On the angular momentum evolution of fully-convective stars: rotation periods for field M-dwarfs from the MEarth transit survey

Abstract

We present rotation period measurements for 41 field M-dwarfs, all of which have masses inferred (from their parallaxes and 2MASS K-band magnitudes) to be between the hydrogen burning limit and 0.35 Msol, and thus should remain fully-convective throughout their lifetimes. We measure a wide range of rotation periods, from 0.28 days to 154 days, with the latter commensurate with the typical sensitivity limit of our observations. Using kinematics as a proxy for age, we find that the majority of objects likely to be thick disk or halo members (and hence, on average, older) rotate very slowly, with a median period of 92 days, compared to 0.7 days for those likely to be thin disk members (on average, younger), although there are still some rapid rotators in the thick disk sample. When combined with literature measurements for M-dwarfs, these results indicate an increase in spin-down times with decreasing stellar mass, in agreement with previous work, and that the spin-down time becomes comparable to the age of the thick disk sample below the fully-convective boundary. We additionally infer that the spin-down must remove a substantial amount of angular momentum once it begins in order to produce the slow rotators we observe in the thick disk candidates, suggesting that fully-convective M-dwarfs may still experience strong winds. Comment: 17 pages, 16 figures, 3 tables, emulateapj format. Accepted for publication in ApJ

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arXiv:1011.4909v1 [astro-ph.SR] 22 Nov 2010
Draft version November 23, 2010
Preprint typeset using LATEX style emulateapj v. 03/07/07
ON THE ANGULAR MOMENTUM EVOLUTION OF FULLY-CONVECTIVE STARS: ROTATION PERIODS
FOR FIELD M-DWARFS FROM THE MEARTH TRANSIT SURVEY
Jonathan Irwin, Zachory K. Berta, Christopher J. Burke, David Charbonneau and Philip Nutzman
Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA
Andrew A. West
Department of Astronomy, Boston University, 725 Commonwealth Ave, Boston, MA 02215, USA
Emilio E. Falco
Fred Lawrence Whipple Observatory, Smithsonian Astrophysical Observatory, 670 Mount Hopkins Road, Amado, AZ 85645, USA
Draft version November 23, 2010
ABSTRACT
We present rotation period measurements for 41 field M-dwarfs, all of which have masses inferred
(from their parallaxes and 2MASS K-band magnitudes) to be between the hydrogen burning limit
and 0.35 M⊙, and thus should remain fully-convective throughout their lifetimes. We measure a wide
range of rotation periods, from 0.28 days to 154 days, with the latter commensurate with the typical
sensitivity limit of our observations. Using kinematics as a proxy for age, we find that the majority
of objects likely to be thick disk or halo members (and hence, on average, older) rotate very slowly,
with a median period of 92 days, compared to 0.7 days for those likely to be thin disk members
(on average, younger), although there are still some rapid rotators in the thick disk sample. When
combined with literature measurements for M-dwarfs, these results indicate an increase in spin-down
times with decreasing stellar mass, in agreement with previous work, and that the spin-down time
becomes comparable to the age of the thick disk sample below the fully-convective boundary. We
additionally infer that the spin-down must remove a substantial amount of angular momentum once
it begins in order to produce the slow rotators we observe in the thick disk candidates, suggesting
that fully-convective M-dwarfs may still experience strong winds.
Subject headings: stars: rotation – starspots – stars: low-mass, brown dwarfs – stars: evolution
1. INTRODUCTION
The rotational evolution of low-mass stars is predom-
inantly governed by two competing processes. During
the pre-main-sequence (PMS) phase, these stars are still
collapsing, and thus, the moment of inertia decreases as
a function of time. If angular momentum is conserved,
the angular velocity must correspondingly increase. This
spin-up, which persists until the star reaches the zero
age main sequence (ZAMS), is counteracted by angular
momentum losses, which are thought to be related to
the star-disc interaction at early times (e.g. accretion-
driven winds; Matt & Pudritz 2005, or “disc locking”;
K¨ onigl 1991; Collier Cameron, Campbell & Quaintrell
1995), and stellar winds at late times, particularly after
the star reaches the ZAMS.
It is well-established that for solar-type stars, it is
possible to reproduce the observed evolution reasonably
well within this framework, taking as theoretical inputs
a range of disc lifetimes in reasonable consistency with
those observed in young clusters, a wind loss law involv-
ing saturation of the angular momentum losses past some
critical angular velocity ωsat(e.g., Stauffer & Hartmann
1987; Barnes & Sofia 1996), and allowing the radiative
core to decouple in angular velocity from the convective
envelope (e.g., Krishnamurthi et al. 1997; Allain 1998;
Denissenkov et al. 2010). A key feature of this solar-type
evolution is that the spread in disc lifetimes gives rise to a
Electronic address: jirwin -at- cfa.harvard.edu
spread in rotation rates in young clusters, with the max-
imal rotation rate being attained as the stars reach the
ZAMS, after which they begin to spin down, and con-
verge toward a narrow range of rotation rates. Around
the age of the Hyades (625 ± 50 Myr from isochrone fit-
ting; Perryman et al. 1998), the convergence is complete
and all the stars follow a t1/2type spin-down thereafter,
with rotation rate being a well-defined function of mass
and age for F, G and K stars (Barnes 2003, 2007).
Stellar winds depend on the magnetic dynamo that
drives field generation, so the magnetic topologies of the
stars are important in determining the angular momen-
tum loss rates. For solar-type stars, this is thought to
be an αΩ dynamo (where α and Ω refer to the “α ef-
fect”, the twisting of the magnetic field lines caused
by rotation, and the “Ω effect”, driven by differential
rotation) operating at the interface between the radia-
tive core and the convective envelope.
convective star, it is thought that this can no longer
operate, and it has been suggested that a turbulent dy-
namo (Durney, De Young & Roxburgh 1993) or α2dy-
namo (R¨ adler et al. 1990) may dominate. The former
produces small-scale fields that would yield inefficient an-
gular momentum losses through winds.
In recent years, magnetic field measurements of stars
on both sides of the fully-convective boundary have
become available (e.g., Donati et al. 2008; Morin et al.
2008, 2010; Reiners & Basri 2009). These studies indi-
cate there is indeed an abrupt change in field geometries
For a fully-

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2 Irwin et al.
moving across the fully-convective boundary, with fully-
convective objects storing more magnetic flux in large-
scale fields than partially-convective objects. The effect
of these changes on winds is not yet clear, but at any rate,
we do not necessarily expect that the same wind formal-
ism that works for solar type stars or partially-convective
M-dwarfs should necessarily reproduce the observations
for fully-convective M-dwarfs.
While several authors have attempted to extend the
analysis of rotational evolution to M-dwarfs, and par-
ticularly to masses below the fully-convective bound-
ary (≈ 0.35 M⊙; Chabrier & Baraffe 1997), the lack
of observations in this mass domain has made such
an endeavor problematic. While observations in young
open clusters have improved dramatically over the
last decades, now yielding large samples of rota-
tion periods below the fully-convective boundary (e.g.,
Stassun et al. 1999; Herbst et al. 2001; Makidon et al.
2004; Lamm et al. 2005; Scholz & Eisl¨ offel 2004a,b,
2005, 2007; Scholz, Eisl¨ offel & Mundt 2009; Cohen et al.
2004; Littlefair et al. 2005, 2010; Cieza & Baliber 2006;
Irwin et al. 2007b, 2008a), these extend typically only to
a few 100 Myr, and at present only a handful of periods
have been measured for older clusters, where the intrin-
sic faintness of M-dwarfs makes determination of peri-
ods difficult. Furthermore, such objects reach the ZAMS
at much later ages (a few hundred Myr; Baraffe et al.
1998) than solar-type stars, so older clusters are needed
to probe similar stages in the evolution.
Additionally, the Sun has been used in all the studies of
solar-type stars as a reference point to tie down the evo-
lutionary models, particularly the normalization of the
wind loss law, which relies on having data at late-times.
For fully-convective stars, there is as yet no such object:
a star with well known mass, age, and a robustly mea-
sured rotation period (although Proxima comes closest to
providing it; see later in this section). The best progress
below the fully-convective boundary toward determin-
ing the behavior at late times has been made through
v sini observations of field stars (e.g., Delfosse et al.
1998; Mohanty & Basri 2003; Reiners & Basri 2008;
Browning et al. 2010). While this information is invalu-
able, and has led to important insights into the physical
processes at play in these stars, it is important to note
that v sini observations are not sensitive to the slowest
rotators, with typical spectral resolutions yielding a limit
of 3 km s−1. Due to the small radii of M-dwarfs, this cor-
responds to rather short rotation periods, e.g. 3.3 days
for a 0.2 R⊙star.
Rotation period measurements of field M-dwarfs have
provided a few clues as to what might lie below the sensi-
tivity limits of the v sini surveys. Benedict et al. (1998)
report a rotation period of 83 days for Proxima Centauri,
a star which is highly likely to be fully-convective (they
also report a weaker detection of a ≈ 130 day periodicity
in Barnard’s star). Kiraga & St¸ epie´ n (2007) presented
a comprehensive survey of rotation periods for field M-
dwarfs, and whilst their sample is limited mostly to ob-
jects above the fully-convective boundary (they present
periods for two new objects below it), they were able to
confirm a long period for Proxima, obtaining 82.5 days.
Hartman et al. (2009b) also report a number of long rota-
tion periods for field M-dwarfs from the HATNet transit
survey. Finally, we see evidence for a comparably long
rotation period in the transiting exoplanet host GJ 1214
(Charbonneau et al. 2009).
The MEarth transit survey (Nutzman & Charbonneau
2008; Irwin et al. 2009) targets nearby (< 33 pc) north-
ern hemisphere mid to late M-dwarfs to search for tran-
siting super-Earth exoplanets in the habitable zones
of their parent stars.The targets for this survey
are selected to have inferred radii < 0.33 R⊙ be-
cause this is highly advantageous for a transit survey
(Nutzman & Charbonneau 2008), and thus should all lie
below the fully-convective boundary. Each star is typi-
cally observed at moderate (20 minute) cadence on every
clear night for a long period of time (≈ 6 months with
the current implementation of the survey).
therefore possess favorable sampling to search for long
rotation periods.
The data
2. OBSERVATIONS AND DATA REDUCTION
MEarth is a targeted survey, and thus, has observa-
tional properties rather different from the majority of
transit surveys. We summarize the salient features in
this section, particularly as they relate to the detection
of continuous photometric modulations, rather than the
discrete transit events for which the survey was designed.
The data for the present work were gathered us-
ing all 8 telescopes of the MEarth array, which is
located within a single roll-off roof enclosure at the
Fred Lawrence Whipple Observatory on Mount Hop-
kins, Arizona. Data were from two full observing sea-
sons, 2008/2009 and 2009/2010, which run from approx-
imately mid-September to mid-July, where the observa-
tory is shut down during the remaining ≈ 2 month period
corresponding to the summer monsoon season in south-
ern Arizona. Each star was generally observed during
only one of these two seasons, as our target list is cycled
annually to increase the sample size searched for tran-
sits. During each year of operations, hardware changes
were minimal except for necessary repairs, and we used
a fixed 715 nm long pass filter combined with a thinned,
back illuminated 2048 × 2048 CCD on each telescope.
The pixel scale is 0.′′76 per pixel, yielding a field-of-view
of 26′on a side.
Targets for the MEarth survey were selected by
Nutzman & Charbonneau (2008) from a subsample of
stars from the LSPM-North catalog (L´ epine & Shara
2005) with trigonometric parallaxes or spectroscopic or
photometric distance moduli indicating they are within
33 pc (L´ epine 2005).
For the purposes of this work, it is important to be able
to, at a minimum, assign a reliable estimate of mass or
spectral type to each target, and produce some estimate
of the age. The majority of the MEarth targets are quite
poorly characterized from existing data in the literature,
often possessing only near-IR JHK magnitudes from the
2MASS all-sky survey, photographic magnitudes from
the Palomar Sky Surveys1, and proper motion informa-
tion. We therefore elected to analyze only the ≈ 1/3 of
the sample with trigonometric parallaxes. There were
273 such stars possessing more than one observation (in
practice, more observations are of course needed to detect
1Note that although L´ epine & Shara (2005) provide V -band
magnitude estimates, the majority of these are from photographic
plate measurements from the USNO-B1.0 catalog, and therefore
have potentially large systematic uncertainties.

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Rotational evolution of fully-convective stars3
a period; the smallest number of measurements for our
selected rotation candidates was 207 for GJ 51, see also
§3.2). The combined parallax and K-band magnitude in-
formation provides one of the best methods to estimate
the masses of single field M-dwarfs (Delfosse et al. 2000),
and in combination with the proper motion information,
can be used on a statistical basis to constrain the Galac-
tic population to which the target belongs, and hence the
age. Note that full kinematic information is not available
for the majority of our targets as they lack radial veloc-
ities, so this population assignment is necessarily quite
crude at the present time.
During the night, each telescope observes 20−30 fields,
where the vast majority of fields contain only a single
target M-dwarf. These are observed sequentially for the
entire time they are above airmass 2, cycling around the
list to yield a cadence per-field of ≈ 20 minutes. Expo-
sure times are selected to yield a signal to noise ratio for
the target star sufficient to detect a 2 R⊕ planet tran-
sit at 3σ per data point, for the assumed stellar radius,
which is based on the Delfosse et al. (2000) K-band mass-
absolute magnitude relation, and a polynomial fit to the
empirical mass-radius data of Ribas (2006) to convert
mass to radius.
Basic reductions and light curve generation were per-
formed using an automated pipeline, based on that used
by the Monitor project (Irwin et al. 2007a). A number
of instrument-specific refinements have been made, and
these will be described in full in a forthcoming publica-
tion (Berta et al., in preparation), along with our transit
search procedures. There are, however, known to be two
important systematic effects remaining after this stan-
dard processing, which we will now describe as they are
important for rotation period detection.
MEarth uses German Equatorial Mounts, which ne-
cessitate effectively rotating the telescope through 180◦
relative to the sky when crossing the meridian. Thus,
each target samples two regions of the detector, one for
negative hour angle and one for positive hour angle. Flat
fielding errors are manifest as different base-line magni-
tudes on each side of the meridian, so we solve for a
“meridian offset” for each object to remove this effect.
Secondly, we discovered correlations between the mea-
sured differential magnitudes of the target M-dwarfs and
weather parameters, specifically the relative humidity.
This effect has been investigated in some detail and will
be described by Burke et al. (in preparation). It results
from a mismatch in spectral type between the target star
and the comparison stars, where typically the compari-
son stars are much bluer than the targets (usually, they
are close to solar-type for the majority of our fields).
Variations in the precipitable water vapor (PWV) con-
tent of the atmosphere along the line of sight cause the
strength of the telluric water vapor absorption in the
MEarth bandpass to vary, which affects the photometry
of the targets in a way which is not corrected by stan-
dard differential photometry procedures (Angione 1999;
Bailer-Jones & Lamm 2003). This effect can reach sev-
eral percent over the course of an observing season for
the later M-dwarf spectral types.
Since we lack sufficiently red comparison stars with
good signal to noise ratios for the majority of our tar-
get fields, we adopt an alternative method to derive the
required photometric correction. We create a “common
mode” light curve by taking the median of all of the M-
dwarf differential magnitudes observed by all 8 MEarth
telescopes in a half hour time bin (where the size was
chosen to ensure all stars being observed at a given time
are included). The averaging is necessary both to im-
prove the signal to noise ratio, and to remove the effects
of any real variability or transits. Since the precipitable
water vapor usually varies slowly (the majority of the
variations are from night to night) this is sufficient to re-
move most of the effect. The mismatch in spectral type
between the target and the comparison stars varies for
each target, and therefore so does the amplitude of the
effect, so in practice we subtract a scaled version of the
“common mode”, determining the scale factor for each
object by standard least-squares methods. The scale fac-
tor is generally found to be very well-correlated with the
colors (and thus, presumably, spectral types) of the tar-
gets.
3. PERIOD DETECTION
3.1. Method
In order to properly account for the “common mode”
systematic effect and the magnitude offsets between the
two sides of the meridian, we adapt the method described
in Irwin et al. (2006), which uses least-squares fitting of
sinusoids to the observed time-series m(t).
We adopt as the null hypothesis a model of a constant
(real) magnitude, modulated by the systematics correc-
tions:
?m−+ k c(t),h < 0
m++ k c(t),h ≥ 0
where m−and m+are separate (constant) baseline flux
levels for the two sides of the meridian, and k is a scale
factor multiplying the “common mode”. h is hour an-
gle, and c(t) is the “common mode”, determined from a
time-binned median of all the M-dwarfs being observed
by MEarth at any given moment. The scale factor k al-
lows for the variable amplitude of the “common mode”
component in each M-dwarf due to the differing spectral
type mismatch of our targets and comparison stars.
This null hypothesis is compared to the alternate hy-
pothesis that the light curve contains a sinusoidal mod-
ulation, again modulated by the systematics corrections,
of the form:
?m−+ k c(t) + asin(ωt + φ),h < 0
m++ k c(t) + asin(ωt + φ),h ≥ 0
where a and φ represent the semi-amplitude and phase of
the sinusoid, and ω = 2π/P is the angular frequency cor-
responding to rotation period P. In order to determine
ω, we fit this model using standard linear least-squares,
at discrete values of ω sampled on a uniform grid in fre-
quency from 0.1 to 1000 days. By comparing the best-
fitting χ2values for the two hypotheses as a function of
frequency, we derive a “least-squares periodogram” that
accounts for the effect of the systematics.
Due to the small sample size, we omitted the cut in
χ2improvement used by Irwin et al. (2006), and simply
subjected all of the light curves to visual inspection, to
define the final sample of periodic variables, although in
practice all of the objects selected pass the ∆χ2
terion used by Irwin et al. (2006). 41 light curves passed
this selection, where the remainder were consistent with
m0(t) =
(1)
m1(t) =(2)
ν> 0.4 cri-

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4Irwin et al.
no detectable variation, had excessive systematics, or had
insufficient data to determine a period. A number of ob-
jects appeared to have monotonic trends in time, but it is
not clear for many of these if they are due to systematics
or to variability at present.
3.2. Sample properties
Figure 1 summarizes the overall properties of the sam-
ple of 273 stars on which the period search and selection
of rotation candidates was performed. As discussed in
§2, all of these stars have trigonometric parallaxes, and
Figure 1 shows that all of the objects selected as rotation
candidates also have more than 200 data points taken on
≥ 10 nights (for all but one object on ≥ 28 nights), span-
ning at least 120 days.
It is important to note that the selection we have per-
formed does not remove close binaries from the sample.
Tidal effects will modify the rotation rates of the com-
ponents of such binaries, by transferring angular mo-
mentum between the stellar spin and the binary orbit.
The dependence of tides on binary semimajor axis is
very strong (e.g. Zahn 1977) so the objects showing the
largest tidal effects will be in the shortest period systems.
It is therefore possible that some of the objects in the
present sample have had their rotation rates altered by
a binary companion, with the objects rotating at shorter
periods being more likely to have been affected.
We also note that the use of K-band absolute magni-
tude to estimate masses means that these masses will be
overestimated for any unresolved binary or multiple star
systems due to the extra light from the companion(s).
These issues are best resolved by performing spectro-
scopic follow-up at high resolution, to search for double-
lined objects and for radial velocity variability in order
to identify any spectroscopic binaries. This has not yet
been done for the present sample, and it is important to
bear the caveats discussed in this section in mind in the
interpretation of the rotation periods we measure.
3.3. Effect of the “common mode” correction
Since we are solving for the “common mode” ampli-
tude simultaneously with the sinusoidal variability, it is
important to evaluate the effect of this correction on our
period sensitivity.
Figures 2 and 3 show periodograms of the “common
mode” light curve, calculated in the same way as those
we use for period detection by least-squares fitting of
sinusoids. The dominant power in the “common mode”
is on long timescales, and at the corresponding 1 day−1
alias frequencies, with the highest peaks for the two years
of observations corresponding to periods of 25.1 days for
2008/2009 and 14.5 days for 2009/2010.
The presence of power at such frequencies could af-
fect our sensitivity to rotation periods in this range. We
therefore proceed in the next section to simulate the full
end-to-end period detection process, including the ef-
fect of the “common mode” and of the visual inspection
stages, to evaluate the survey sensitivity.
3.4. Simulations
In order to evaluate the sensitivity in period and ampli-
tude, simulations were performed following the method
detailed in Irwin et al. (2006), injecting sinusoids with
Fig. 1.— Summary of sample properties. In each panel, the open
histogram shows the full sample of 273 stars with more than one
observation, and the solid histogram shows the 41 stars for which
we detect periods. The panels show the number of observations
for each target (top), number of nights during which data were
gathered (center), and the range of Julian dates spanned by the
observations (bottom).
periods from 0.1 to 200 days following a uniform distri-
bution in log period into only the light curves rejected
in the previous visual inspection stages to reduce con-
tamination by any real variability. Two semi-amplitudes
were simulated, 0.005 and 0.01 mag, corresponding to the
range of typical amplitudes of our rotation candidates.
The results are summarized in Figures 4 and 5 for the
0.005 and 0.01 mag semi-amplitudes, respectively. We
also show plots of the recovered period versus injected
period, in Figures 6 and 7, respectively. Whilst the over-
all completeness of rotation period detections is quite low
(around 50−60% in most period bins), there is no clear
bias except in the longest-period bin, where the com-

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Rotational evolution of fully-convective stars5
Fig. 2.— Periodogram of the “common mode” light curve for
the first year of observations. Plotted as the square root of the
reduced χ2of the sinusoidal fit as a function of frequency, with
the vertical axis inverted such that higher peaks correspond to
more significant detections as for a conventional periodogram. The
period of the highest peak is 25.1 days. When computing the χ2
values we used the scatter in each time-bin to compute the weights;
this appears to significantly underestimate the scatter between the
bins, hence the large values of the reduced χ2. The peaks around
1 day−1appear to be caused by aliasing, and probably do not
represent real variations in the PWV on these timescales. This has
been confirmed by analyzing contemporaneous GPS-based PWV
measurements (which are not restricted to hours of darkness) from
Flagstaff, Arizona.
Fig. 3.— As Figure 2 only for the second year of observations.
The period of the highest peak is 14.5 days.
pleteness drops to 25−30%. This is expected due to the
limited survey duration.
The reliability and contamination histograms, and Fig-
ures 6 and 7 provide an indication of the reliability of
period recovery in the cases where a significant modula-
tion was detected. For 0.01 mag, these statistics indicate
very good period recovery, with reliability (measuring
the fraction of objects detected with the correct period)
above 80% in all period bins, and contamination below
20%. For 0.005 mag the performance is substantially
worse, with a large drop in reliability around 1 day and
in the longest-period bin. The contamination statistic
shows a similar effect. By examining Figure 6, it is clear
that some of the scattering of objects in and out of the
1 day bin is due to aliasing, and this diagram also indi-
cates that many of the “incorrect periods” contributing
to the poor performance for the longest periods simply
have larger period errors than our 10% threshold. This
is probably the reason for the apparently small change
Fig. 4.— Results of the simulations for 0.005 mag semi-amplitude
expressed as percentages, plotted as a function of period. Top
panel: completeness as a function of real (input) period. Center
panel: Reliability of period determination, plotted as the fraction
of objects with a given true period, detected with the correct period
(defined as differing by < 10% from the true period). Bottom
panel: Contamination, plotted as the fraction of objects with a
given detected period, having a true period differing by > 10%
from the detected value.
in the completeness statistic between the two bins: this
merely counts detections, without regard to whether the
period was correctly determined.
While it is important to perform the simulation on M-
dwarf target stars to account for the systematics, it is
expected that in reality, many or even the majority of
these should show real, astrophysical variability at some
level. It is therefore likely that we have injected our sim-
ulated signals into objects which also have astrophysical
signals. For the 0.01 mag amplitude, this does not ap-
pear to be a serious problem, with the injected signal
generally overwhelming anything already present. How-
ever, this could contribute to explaining the apparently
high contamination and the lack of a significant drop in
completeness in the 0.005 mag amplitude sample.
In order to investigate these effects, and in particular
to also test the influence of systematics on the detec-
tion rate, we have performed an additional set of sim-
ulations for 0.005 mag semiamplitude but using white,
Gaussian noise of standard deviation set by the estimated
observational errors, rather than using the observed light
curves. This procedure eliminates systematics (“corre-
lated noise”) and any existing astrophysical variability
but should otherwise have a comparable noise level to