘ . The continuous (sqrt) and dotted (exp) lines are the inverse square root and exponential ts as given by equations 11 and 12. The blue (GB13) and red (9s) tracks are from ^{Amard et al. (2016)}. The color gure can be viewed online. 

The temporal evolution of the absolute value of the torque under the inverse square root and exponential approximations is plotted in Figure 9. As can be seen, there is a conspicuous difference between the inverse square root and exponential approximations when stars are younger than a few hundred million years. This disparity extends for a longer time and is more pronounced in low mass stars. Since an in-verse square root law is at odds with the observed evolution of angular momentum in young stars, particularly young low mass stars (see § 3.2), it follows that equation 11 cannot be a close description of the torques applied to these objects. On the other hand, the exponential t is a reasonably close approximation to the angular momentum evolution of young stars, so it should lead to a more accurate depiction of the real torque for slowly rotating stars that are at least as old as the Pleiades cluster, as long as radial differential rotation is nearly absent. Notice that the magnitude of the torque at any given age is much smaller when the exponential t is used, and that the demands imposed by a smaller torque are more easily met.

Fig. 9 Absolute value of the torque in units of 10⁴⁶ and 10⁴⁴ gr cm² s ¹ Myr ¹ ( 46 and 44) as a function of time, for stellar masses equal to 0.8, 1.0 and 1.2 M . The upper age limit on the left hand side figures is 750 Myr, and the lower age limit on the right hand side figures is 500 Myr. The continuous and dotted lines are torques produced by the inverse square root and exponential fits to the evolution of the angular momentum (equations 11 and 12). 

Figure 9 also confirms that the torque is very powerful and variable at least during the first ≈500 Myr. After dropping by a couple of orders of magnitude, it is increasingly weaker and stable. Thus, it seems that at least one of the agents driving angular momentum losses, be it the magnetic field strength and/or the mass loss rate, is extremely robust and mutable during the early stages of main sequence evolution. Later on, when the torque is less effective and variable, at least one of these agents must be considerably more stable and moderate. No-tice that after ≈500 Myr, the torques inferred from inverse square root and exponential relations for the angular momentum are almost indistinguishable.

3.6. A Simple Model for the Torque and Mass Loss Rate as a Function of Time

These empirical approximations and an idealized model of the torque, can be used to explore the mass loss rate as stars evolve during the main sequence. At any given point and time, the torque is given by

Where M is the mass loss rate, Ω is the rotational frequency and the lever arm R_c - known as the co-rotation or Alfven radius - is the distance between the stellar surface and the point where angular momentum is being lost. The co-rotation radius is determined from the proposition that angular momentum is lost when the stellar wind is detached from the magnetic field. This happens when fluid pressure equals magnetic pressure,

where B_c is the magnetic flux density at R_c. Since the mass loss rate and the magnetic field strength are latitude, longitude and time dependent, and the angular velocity is latitude and time dependent, it follows that the co-rotation radius cannot be uniform and stable. Thus, a continuously changing crumpled ball of paper - definitely not a smooth and steady regular figure - is a plausible visual representation of the co-rotation surface.

The detailed configuration of the magnetic field above the stellar surface has long been recognized as a very di cult problem (^{Mestel 1968}), since it is an extremely complex mixture of small, medium and large scale dynamic structures rising through-out the convective envelope (e.g., ^{Lang et al. 2014}).

But it is worth remembering that the dipole is the constituent with the slowest radial decay and there-fore presides over the strength of the magnetic field at the co-rotation radius, as was shown by ^{Finley & Matt (2018)} after analyzing the combined effect that dipolar, quadrupolar and octupolar geometries had on the magnetic braking mechanism.

In the present day Sun, magnetic fields are produced close to the tachocline, the thin layer between the radiative core and the convective envelope (^{Charbonneau 2010}). But different processes are involved in other stars. In fully convective stars, magnetic fields cannot be generated by the classical process, and it is possible that small scale magnetic fields are produced at various depths by turbulent velocity fields (^{Durney, De Young & Rox-burgh 1993}). Additionally, strong toroidal azimuthal fields can show up directly at the stellar surface of stars with a mass between 0.1 and 1.5 M_ʘ , and it has been suggested that these are produced by dynamos distributed throughout the convection zone (^{Donati et al. 1992}; ^{Donati & Collier Cameron 1997}; ^{See et al. 2015}), an idea supported by ^{Brown et al. (2010)}. Thus, different magnetic energy sources can be located relatively close to the stellar surface but not beyond the tachocline (if there is one).

An extreme, manageable and sensible simplication is to assume that the magnetic field is produced by a collection of dipoles placed at various distances h under the stellar surface. If h is much larger than the physical size of each dipole,

where B_s is the dipole’s magnetic field density emerging at the stellar surface. Magnetic braking is effective if the co-rotation radius is at least a few stellar radii from the surface, so that it is quite likely that h=R_c < 0:01. If this is so, and the stellar surface is covered by any number of magnetic active regions produced by identical dipolar fields, the total torque is

where the filling factor, f_s, is the area covered by the regions that contribute to rotational braking, over the stellar surface area. Notice that the sur-face field, the mass loss rate and the angular velocity (B_s; M and Ω) are averages of these quantities over a time period where all longitudinal and latitudinal fluctuations have been smoothed out, i.e., the twisted short-time dependent co-rotation surface has been idealized as a smooth long-time dependent spherical figure. This implies that the long-time period must comprise several stellar cycles and, even so, it is almost certain that the distance between the stellar and co-rotation surfaces depends on latitude. Consequently, these averages cannot be expected to be equal or similar to values obtained from observations, since these are completed in shorter time spans.

Mean values of the total field strength can be obtained by measuring Zeeman splitting of unpo-larized spectral lines, a procedure known as the ZB technique. This method provides no information on the magnetic field geometry, but it includes the contribution of magnetic field structures of all magnitudes and sizes. Notice that some of these structures may not participate in the magnetic braking process. With a data base of close to a couple of dozen late type main sequence stars, ^{Saar (1996}, ²⁰⁰¹⁾ found that, except for the most active, their magnetic field density was close to the photospheric equipartition value (same magnetic and thermal pressures), f_sB_s is between ≈20 and 4000 G and f_sB_s is nearly pro-portional to Ω⁵⁼³ if the rotational period is more than 3 days (the reported exponent is 1.7). The data show that there is an almost linear relation between the X-ray flux and f_sB_s, but there is no mention of possible correlations between magnetic field and age or spectral type.

The ZDI technique has been applied to Stokes V profiles, but in this case the magnetic field flux density, BV, does not include the longitudinal component and small scale elds. Using this tech-nique, ^{Vidotto et al. (2014)} calculated the large scale surface magnetic eld densities of some 60 main sequence stars, and found that BV is no larger than ≈100 G when the spectral type is between F7 and M3, but larger than this and up to 1580 G in later spectral types. From a sample of close to 60 stars, they worked out that is nearly proportional to Ω^4/3 (their exponent is 1.32), though there is a very large scatter and the correlation coeficient is not particularly good. Using age estimates based on different methods, they conclude that BVαt-0.655. They also suggest that small and large scale elds could share the same dynamo generation process, but no mention is made of a detailed connection be-tween spectral type and magnetic field in main sequence stars.

Taking these relations between magnetic field and angular velocity into account,

with β≃4/34/3 for the global field (ZDI technique) or 5/3 when all magnetic structures are included (ZB technique). Combining this with equation 13 for the torque, the evolution of the mass loss rate is given by

where A(M) is a mass dependent function.

Two solutions for the ratio of the mass loss rate at any time, with respect to the mass loss rate at t = 100 Myr, Ṁ(100), are displayed in Figure 10. These figures are for β = 5/3 (very similar figures are obtained when β = 4/3). Being a much better approximation to the evolution of angular momentum during the entire main sequence, the exponential t (equation 7, γ = C 1 and C as given in Table 5) was used to compute the mass loss rate. The moment of inertia is taken from YaPSI models with [F e/H] = 0:0 and Y₀ = 0:28. On the left hand side, the average depth of the magnetic field source (h) is assumed to be constant in time, and is therefore included in the empirical mass dependent function A(M). On the right hand side, h does change in time but not the ratio h=h_t, where h_t is depth of the convective envelope. In this case, h=h_t is included in A(M) and the depth of the convective envelope is an additional mass and time dependent parameter. As the moment of inertia, h_t is taken from YaPSi models with [F e/H] = 0.0 and Y₀ = 0.28.

Fig. 10 Ratio of the mass loss rate at time t, Ṁ^_(t), with respect to the mass loss rate at t = 100 Myr, Ṁ(100). Dotted lines are for M/M_ʘ = 0.5 (green), 0.6 (black), 0.7 (red) and 0.8 (blue). Continuous lines are for M/M_ʘ = 0:9 (green), 1.0 (black), 1.1 (red) and 1.2 (blue). As indicated, the black dashed line shows the 1=t function, which is a good approximation to the mass loss rate when the moment of inertia is constant and the evolution of the angular momentum is described by an inverse square root law. The color figure can be viewed online. 

If this simple model for the torque is basically correct and the magnetic field behaves as has been discussed, these figures call upon these outcomes:

In either one of these scenarios (h or h/h_t constant in time) the mass dependent function, A(M), can be calibrated using a star with known mass, age and mass loss rate. At present, these quantities are known with any certainty for only one star, the Sun. Assuming that it did not have a unique rotational history, the magnetic torque model where h/h_t is constant implies that the present solar mass loss rate is very close to its lowest value (2 - 3 x 10 ¹⁴ M_ʘ yr-¹, ^{Wang 1998}) and, if magnetic braking is still dominant, leads to a mass loss rate that may be 5 - 8 10 ¹⁴ M_ʘ yr^-1 when our star is 9000 Myr old. Under this scenario, the mass loss rate may have been 2 to 3 10 ¹³ M_ʘ yr ¹ when the Sun was around 100 Myr old, if its rotational period was similar to the rotational period of solar type stars in the Pleiades cluster (as discussed in § 3.1, this is not unlikely).

Mass loss rates of a handful of late type main and post-main sequence stars have been calculated from astrospheric absorption in the stellar Lyman-α emission line (^{Wood et al. 2005}; ^{Linsky & Wood 2014}), the only available technique to measure this quantity. There is a good looking correlation between mass loss rate and X-ray flux, as long as the latter is smaller than ≈10⁶ erg cm^-2 s^-1. Combining this correlation with an X-ray flux vs. age re-lationship (^{Ayres 1997}), ^{Wood et al. (2005)} affirm that Ṁ α t^{-2:33 ±0:55} in stars where the X-ray flux is smaller than the aforementioned limit. Except for stars younger than ≈100 Myr, this relationship is at odds with all the estimates that have just been mentioned.

Different explanations can be tried to account for these discrepancies. But the problem may be directly associated to the mass loss rate vs: time relationship put forward by ^{Wood et al. (2005)}, since there is no connection between the mass loss rate and the rotational period of these stars, as can be seen in Figure 11. Under this circumstance, it is unlikely that a relation between mass loss rate and age is hidden in their data set. This result should not be unexpected, since the mass loss rate data set stems from a very diverse group of single and bi-nary stars: giant, sub-giant and G2 to M5.5 dwarf stars. Furthermore, there is no way to know the individual mass loss rates in the binary systems α Cen, 36 Oph, λ And, 70 Oph and ξ Boo, since the size of the modeled astrosphere includes both stars. More measurements of the mass loss rate of late type main sequence stars are required to have a better under-standing of this quantity as a function of mass and age. An important addition to this discussion is the effect that coronal mass ejections may have on the mass loss rate and rotational braking (e.g., Cranmer 2017).

Fig. 11 Rotational periods as a function of mass loss rates. Main sequence stars in black, post-main sequence stars in red. Triangles stand for single star measurements and starred symbols (linked by a line when more than one period has been measured) for measurements including both stars. All mass loss rates are taken from ^{Wood et al. (2005)}. Rotational periods stem from a variety of sources: Proxima Cen (^{Collins, Jones & Barnes 2017}), α Cen A (^{Bazot et al. 2007}), α Cen B (^{Dewarf, Datin & Guinan 2010}), ϵ Eri (^{Fröhlich 2007}), 61 Cyg A, 36 Oph A and B, 70 Oph A and B, 61 Vir and δ Eri (^{Baliunas, Sokolo & Soon 1996}), λ And (^{Kukarkin et al. 1971}), EV Lac (^{Contadakis 1995}), ξ Boo A and B (^{Noyes et al. 1984}) and DK UMa (^{Gondoin 2005}). The color figure can be viewed online.