We utilize the SUPERBLINK proper motions to produce a reduced proper motion diagram (RPM) of all the SUPERBLINK stars monitored by K2 (Figure 3.1). The reduced proper motion diagram is similar to an HR diagram and the locations of stars on the diagram can be interpreted in a similar way. The main sequence makes an S-shape diagonally across the diagram, the red dwarfs lie in the upper right corner, and the white dwarfs in the lower left. However, the similarities end there because the RPM diagram is not a plot of absolute magnitude or luminosity. An RPM diagram plots the reduced proper motion, H, given by the equation
HG=G+ 5 log(µ) + 5 (3.1)
where G is the apparent magnitude (in this work we utilize theGAIA G magnitude) and µ the total proper motion. The RPM value is plotted against an astronomical color term such as G-J utilized here. The total proper motion, µ, can be written as the ratio between the velocity of a star its distance from the Sun,
µ= 0.211∗Vt
where Vt is the transverse velocity in units of kms−1, d is the trigonometric distance in units of parsecs, and µ is the total proper motion in units of seconds of arc year−1.
Trigonometric distance is calculated from the inverse of the parallax (d= 1/π). Substituting this expression for parallax in to Equation 3.1 gives
µ= 0.211∗Vt∗π, (3.3)
which can be substituted in to Equation 3.1 to give
HG=G+ 5 log(0.211∗Vt∗π) + 5, (3.4)
which becomes
HG =G+ 5 log(0.211) + 5 log(Vt) + 5 log(π) + 5. (3.5)
The absolute magnitude of a star, a measure of its luminosity, is given by the equation MG =G+ 5 log(π) + 5. Therefore, the reduced proper motion effectively is:
HG=MG+ 5log(Vt) + 5log(0.211) =MG+ 5log(Vt)−3.38 (3.6)
So the reduced proper motionHGdepends on the absolute magnitude,MG, i.e. the star’s luminosity, and the star’s transverse motion, Vt, in the plane of the sky.
The RPM diagram is therefore a useful tool for separating stars by how fast they move through space. In particular, this separates out the stars from the local Galactic disk (low velocity) and from the local Galactic halo (high velocity) populations. The reduced proper
Figure 3.1 Reduced proper motion diagram for all 58,484 high proper motion stars observed in the first 15 K2 campaigns with light curves analyzed in the present study. The fast rotators are indicated by red circles. The blue dashed box represents a “by eye” boundary identifying the locus of main-sequence stars of the old, Galactic halo population; stars of the Galactic disk fall above this box. The M dwarf fast rotators, especially at the red end of the main sequence (G-J>2.0), tend to be slightly elevated above the main locus of the Galactic disk population, which is a typical feature of nearby young stars. The rapidly rotating halo objects (red dots inside the box) may be interacting binary systems, in which the rotation rates have been spun up by tidal interactions, this idea is explored further in Chapter 6.
motion diagram is particularly effective at separating out very young stars, due to a combi- nation of two factors. First, young stars tend to be overluminous at a given color, because they are still contracting and are thus bigger; this shifts young stars up on the reduced proper motion diagram (due to them having lower absolute magnitudes); the same shift would actually be noticed in the regular HR diagram. Second young stars tend to live in the “kinematically cold” populations, meaning that their space motions are generally small, and this also decreases their reduced proper motion, also shifting the stars up on the diagram. Very old stars, on the other hand, tend to live in the “kinematically hot” population of local stars, and have large transverse motions which increase the RPM value (shifting the stars down on the diagram). In addition, old stars are generally metal poor, so they appear bluer than metal-rich stars of equal mass and luminosity; as a result, halo stars tend to be shifted blueward on the diagram (to the left). All of these factors work to clearly separate out the youngest stars in the local Galactic disk population, from the oldest stars in the local Galactic halo population. In general, young stars of a given color will be higher on the diagram than old stars of the same color. This is very useful tool for the identification of young stars in particular.
The box in Figure 3.1 represents a “by eye” boundary identifying the locus of the main- sequence halo stars, i.e. the oldest stars in the sample. This box is consistent with the metallicity and proper motion analysis for old, metal-poor stars described in L´epine et al. (2007). The fast rotator candidates identified in our analysis are plotted as large red circles, with all other stars shown as small black dots. Strikingly, the vast majority of the fast rota-
tor candidates are found within the locus of the disk stars, with only 32 candidates falling within the locus of halo stars. In addition, the M dwarf (V −J > 2.7) fast rotators as a group are “elevated” (i.e. they have lower reduced proper motion values on average) within the locus of the disk stars compared to other disk stars of similar colors. For example, M dwarfs with colors 2.5<G-J<3.0, the average HG value for all SBK2 stars is 14.9, while the fast rotators the average HG is 13.8, i.e. the fast rotators are shifted up.
In the RPM diagram, this either means that these rapidly rotating M dwarfs tend to have brighter absolute magnitude at a given color or that they tend to have lower average trans- verse motions. As explained above, a low average motion is consistent with youth because nearby star forming regions and young moving groups have relatively low space motions relative to the Sun. A higher absolute magnitude is also consistent with youth because very young (< 100 Myr) M dwarfs have not fully contracted onto the main sequence (Baraffe et al. 2015). As a result, young M dwarfs are larger and more luminous for their color com- pared to older main sequence stars of the same colors and/or mass. Again, either one or a combination of both of these factors results in a smaller reduced proper motion value in young, local stars. The fast rotators identified in our analysis, therefore, show exactly the trends one would expect if these fast rotators were indeed part of the local young population. Of the 58,484 SBK2 stars, 12,470 are candidate halo objects based on their location in the RPM diagram (the stars inside the blue dashed box in Figure 3.1), thus representing about 21% of the sample. Of the 1,113 rapid rotators we identify, only 32 have reduced proper motions consistent with halo objects, which is less than 3% of the fast rotators identified.
The identification of even a few fast rotators among stars associated with the Galactic halo population is surprising, because halo stars are expected to be relatively old (& 10 Gyr). Gyrochronology predicts that all stars of that age should have slowed to a very low rotation period. However, one way that such old stars could still be rotating fast is if they are in a binary system. In the presence of a close companion, a high rotation rate can be driven by tidal interactions, which will tend to synchronize the rotation period of each individual star with the orbital period of the system. If the orbital period of the system is short (less than a few days), then both stars will maintain a fast rotation rate. According to the data on nearby binaries and multiples presented in Raghavan et al. (2010), we would expect to find ∼0.4% of the G stars in our sample to be short period binaries, with separations small enough to make them tidally interacting binary stars; this is based on the observation that only 2 binaries with orbital periods <5 days are found among the 454 local stars that were surveyed in their study. Additionally Fischer & Marcy (1992) found 0.42% of M dwarfs to have stellar companions. For our subset of halo stars, we would thus expect∼0.4% of 12,470 stars to be close binaries with short orbital periods and potentially interacting, which would amount to about 50 objects. This assumes that the statistical distribution of orbital periods is the same for the stellar populations of the disk and halo. The 32 candidate halo fast rotators we identify are reasonably within a factor 2 of the predicted number. The slightly lower than expected value may indicate that the binary fraction is marginally lower for the local halo population, compared to the nearby disk population of G stars surveyed in the Raghavan study. Alternatively, we may be missing some close binary stars that may not be
close enough to sufficiently interact or that have inclinations that make it difficult to detect rotation modulation, for example if their axis of rotation is tilted perpendicular to the plane of the sky. It is worth mentioning that the Raghavan study only found 2 systems with these short orbital periods. The uncertainty in their sample is on the order of±√2 =±1.4. Our slightly lower number could still be in agreement with the Raghavan results, considering the small number statistics.
The fact that the vast majority of the halo stars are generally non-variable suggests that these stars, as a group, may be ideally suited for use as “calibration” sources to refine the reduction of K2 photometric data. A calibration sample is typically used to test the uncer- tainty in a data reduction or analysis algorithm. In this case, we do not expect to find a modulation in most of the halo stars, at least on timescales of a few days. A consequence of this is that our halo sample can in fact be used to place an upper limit on the number of false positives in our sample. In the worst case, if we assume that all 32 candidate fast rotators identified among the halo stars are false positives, and not actual fast rotators in the halo, then this would suggest a false positive detection rate of 0.3% for all the stars in our database. If this false positive rate were applied to the subset of disk stars, this means that only about 118 of the fast rotators in the disk population might be false positives, or
∼ 10% of all the candidate fast rotators identified in our study. We will see in Chapter 6 that only 10 of the 32 halo detections are actually false positives, indicating an actual false positive rate of 0.1% implying that at most only about 40 of the 1,113 SBK2 fast rotators may be false detections.
In any case, the much lower detection rate of fast rotators among the halo stars provides a convincing validation of our ACF analysis. Indeed extrinsic signals such as instrumental artifacts in theK2 light curves should be affecting all light curves equally regardless of what the disk/halo status of the star is. The very fact that the great majority of our candidate fast rotators are identified among “disk” stars shows that these modulations must be intrinsic signals, and thus most likely genuine modulations in those stars.