Observations of Spin-down in Post-mass-transfer Stars and the Possibility for Blue Straggler Gyrochronology

To provide a more robust comparison between the rotational evolution of BSSs and other post-mass-transfer systems to models of stellar angular momentum evolution, we assemble a sample of wide (P_orb > 80 days) post-mass-transfer binaries from the literature consisting of FGK-type primaries with detected WD companions. These WDs all have temperature measurements enabling age estimates for the post-mass-transfer systems from WD cooling models. The primaries in these systems also have rotational measurements from spot modulation or from spectroscopic v sin i measurements.

Our sample is composed of 12 binaries from the literature containing a WD and a BSS or MS star, all in close enough orbits to infer that mass transfer would have taken place in their past, but not so close that current tidal effects would affect their rotation rates. In most cases, orbital periods have been measured for these systems from radial velocities or eclipses (Geller et al. 2009; Kruse & Agol 2014; Gosnell et al. 2015; Kawahara et al. 2018). In a few cases, precise orbital periods are not known, but constraints from radial velocities and/or astrometry indicate likely periods on the order of months or years (Kellett et al. 1995; Jeffries & Stevens 1996; Holberg et al. 2014).

This sample includes photometric WD detections to BSSs in the old (7 Gyr) open cluster NGC 188 (Gosnell et al. 2015), extreme-UV (EUV) detections of WD companions to field K-dwarfs (Kellett et al. 1995; Jeffries et al. 1996; Holberg et al.2014), and Kepler detections of self-lensing binary systems containing WDs (Kruse & Agol 2014; Kawahara et al. 2018). These varying detection methods allow us to span an age range from hot and young (detectable with EUV surveys), to intermediate age (requiring high-precision Hubble Space Telescope (HST) UV photometry), to quite old and cool (undetectable photometrically in binaries with solar-type primaries, but discovered in microlensing surveys).

For uniformity, we adopt the WD temperature estimates for our sample from the literature, but determine our own WD cooling ages using the models of Holberg & Bergeron (2006) and Tremblay et al. (2011),⁶ except for WOCS 5379 where we use the cooling models of Althaus et al. (2013) because the WD has an He-core instead of a CO-core (N. M. Gosnell et al. 2018, in preparation). We adopt the $\mathrm{log}g$ values for the WDs from the literature when available. Five sources only have photometrically detected WDs, and for those we assume a surface gravity of $\mathrm{log}g=7.8$, corresponding to an approximate WD mass of 0.5 M_⊙. These $\mathrm{log}g$ values are shown in italics in Table 1. The ages are determined using a bilinear interpolation in T_eff and $\mathrm{log}g$.

Table 1.  Post-mass-transfer BSS (or MS)-WD Binaries

ID	Source	MS/BSS	WD Teff	WD $\mathrm{log}g$	WD age	Prot
		Spectral Type	(K)		(Myr)	days
WOCS 5379a	Gosnell et al. (2018)	F7V	${15400}_{-250}^{+280}$	${7.5}_{-0.05}^{+0.06}$	${230}_{-30}^{+40}$	>2.5
WOCS 4540a	Gosnell et al. (2018)	F6V	${17100}_{-100}^{+150}$	${7.7}_{-0.02}^{+0.04}$	${95}_{-5}^{+7}$	${1.8}_{0.92}^{2.3}$
WOCS 4348	Gosnell et al. (2015)	F5V	13000 ± 500	7.8	${245}_{-25}^{+30}$	${1.2}_{0.6}^{1.5}$
WOCS 5350	Gosnell et al. (2015)	F5V	13200 ± 500	7.8	${235}_{-25}^{+30}$	${5.3}_{2.9}^{7.1}$
WOCS 1888	Gosnell et al. (2015)	F6V	11200 ± 500	7.8	${370}_{-40}^{+50}$	${3.6}_{1.9}^{4.7}$
WOCS 2679	Gosnell et al. (2015)	F6V	11300 ± 500	7.8	${360}_{-40}^{+50}$	${1.4}_{0.8}^{1.9}$
WOCS 4230	Gosnell et al. (2015)	F8V	11800 ± 500	7.8	${320}_{-35}^{+40}$	${1.0}_{0.6}^{1.3}$
RE 0044+09	Kellett et al. (1995)	K2V	28700 ± 1500	8.41	${51}_{-12}^{+13}$	0.4
KOI-3278	Kruse & Agol (2014)	GV	10000 ± 750	8.14	${840}_{-160}^{+220}$	12.5
KIC 6233093	Kawahara et al. (2018)	GV	<10000	8.0	>1000	17.1
2RE J0357+283	Jeffries et al. (1996)	K2V	35000 ± 5000	8.0	${6.3}_{-2.3}^{+2.9}$	0.4
HD 217411	Holberg et al. (2014)	K0V	37200 ± 300	7.8	${4.8}_{-0.12}^{+0.12}$	0.6

Note.

^aWD atmosphere fits found assuming a cluster distance to NGC 188 of 1950 pc with a Plummer radius of 11 pc.

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In Table 1, we list each source in our sample, along with the literature reference for the system, the primary's spectral type, the literature values for WD temperature and $\mathrm{log}g$ value, our age estimate from WD cooling, and the primary star's rotation rate (which we discuss in Section 2.3). This range in spectral types spans masses from ∼0.8 M_⊙ to 1.5 M_⊙.

The field stars in our sample have rotation-period measurements from photometric modulation. We adopt these from the literature source in Table 1, except in the case of the detection in Kawahara et al. (2018), where we use the McQuillan et al. (2014) rotation period of 17.1 days measured from the Kepler light curve. McQuillan et al. (2014) classify this as a marginal detection, but we confirm this detection with visual examination of the light curve. Our Lomb–Scargle periodograms (Hartman et al. 2008) also confirm the ∼17 day period.

For BSSs in the cluster NGC 188 we have only v sin i measurements from the spectral archive of the WIYN Open Cluster Study (Geller et al. 2008). For these systems, we convert the v sin i measurement to a rotation period. First, we calculate photometric radii for the blue-straggler primaries. To do this, we adopt the temperatures from Gosnell et al. (2015), V-band magnitudes from Sarajedini et al. (1999), and a distance modulus and reddening for the cluster of (m − M)_V = 11.44 and E(B–V) = 0.09 (Sarajedini et al. 1999).

We use this radius to convert the observed rotational v sin i to a distribution of possible periods assuming a random, uniform distribution of possible inclinations. We adopt the median value of this period distribution in Figure 1, and also show error bars corresponding to the interquartile range.

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We report these periods in Table 1.

In Figure 1, we show the relationship between the age (i.e., time since mass transfer ended) and the rotation period of the primary (i.e., the mass accretor) for the post-mass-transfer binary systems in our sample. The youngest post-mass-transfer stars in our sample (t < 100 Myr) are rotating with short periods of 0.4–0.6 days. The intermediate-aged systems of 100–400 Myr have rotation periods ranging from about 1–10 days. The oldest stars in our sample with ages >660 Myr have the slowest rotation periods of more than 10 days.

The rotation periods match the spin-down models of Gallet & Bouvier (2015; hereafter GB2015) for single solar-type stars strikingly well.

In Figure 1, we compare our observations to two sets of models: (1) spin-down models from GB2015 developed to match observed rotation rates of normal solar-type stars, and (2) spin-down models for BSSs formed from collisions between two MS stars. These two models offer two visions of angular momentum evolution in a post-mass-transfer stars: similar to typical spin-down on the MS, or dramatically different because of significant structural differences that might arise from stellar interaction. We briefly describe the evolution of these models below, and refer the reader to the original papers for further detail.

3.2.1. Gallet & Bouvier (2015)

3.2.2. Sills et al. (2001)

The spin-down of post-mass-transfer stars is remarkably similar to the spin-down models of normal main-sequence stars.

It is not obvious why the angular momentum evolution of mass-transfer products should match the spin-down behavior of standard MS stars. For example, mass transfer might alter the mass accretor's structure and interior angular momentum distribution, or change the magnetic field strength or configuration. Here we discuss the physical implications of this result.

First, let us consider the youngest stars in our sample with ages <100 Myr. These stars have rotation rates of 0.4–0.6 days, corresponding to  ∼ 30% of break-up velocity. On the one hand, rapid rotation is unsurprising as mass-transfer models predict substantial spin-up as a result of mass accretion (Packet 1981; de Mink et al. 2013; Matrozis et al. 2017). On the other hand, it is notable that these stars are rotating at similar rates. Given that the youngest of these systems is <5 Myr old, these stars have not had a chance to substantially spin down via the standard magnetized wind. Instead, whatever set the angular momenta of these objects must have occurred during or shortly after mass transfer. Packet (1981) argued that accretion from a disk should be limited due to spin up, as the surface of an accreting star should reach Keplerian rotation after accreting just a few percent of its mass, preventing any further accretion onto the star. More recently, Matrozis et al. (2017) argued that given the masses observed among CEMPs and barium stars, these stars must be able to accrete several tenths of a solar mass of material from their AGB companions. To accrete so much mass, they argued, would require an efficient angular momentum loss mechanism to act during the mass transfer process—perhaps disk locking or ejection of significant material through a strong wind—effectively capping the star's angular momentum so that more material can be accreted. It seems plausible, then, that the maximum rotation rate for a post-mass-transfer star might be limited by such angular momentum regulation, and thus all three of our young systems are rotating near this maximum. Adding more young systems to this sample is necessary to further explore this idea and determine if young FGK-type post-mass-transfer stars are indeed all rotating with similar periods, or if there is actually a larger spread in rotation rate than is evident in our small sample.

The collisional models, in comparison, have slightly faster initial rotation rates, but we caution against reading too much into this comparison. Given the wide range of initial conditions and the rescaling of the initial angular momentum that has been applied to theses models, one could tune the collision models to have a wide range of rotation rates.

Notably, these 0.4–0.6 day rotation rates are comparable to the fastest rotation rates observed among solar-type stars in young clusters (gray bars in Figure 1) and also to the the peaks in the GB2015 models. Perhaps angular momentum growth during pre-main-sequence accretion is similarly capped.

The older (>100 Myr) systems in our sample allow us to explore spin-down behavior well after mass transfer has ended. Here the spin-down rate is primarily determined by magnetic braking via a stellar wind (parameterized in the GB2015 models by a scaling factor K₁ to the Matt et al. (2012) law). The post-mass-transfer stars are therefore spinning down at the rate expected for their mass. A straightforward interpretation is that magnetic braking must be operating as usual in these stars, suggesting that these stars have normal convective envelopes and magnetic fields. In contrast, the Sills et al. (2001) collision models do not spin down via magnetic braking as they age. While typical stars in this mass range do spin down due to magnetic braking, the collision products are slightly hotter and brighter, and so never develop a convective envelope. Thus, magnetic braking is never expected to take affect, and the stars maintain rapid rotation rates throughout their main-sequence evolution.

The similarity of the GB2015 models and the spin-down behavior observed by our sample of FGK-type post-mass-transfer binaries suggests that gyrochronology relationships developed for standard solar-like stars are also applicable to post-mass-transfer systems. Rotation rates among post-mass-transfer systems like the BSSs, then, may be a useful proxy for time since formation, with recently formed systems rotating at a large fraction of break-up velocity, and older systems converging to rotation rates reflecting their age. Even in the absence of direct detections of WD companions, then, comparing the rotation rate of a post-mass-transfer star to models like GB2015 could allow us to infer the time since the mass-transfer event. This possibility is particularly useful given the difficulty of detecting older, fainter WD companions around FGK main-sequence stars, and opens up a new method to determine formation rates and lifetimes for these systems, timescales that remain uncertain.

Gyrochronology ages are generally not precise for young (<1 Gyr) main-sequence stars due to the large spread in their rotation rates at young ages. However, our post-mass-transfer sample has little scatter at young ages, suggesting gyrochronology may by a much more useful age-dating technique for young post-mass-transfer systems.

While direct detection of a WD companion is helpful in establishing that a star has a mass-transfer origin, there are many systems in which mass-transfer can be reasonably assumed without detecting the WD companion directly. Stars with abundance anomalies—e.g., barium stars or CEMP stars—are good examples. In addition, post-mass-transfer systems have distinctive orbital properties (∼1000 day periods, near circular, mass functions indicating WD companions; e.g., Jorissen et al. 1998; Carney et al. 2005). Many blue stragglers are found to have these orbital properties (e.g., Gosnell et al. 2015), and gyrochronology might reasonably be applied to any of these systems.