An unsupervised method for identifying X -enriched stars directly from spectra: Li in LAMOST
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An unsupervised method for identifying X -enriched starsdirectly from spectra: Li in LAMOST
Adam Wheeler, David W. Hogg,
2, 3, 4 and Melissa Ness Department of Astronomy, Columbia University, Pupin Physics Laboratories, New York, NY10027, USA Center for Cosmology and Particle Physics, Department of Physics, New York University, 726Broadway, New York, NY 10003, USA Center for Computational Astrophysics, Flatiron Institute, 162 5th Av., New York City, NY10010, USA Max-Planck-Institut f¨ur Astronomie, K¨onigstuhl 17, D-69117 Heidelberg, Germany
ABSTRACTStars with peculiar element abundances are important markers of chemical enrich-ment mechanisms. We present a simple method, tangent space projection (TSP),for the detection of X -enriched stars, for arbitrary elements X , from blended lines.Our method does not require stellar labels, but instead directly estimates the coun-terfactual unrenriched spectrum from other unlabelled spectra. As a case study, weapply this method to the 6708 ˚A Li doublet in LAMOST
DR5, identifying 8,428 Li-enriched stars seamlessly across evolutionary state. We comment on the explanationfor Li-enrichement for different subpopulations, including planet accretion, nonstan-dard mixing, and youth. INTRODUCTIONRecent years have seen many resolved-star spectroscopic surveys, and correspond-ing growth in the use of data-driven spectral models, e.g.
The Cannon (Ness et al.2015), the DD-
Payne (Xiang et al. 2019), kernel principal component analysis ap-proaches (Xiang et al. 2017), and models using deep convolutional neural nets (Leung& Bovy 2019). These models have pushed astronomers into new regimes of precision(Jofr´e et al. 2018), and allowed us to infer evolutionary state, mass, and detailed abun-dances from low-resolution spectra (Ho et al. 2017a; Ting et al. 2018; Xiang et al.2019; Wheeler et al. 2020; Sandford et al. 2020), previously considered the domain ofhigh-resolution spectra.Given a complete enough understanding of stellar atmospheres, the interstellarmedium, the earth’s atmosphere, and our instrumentation, data-driven spectral mod-els would be redundant, but we are far from such total knowledge. Most data-drivenmethods applied to stellar spectra (including all mentioned above) are concerned withsupervised regression. They use a set of spectra labelled with a-priori atmospheric pa-
Corresponding author: Adam [email protected] a r X i v : . [ a s t r o - ph . S R ] S e p rameters, surface abundances, and reddening parameters, to calibrate a model whichis then applied to unlabelled spectra. These approaches are limited by their quantityand precision of the training data, which restricts the datasets, labels, and regimesto which they can be applied. Furthermore, their reliance on labeled data means thatthey can be limited by the strong biases and systematic errors introduced by physicalmodels.In this work, we pursue an alternative approach. Rather than estimate abundances,we endeavour only to identify stars that are strongly enriched in a particular element.This relaxed goal permits approaches which require no labelled training data andminimal researcher input, but which retain high scientific payoff, since X -enrichedstars often have undergone uncommon events. Unusual abundance patterns may alsosignal birth in an accreted galaxy with an enrichment history different from the MilkyWay’s (e.g. Horta et al. 2020; Hawkins et al. 2020; Molaro et al. 2020; Vincenzo et al.2019).We take advantage of the fact that stellar spectra, which are naturally expressed ashigh-dimensional flux vectors, are embeddable on (or near) a lower-dimensional mani-fold. We censor the relevant absorption region of a spectrum, then use its neighbors ona local (euclidean) patch of the spectral manifold to impute the masked pixels. Thisserves as a (possibly) couterfactual unenriched realization of the spectrum, againstwhich we examine the residuals to identify “unexpectedly” strong absorption.This work is related to ideas in manifold learning and nonlinear dimensionalityreduction, especially local linear embedding (Roweis & Saul 2000) and isomap (Tenen-baum 2000). Hessian local linear embedding (Donoho & Grimes 2003) shares with ourmethod the use of singular value decomposition to estimate the tangent space of thedata-manifold. Unlike these methods, we never explicitly construct global nonlinearcoordinates since all of our calculations can be performed in a small patch on the spec-tral manifold. While we leverage the embeddability of spectra in a low-dimensionalspace, we never construct a continuous low-dimensional representation.This is not the first unsupervised model deployed an a large spectral survey.Feeney et al. (2019) used a fully probabilistic nonparametric model to characterizethe spectra of APOGEE red clump stars, denoising them by a factor of a few anddemonstrating the mutual information present in features of elements belonging tothe same nucleosynthetic family. Though their model could, in principle, be usedto impute masked pixels, it is too expensive to deploy across all wavelengths of allobserved stars. ˇZerjal et al. (2019) used a simple nearest-neighbor method to identifyLi-enriched KM dwarfs in
GALAH . ˇCotar et al. (2020) used autoencoders (a familyof neural-net architectures) to identify emission stars.We turn our attention to stars that are enriched in Li. Li-7 burns at a mere2 . × K (Bodenheimer 1965), and is depleted at all stages of stellar evolution. It isthought to be replenished in myriad ways. Its low ionization potential means that instellar atmospheres Li exists mostly as Li-II, which is not detectable. Li-I’s strongest Throughout this paper, we use “enriched” or X -enriched specifically to refer to stars with a high X abundance relative to stars with similar spectra —and hence parameters and abundances. A starwhich is X -rich may not be X -enriched, and vice versa. feature (the 6708 ˚A doublet) is thus very weak and T eff -sensitive. It is also sensitive tonon LTE (local thermodynamic equilibrium) effects Lind et al. (2009). All this meansthat Li can be challenging to study with physical models. DATAWe use LAMOST DR 5 (Deng et al. 2012; Zhao et al. 2012), which contains 5 . × low-resolution ( R = 1800) spectra of 4 . × unique stars. It covers wavelengthsfrom 3800 ˚A to 9000 ˚A. We analyze repeat observations independently with TSP,but stack the resulting residuals before identifying strong absorption. We pre-treatall LAMOST spectra by interpolating to a common rest-frame wavelength grid, thenapplying the approximate continuum normalization first used in Ho et al. (2017b),that is, that is, dividing the spectrum by a itself smoothed with a 25 ˚A Gaussiankernel. While this transformation distorts broad features, it is applied homogeneouslyacross all spectra and thus will not introduce biases. We impute any pixels withnormalized flux, f , outside of 0 ≤ f ≤ by setting f = 1, and setting the associated uncertainty to inf . METHODSThe inputs to our algorithm are the following: • The data, assumed to be a set of uniform vectors. In this work these are rest-frame spectra, interpolated onto a common wavelength grid. • The reference set, a subset of the data well-distributed throughout the under-lying parameter space • Integers k and q which specify the number of neighboring data points to useand the dimensionality of the manifold, respectively. • The components of each data point that are of interest. For stellar spectra,these are the pixels containing the spectral feature(s) under investigation. Wewill refer to data as censored when these components are dropped. We refer tothe components themselves as masked . We take n to be the number of unmaskedpixels, and m to be the number of masked pixels.We take each spectrum to have flux uncertainty described exactly by a multivariatenormal distribution with known covariance. The most naive version of this algorithmrequires complete data. As discussed in section 2, we obtain complete data by imput-ing bad pixels with f = 1. Note, however, that TSP could in principle itself be usedto impute bad pixels more robustly.Ideally, the reference set would include every spectrum available (perhaps exclud-ing those with strong absorption features, see discussion in section 5). Using a randomsubset of spectra instead speeds up computation. In machine learning terminology, this might be called the training set. In the L-ISOMAP dimen-sionality reduction algorithm (de Silva & Tenenbaum 2002), these data are called “landmarks”.
For each spectrum, f :1. Compare f to all spectra in the reference set (with the region of the absorptionfeature masked) to find its k nearest neighbors.2. Compute a basis for the q -hyperplane that captures as much variance amongstthe neighbors as possible, the approximate tangent space.3. Impute the masked pixels by projecting f onto the approximate tangent space.4. Determine if the residuals corresponds to excess absorption. Figure 1.
Algorithm summary.
We need to know the expected profile of the spectra feature to identify enrichedstars after imputing. For all but the strongest absorption lines in low-resolution spec-tra, knowing the instrument resolution is sufficient, since the line’s observed profilein dominated by the line spread function.Figure 1 presents an overview of the algorithm, section 3.1 goes into detail, and3.2 describes our approach to identifying enriched stars from imputed residuals.3.1.
Tangent Space Projection
Stated briefly, TSP is imputation of masked data via local principal componentregression. Here we detail how to use TSP to predict masked spectral pixels of anarbitrary target spectrum, using only a reference set of randomly-selected unlabeledspectra. The reference set and target spectra are assumed to be from the same in-strument and interpolated to the same wavelength-grid. Take the target spectrum’s λ th unmasked pixel to be f λ . First find the k nearest neighbors in the reference set(leaving out the masked portion of the spectra), i.e. those that minimize the euclideandistance (L2 norm), d = (cid:118)(cid:117)(cid:117)(cid:116) n (cid:88) λ =1 ( f λ − f ref λ ) , (1)for a reference spectrum whose λ th pixel is f ref λ . We exclude those reference spectrawith missing data in the masked region.Take F to be the k × ( n + m ) matrix whose rows are the uncensored neighboringspectra in the zero-mean basis (with the neighborhood mean subtracted from eachrow). Next, calculate the q -hyperplane that captures as much variance as possibleamongst the neighbors (using their full, uncensored spectra), that is the first q prin-cipal components of F. Let E be the q × n matrix whose rows are the first q censored n -pixel eigenspectra, and let (cid:101) E q × m be the matrix whose rows are the m -pixel maskedportion of the eigenspectra.Let f be the n -pixel (censored) target spectrum in the zero-mean basis, and let Σbe the n × n covariance matrix describing the uncertainty in f . Then, we can predictthe censored pixels, (cid:101) f , by projecting f onto the row space of E: (cid:101) f pred = (cid:101) E(E T Σ − E) − E T Σ − f . (2)This is the maximum-likelihood estimate for the m masked pixel values given f and Σ,assuming that the ground truth spectrum lies on the hyperplane spanned by the first q eigenspectra. Treating the eigenspectra, (cid:101) E and E, as fixed, the covariance describingthe uncertainty on (cid:101) f pred is (cid:101) Σ pred = (cid:101) E(E T Σ − E) − (cid:101) E T . In practice, we neglect thisuncertainty since when n (cid:29) m (i.e. when a small fraction of all pixels are masked) itis subdominant to the measurement uncertainty in the masked pixels, (cid:101) f .3.2. Identifying Strong Absorption in the Residuals
Having used the spectrum outside the censored region to predict the pixels within,we can examine the residuals between the measured and predicted censored pixels toidentify unusually-strong absorption features and eliminate contaminants. We use asimple likelihood ratio test, i.e. matched filtering.Consider a family of models of the form α m , where α is a scalar amplitude and m is a fixed profile. The value of α which maximizes the likelihood is α ∗ = m T C − (cid:101) rm T C − m , (3)with uncertainty σ α ∗ = ( m T C − m ) − / . (4)For a given m , the optimal-amplitude likelihood islog p ( (cid:101) r | α ∗ m ) = 12 ( m T C − (cid:101) r ) m T C − m + const . (5)If m is a line model with equivalent width w , α ∗ w can be interpreted as best-fit excessequivalent width (EEW), equivalent width in surplus of that found in a counterfac-tually unenriched spectrum.We calculate the likelihood, amplitude (EEW), and amplitude uncertainty fora Gaussian (in wavelength) absorption feature with width given by the instrumentresolution, as well as for two contaminant models: uniform residuals, as might arisefrom a poorly modelled continuum, and residuals with a single non-zero pixel, asmight be caused by a cosmic ray. Our line model and contaminant models are plottedin Figure 2. We require the contaminant models to be less likely than the line modelto identify a star at Li-enriched (see section 4).3.3. Hyperparameter Selection
To inform the choice of hyperparameters k and q , we evaluate TSP’s predictiveaccuracy for arbitrary spectral regions. We take 1000 random LAMOST spectra andpredict a different randomly-selected contiguous region of 100 pixels (roughly an orderof magnitude larger than a mask for a single line) for each. Figure 3 shows the per-pixel χ of the masked pixels as a function of subspace dimension, q , for differentneighborhood sizes, k . Large values of each ( k (cid:38) , q (cid:38)
25) give better predictionsof the censored region, which is consistent with the large number latent parametersfound to meaningfully describe
LAMOST spectra by (Xiang et al. 2017). The nominalmeasurment uncertainty is saturated for nearly all values of k and q , indicating that . . . . . λ [˚A] − . − . − . . flu x contaminant modelline model Figure 2.
Our line model (black) and contaminant models (grey, dashed). Only on single-pixel contaminant model is shown for clarity. it may be overestimated. Using the mean of the k nearest neighbor spectra as aprediction, while simpler than TSP, is not as effective. The horizontal dashed line inFigure 3 shows the predictive accuracy for k = 3, the neighborhood size for whicha local average performs best. TSP achieves better predictive accuracy for nearly allhyperparameter values considered.For our case-study, we choose k = 1000 and q = 50. We remark, however, that pre-dictive accuracy in the Li absorption region is not especially sensitive to this choice,as demonstrated by Figure 4, which shows the masked portion of a spectrum exhibit-ing strong Li absorption with TSP’s predictions for q = 4 and q = 50. TSP imputesthese pixels very similarly regardless of our choice of q . Note that TSP predicts noabsorption, despite the fact that some neighbors have absorption similar to that ofthe spectrum being imputed.We found that increasing the size of the reference set decreased the χ / pixel,but changed the preferred values of k , as expected. We use a reference set of 30,000randomly chosen spectra, but remark that a larger reference set would presumablymarginally decrease predictive χ / pixel at a greater computational cost. Using a refer-ence set composed exclusively of spectra with a high signal to noise ratio ( S/N >
S/N stars are not uni-formly distributed across parameter space. RESULTSWe apply our method to the 6708 ˚A feature, masking wavelengths, λ , in therange 6703 ˚A – 6717 ˚A (vacuum). Table 1 shows the output quantities for eachstar. We mask, but do not use, the 6106 ˚A Li feature. To consider stars to be Li-enriched, we require the detection of an absorption feature with an EEW of at least0.15 ˚A with an uncertainty excluding null EEW at the 3 σ level ( σ EEW < . LAMOST stars). Hereafter, we q . . . . . . χ / p i x e l local mean k=50k=100k=250k=500k=1000k=2000 Figure 3.
Mean squared error of the predicted spectral region as a function of subspacedimension, q , for different neighborhood sizes, k . The dashed grey line marks the best χ / pixel ( k = 3) achieved by using the local mean spectrum as a prediction. We use k =1000, q = 50 for our analysis. λ [˚A]0 . . . . . flu x k=30, q=4k=1000, q=50data Figure 4.
The prediction of the Li absorption region in a spectrum of
LAMOST
J062739.73+463634.4 (Gaia DR2 965846541808816256), a Li-enriched giant. The imputedspectrum for k = 30 , q = 4 is very similar to that for k = 1000 , q = 50. Neither predict theabsorption seen in the data. The 30 nearest spectral neighbors are shown in gray. refer to stars with σ EEW < .
05 as candidates (and those that are both candidatesand have EEW > .
15 simply as
Li-enriched ). Within the set of candidates, there isno dependence of EEW on
S/N . Figure 5 shows portions of 21 Li-enriched spectraon different parts of the Kiel diagram.Figure 6 shows a histogram of the EEW values for all candidate stars, and allLi-enriched stars. Nearly all stars with EEW > .
15 ˚A have their censored-regionresiduals best matched by the line model (see section 3.2), except at EEW (cid:38) . Table 1.
Catalog schema. Data available at https://doi.org/10.7916/d8-3ap9-qe35.column name type description designation string
LAMOST unique star identifier source id string
Gaia identifier diff float array flux residuals for each masked pixel ivar float array inverse variance in the flux residuals max best fit chi2
Float largest whole-spectrum χ for any observation of this star isline bool array true when the line model is more likely than any contami-nant model likelihoods float array the likelihood values (up to an additive constant) of the linemodel and each contaminant model EEW float excess equivalent width [˚A]
EEW err float uncertainty in the EEW, σ EEW [˚A] enriched bool true if the star is Li-enriched per definition in 4 ra float right ascention dec float LAMOST unique star identifier teff float T eff [K] from LAMOST (mean of observations) logg float log g from LAMOST (mean of observations) feh float [Fe/H] from
LAMOST (mean of observations) snrr float
LAMOST r -band ( S/N ) (largest of all observations) λ [˚A]1234 flu x + o ff s e t Li-I 3 . . T eff l og g Figure 5. left:
Portions of 21 spectra of Li-enriched stars spaced roughly evenly on the Kieldiagram and sorted by H α (6565 ˚A) amplitude. right: their positions on the Kiel diagram,with mass tracks for 2 M (cid:12) , 1.5 M (cid:12) , 1 M (cid:12) , 0.8 M (cid:12) , and 0.5 M (cid:12) solar metallicity stars(details in text). can be obtained by assuming that for a chemically typical star, the expected EEWdistribution is symmetric about 0. That is, the stars best fit by a line with negativeEEW, provide an upper bound for the number spurious detections at positive EEW,assuming there are more Li-enriched stars than Li-depleted. This reasoning suggeststhat up to 40% of candidates are not truly Li-enriched, although any amount of Li-depletion uncaptured by the model will artificially inflate the false-positive rate andthe true rate is probably much lower. − . − . . . . N allflagged Figure 6.
The distribution of excess equivalent widths for all candidates (those with σ EEW < . > . There are no obvious spatial trends in the fraction of Li-enriched stars; there areonly trends in stellar parameters and evolutionary state. Figure 7 shows the 8,428Li-enriched stars in the log g – T eff plane, along with the occurrence fraction of Li en-richment, and the number of candidates. Plotted for comparison are solar-metallicitymass tracks from MESA isochrones and stellar tracks (MIST; Dotter 2016; Choi et al.2016; Paxton et al. 2011, 2013, 2015), Li enrichment is especially prevalent amongpre-main sequence stars, stars near the zero-age main sequence (ZAMS), especiallyat larger T eff , subgiant branch stars, and red giants at and above the red clump. Wego into detail in the sections below.4.1. Planet Accretion on the Subgiant Branch
Presumably, stars occasionally consume their satellites. These events may en-rich the photosphere, especially when the satellites are large. Accretion of sub-stellarcompanions has been identified many times as a Li-enrichment mechanism (Alexan-der 1967; Siess & Livio 1999; Villaver & Livio 2009; Adam´ow et al. 2012; Carlberget al. 2012; Oh et al. 2018), and detection of cometary material accreted by whitedwarfs has a long history (van Maanen 1917; Weidemann 1960; Zuckerman et al. 2003;Kepler et al. 2016, e.g.). Recently, Soares-Furtado et al. (2020) found that, while pho-tospheric Li will become unobservable within roughly 10 years for red giants, whichhave deep convective envelopes, it has a lifetime of up to 1 Gyr for 1.4 – 1.6 M (cid:12) stars on the subgiant branch and near the main sequence turn-off. Compounded withthe fact that stars on the subgiant branch are expanding rapidly, this suggests thatplanet accretion could account for a significant fraction of Li-enriched subgiants.Figure 7 shows a moderate enrichement fraction ( ∼ . M (cid:12) subgiant branch. Neither the data-driven abundances of Xiang et al. (2019) (Na, Mg,Al, Si, Ca, Ti, Cr, Mn, Co, Ni, Cu, Ba) nor Wheeler et al. (2020) (O, Sc, Eu, Mg, MIST version 1.2. Mass tracks generated with initial v/v crit set to 0 . . . . . . T eff l og g . M (cid:12) . M (cid:12) . M (cid:12) . M (cid:12) . M (cid:12) . . . . . . . EE W [ ˚A ] . . T eff l og g − − − (fraction enriched) 3 . . T eff l og g N Figure 7. top:
The 8,428 Li-enriched stars plotted on the Kiel diagram with MIST solar-metallicity mass tracks. bottom left: fraction all stars with σ EEW < .
05 ˚A with areLi-enriched. bottom right: the number of of stars with σ EEW < .
05 ˚A. Dashed verticallines are the same as in Figure 9
Li-enriched Red Giants
More than half the stars we identify as Li-enriched are red giants (4459 havelog g < . ,
000 K (cid:46) T eff (cid:46) , . (cid:46) log T eff [K] (cid:46) . A (Li) > .
5; e.g. Gao et al. (2019), Casey et al. 2019). The first Li-enriched red giant is often considered to be Wallerstein & Sneden (1982), although aLi-enriched asymptotic giant branch (AGB) star was reported four decades earlier byMcKellar (1940). The exact mechanisms that produce this enrichment have not beenconclusively identified.As mentioned in section 4.1, sub-stellar companion accretion is unlikely to explaina significant fraction of Li enrichment for red giants. Cosmic ray spallation can accountfor some Li enrichment (Burbidge et al. 1957; Mitler 1964), but only a small fraction,and does not produce the observed isotope ratio (Reeves 1994). Mass-transfer from anAGB star companion can account for Li-enrichment in those that have one, but notin isolated giants. Li-enriched giants are not preferentially found in binaries (Adam´owet al. 2018). Li-enriched material is produced by classical novae (Starrfield et al. 1997;Molaro et al. 2016) and Type II supernovae (Dearborn et al. 1989), but it is unclearwhether, and in what quantity, this material can be accreted onto giant stars.Cameron (1955) and Cameron & Fowler (1971) first suggested that surface Li couldbe enhanced if beryllium-7 is transported via convection from a depth at which PP-IIfusion is occurring and decays via electron capture to Li-7. For the beryllium to betransported to a cool layer of the star before it decays into Li, the convection timescalemust be faster than its decay timescale, thought to be 50 – 100 yr (Cameron 1955).While this process can occur as originally suggested in AGB stars (Deepak & Reddy2019; Singh et al. 2019), whose convective zones reach to sufficient temperatures, itmust be augmented with an additional mixing mechanism for RGB stars.In RGB stars, an additional mixing process is needed. Two possibilities are internalrotation (Sweigart & Mengel 1979; Fekel & Balachandran 1993; Charbonnel 1995),and thermohaline mixing (Sackmann & Boothroyd 1999; Charbonnel & Balachandran2000; Denissenkov & VandenBerg 2003; Lattanzio et al. 2015). Casey et al. (2019)found that tidal interactions where the likely culprit, while Martell et al. (2020) foundthat at least two mechanisms are likely in effect. Recently, Kumar et al. (2020) foundthat a ubiquitous process is in operation for all low-mass stars, on the basis of thehigh Li abundances in the red clump.We see no evidence on the basis of the log g and T eff distributions of enriched starsthat Li-enrichment is more common in the red clump (RC) than on the red giant2 − . − . − . . . c o un t − . − . − . . . . . . . . . . o cc u rr e n ce f r a c t i o n [ % ] EEW > .
15 ˚AEEW > . Figure 8.
The raw count and occurrence fraction of Li-enriched giants as a function ofmetallicity. Errorbars are determined assuming Poisson uncertainty in raw counts. Thefraction of Li-enriched giants increases smoothly with [Fe/H]. branch (RGB), in sharp contrast with Deepak & Reddy (2019); Martell et al. (2020);Casey et al. (2019). This may be partially attributable to our direct use of absorptionfeature, rather than calculating the abundance from the equivalent width. The morelikely possibility is that TSP is able to correctly impute strong Li absorption forRC stars for at least on enrichment channel. If this is the case, understanding whichspectral features contain joint information with the Li doublet might indicate themixing mechanism driving photospheric Li-enrichment. Notably, we identify few Li-enriched giants below the red clump.In agreement with other recent work (Casey et al. 2019; Martell et al. 2020; Deepak& Reddy 2019), we see that the occurrence rate of Li-enriched giants increases stronglywith metallicity (Figure 8). However, we see no evidence for a sharp increase in thewoccurrence fraction as [Fe/H] increases past the solar value, as noted by Martell et al.(2020) and hinted at in the data of Casey et al. (2019). Curiously, Martell et al. (2020)found that a smooth increase on Li-enrichment was associated with RC stars, andthat RGB stars only had enrichment “turning on” at super-solar metallicties. Thosedistinct trends are likely attributable to a multiplicity of enrichment mechanisms,further suggesting that our analysis is sensitive some but not others.4.3.
Young and pre-Main Sequence Stars
Some Li-enriched stars are enriched by virtue of having not yet depleted theirbirth Li. We are more likely to identify pre-main-sequence stars as Li-enriched thanstars at any other evolutionary stage, with the enriched fraction exceeding 50% forthose with log g >
4. These stars are exhibit some of the largest EEWs (see Figure 7),with some larger than 0.5 ˚A, and roughly half have H α emission, indicating magneticactivity.We know from studies of open clusters that on and near the zero-age main sequence(ZAMS), the Li abundance is only mildly T eff -dependent for stars with T eff (cid:38) T eff [K] (cid:38) . T eff past that value (e.g.Sestito & Randich 2005). Together with the fact that the 6708 ˚A Li doublet is T eff -3 . . . . .
80 log T eff [K] − . − . . . . . EE W [ ˚A ] Figure 9.
Li EEWs as a function of T eff for the Pleiades stars observed by LAMOST . Notethat the x-axis is reversed. The grey dashed line marks the threshold at which we considera star Li-enriched. The decrease in EEW with increasing effective temperature is due toLi’s low ionization potential. The decrease with decreasing effective temperature is becausethose stars have depleted more of their Li. sensitive, this means that we primarily identify your main sequence stars of moderateeffective temperature. These effects can be seen clearly in the Pleiades, the opencluster in which
LAMOST has observed the most stars. Figure 9 shows the EEWfor each. Dashed lines mark EEW = 0 .
15 ˚A, our Li-enrichment threshold, and thetemperatures within which Pleiades stars exceed this threshold: 4650 (cid:46) T eff [K] (cid:46) . (cid:46) log T eff [K] (cid:46) . DISCUSSIONWe have presented an unsupervised method, tangent space projection (TSP), foridentifying stars which are high X -enriched, for X corresponding to any spectrafeature(s). It uses the fact that most stellar spectra lie on a low-dimensional manifold,but those of chemically aberrant stars often don’t. We applied TSP to the 6709 ˚A Lidoublet in LAMOST
DR5, identifying 8,428 Li-enriched stars.The fact that TSP has predictive accuracy better than the nominal
LAMOST measurement uncertainties indicates that those uncertainties may be over-estimated.They may also be the best independent approximation of the true joint uncertainties,which include some correlations between pixels.TSP is applicable to any homogeneous catalog of spectra. Its most useful applica-tion is to blended features and those for which physical modelling remains a challenge.For low-resolution surveys like
LAMOST , essentially all spectral features fall into thiscategory, but medium- and high- resolution surveys, such as
Gaia
DR4,
RAVE (Stein-metz et al. 2020; Casey et al. 2017),
APOGEE (Majewski et al. 2017; Garc´ıa P´erez4et al. 2016), and Sloan V (Kollmeier et al. 2017) also contain many such features, e.g.the recently noted Ce and Nd lines (Hasselquist et al. 2016; Cunha et al. 2017) in
APOGEE . In the disk, we expect to find few stars with unusual enrichment patterns,but they have to potential to give us unique insight into nucleosynthetic processes(Weinberg et al. 2019). In the halo, the chemical signatures are strongly linked to thedynamical history of the Milky Way (e.g. Ji et al. 2020; Das et al. 2020; Naidu et al.2020). 5.1.
Algorithm
We see the strengths of TSP not in sophistication, but in simplicity and suitabil-ity to the problem addressed. Further simplifications would come at the expense ofpredictive accuracy, as discussed in section 3.3.There are several elaborations on TSP potentially appropriate for future work. Wefound that predictive accuracy increase with the number of stars in the reference set,but we truncated our parameter search at a reference set of 30,000 stars for the sake ofspeed. Identifying the nearest spectral neighbors, currently the most computationallyexpensive step, could be accelerated in a variety of ways, e.g. by making a pre-passwith the spectra in compressed form. Presumably, the presence of Li-enriched stars inour reference set prevented us from identifying a fraction of enriched stars. RemovingLi-enriched reference stars from the reference set via iterative application of TSPcould address this potential problem and give more complete results.Equation (2) is a form of unregularized regression. Ideally, all projection weightswill be small, since large weights correspond to a part of the tangent space thatdoesn’t overlap with the spectral manifold. We have found that this is true for ourdata, but a regularized form of Equation (2) could help ensure that is is more oftenthe case.In this work, we separate calculation of the imputed prediction, (cid:101) f pred , from exam-ining residuals to identify strong absorption. Jointly fitting for EEW simultaneouslywith projection onto the approximate tangent space would be more principled, andpotentially more effective. Using a probabilistic or robust form of PCA (e.g. Bishop2006 chapter 12) could give a more principled estimate of the tangent space. Otherneighbor selection schemes are also possible, such as using all neighbors within agiven distance hypersphere. An adaptive selection scheme could potentially adjustdynamically for the changing density of the reference set across the manifold.In section 3.3, we showed that the imputed prediction in the region of the Lidoublet is remarkably insensitive to the choice of k and q . In fact, we found this tobe the case even when we didn’t mask the Li doublet region, presumably because thenumber of pixels outside the doublet region dwarf the number of Li-sensitive pixels,and dominate equation 2. In addition, if the subspace dimension, q , is small, the modelwill have too few degrees of freedom to capture Li variation independently from otherabundances.One might ask what advantages TSP has over any established nonlinear dimen-sionality reduction algorithm to lossily compress and decompress spectra. While suchapproaches would likely work, our censor-and-predict scheme is not naturally sup-5ported by these tools. Even if censoring turns out to be unnecessary for real data, weavoid a large computational optimization problem because we don’t need a global andcontinuous low-dimensional basis. We incorporate uncertainty during the predictionstep, and the simplicity of TSP means that a justified uncertainty estimate is easy tocompute. 5.1.1. What could possibly go wrong?
Here we list situations and ways in which TSP can fail to identify X -enrichedor X -depleted stars. Visualizing all steps of the analysis is the best way to diagnosethese kinds of problems. • The manifold is not well-sampled.
If there are not enough points in the referenceset, the k -nearest points may “jump across a wrinkle in the manifold”. This willresult in a poor fit to both the masked and unmasked pixels. • The manifold is sampled very non-uniformly.
In general, the reference set willnot be uniformly distributed across the manifold, meaning that the above issuecan arise in some regimes, but not others. Using an adaptive method to pickthe neighborhood size, k , may help in these situations. • The mask is wrong, i.e. the abundance effects the spectrum in many places . Ifthe abundance X effects stellar spectra in regions outside of the mask, TSP maybe able to predict the masked pixels even of chemically aberrant stars. This mayresult in identified outliers disappearing for large subspace dimension, q , whenthe model has enough degrees of freedom to capture A ( X ) as an independentfactor of variation. • The line model is wrong.
If the line location or profile are wrong, some en-riched stars may fail to be identified, and EEWs may be misestimated. Thiswill typically be identifiable by visualizing residuals, (cid:101) r , and best-fit line models, α ∗ m . • The contaminant models are incomplete or too eager.
Similarly, an unaccounted-for or misspecified contaminant can cause stars to be misclassified. Again, vi-sualization is the best way to identify this situation. • The method underperforms relative to a supervised model . Finally, a superviseddata-driven model, or a completely ab-initio physical model will be a moreappropriate choice, in cases where the physical models are fast and accurate ortraining labels are abundant, precise, and accurate.We believe that this work is not hampered significantly by any of these issues. Vi-sualization indicates that the model fits the data closely and can accurately predictheld-out data. The tests described in section 3.3 shows that the typical predictiveaccuracy is better than the nominal measurement uncertainty. The Li doublet’s pro-file is dominated by instrumental dispersion, and it’s profile is well-described by aGaussian, and its wavelength precisely known.6 CONCLUSIONSOur chief findings are the following: • We introduce TSP, a method for imputing data using ideas from manifoldlearning which can be used to identify stars enriched in a given element fromtheir spectra without a physical model. • We apply TSP to the 6708 ˚A Li doublet in
LAMOST , identifying 8,428 Li-enriched stars. • We examine the abundances of Li-enriched stars near the 1.5 M (cid:12) subgiantbranch, the regime where Li-enrichment is thought to be most likely to bedue to planet accretion. We find that Li-enriched and Li-normal subgiants havenearly identical individual abundance distributions in the 10 elements examinedand are thereby not distinguished by any signature of potential engulfment inother abundances. • We do not see a sharp increase in the fraction of Li-enriched red giants at solar[Fe/H] and we identify few Li-enriched red giants with log g above that of thered clump. Furthermore, we see no surplus of Li-rich giants on the red clump, incontraction to prior studies. This suggests that TSP are sensitive to a differentset of enrichment mechanisms than abundance-based searches. • Using observations of stars in the Pleiades, we demonstrate that we reliablyidentify young main sequence stars as Li-enriched, within a temperature rangeof 4650 (cid:46) T eff [K] (cid:46) Software: matplotlib (Caswell et al. 2020)ACKNOWLEDGMENTSAJW is supported by the National Science Foundation Graduate Research Fel-lowship under Grant No. 1644869. MKN is in part supported by a Sloan ResearchFellowship. REFERENCES
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