Predicting Granulation "Flicker" and Radial Velocity "Jitter" from Spectroscopic Observables

TL;DR

This study recalibrates the relationship between stellar parameters and granulation flicker using APOGEE-Kepler data, enabling accurate predictions of stellar surface activity and radial velocity jitter from spectroscopic observables.

Contribution

It provides a new, more accurate calibration of flicker prediction models for different star types and demonstrates spectroscopic data's effectiveness in estimating granulation and jitter.

Findings

Flicker relationship differs for dwarfs/subgiants and red giants.

Spectroscopic measurements predict flicker with 7-20% accuracy.

Estimated jitter values for 129,000 stars based on flicker correlations.

Abstract

Surface granulation can be predicted with the mass, metallicity, and frequency of maximum power of a star. Using the orders-of-magnitude larger APOGEE-Kepler sample, we recalibrate the relationship fit by Corsaro et al. (2017) for "flicker", an easier-to-compute diagnostic of this granulation. We find that the relationship between the stellar parameters and flicker is significantly different for dwarf and subgiant stars than it is for red giants. We also confirm a dependence of flicker amplitude on metallicity as seen originally by Corsaro et al. (2017), although the dependence found here is somewhat weaker. Using the same APOGEE-Kepler sample, we demonstrate that spectroscopic measurements alone provide sufficient information to estimate the flicker amplitude to 7 percent for giants, and 20 percent for dwarfs and subgiants. We provide a relationship that depends on effective temperature, surface gravity, and metallicity, and calculate predicted flicker values for 129,000 stars with APOGEE spectra. Finally, we use published relationships between flicker and radial velocity jitter to estimate minimum jitter values for these same 129,000 stars, and we identify stars whose total jitter is likely to be even larger than the granulation-driven jitter by virtue of large-amplitude photometric variability.

Tables

Table 1. Table 1 Basic data for the 2465 stars in our calibration sample. Spectroscopic information comes from the APOGEE survey ( Abolfathi et al. , 2017 ) , asteroseismic values come from Pinsonneault et al. ( 2018 ) for giants and Serenelli et al. ( 2017 ) for dwarfs and subgiants, and flicker information is calculated here using the formula of Bastien et al. ( 2016 ) . The full table is provided in the electronic version of the journal. A portion is shown here for guidance regarding form and content.

KIC ID	2MASS ID	${\nu }_{\max }$	${\sigma }_{\nu max}$	Mass	${\sigma }_{Mass}$	[Fe/H]	${\sigma }_{\left[Fe/H\right]}$	Teff	${\sigma }_{T_{\mathrm{eff}}}$	log (g)	${\sigma }_{log\left(g\right)}$	$F_8$	${\sigma }_{F_8}$	$v_{sc}$	Ev. St.
		$(\mu$Hz)	($\mu$Hz)	(M☉)	(M☉)	(dex)	(dex)	(K)	(K)	(cgs)	(cgs)	(ppt)	(ppt)	(kms)
10000207	2M19052985+4654372	94.6	0.9	1.04	0.04	-0.18	0.03	4703	74	2.79	0.06	0.27	0.01	0	RGB
10000547	2M19062193+4657016	160.5	1.4	1.14	0.05	-0.20	0.04	4968	85	3.11	0.06	0.20	0.01	0	RGB
10001326	2M19081378+4655345	47.6	0.4	1.20	0.05	-0.02	0.03	4656	73	2.57	0.06	0.39	0.03	0	RGB
10003270	2M19122841+4658290	781.1	54.1	1.31	0.07	-0.17	0.13	6199	175	3.89	0.10	0.05	0.00	0	D/S
10003349	2M19123799+4656309	48.8	0.6	1.36	0.06	0.02	0.03	4687	75	2.56	0.06	0.32	0.02	0	RGB
10004825	2M19153149+4657298	57.5	0.5	1.66	0.07	0.25	0.03	4690	77	2.71	0.05	0.29	0.01	0	RGB
10006097	2M19175990+4654261	140.6	1.3	1.11	0.05	0.04	0.03	4855	84	3.02	0.06	0.23	0.01	0	RGB
10007492	2M19202078+4656427	214.9	1.9	1.46	0.06	0.00	0.03	4994	90	3.18	0.06	0.15	0.00	0	RGB
10014959	2M19325640+4654288	174.2	1.6	1.05	0.04	0.15	0.03	4740	72	3.13	0.05	0.22	0.01	0	RGB

Table 2. Table 2 Median values of the inferred parameters ( s , t , u , ln ⁡ β ) 𝑠 𝑡 𝑢 𝛽 (s,t,u,\ln\beta) for both short cadence (343 stars) and long cadence (2122 stars) data, with the physical parameter they relate to also indicated in brackets, as well as the original fit parameters from Corsaro et al. ( 2017 ) (26 stars) for comparison. Bayesian credible intervals of 68.3 % are included.

Model	$s\left({\nu }_{\max }\right)$	$t\left(\text{Mass}\right)$	$u$ ([M/H])	$\ln \beta$
$\mathrm{SC}$	$-{0.885}_{-0.014}^{+0.013}$	$-{0.673}_{-0.047}^{+0.048}$	${0.637}_{-0.034}^{+0.031}$	$-{4.039}_{-0.019}^{+0.018}$
$\mathrm{LC}$	$-{0.467}_{-0.010}^{+0.004}$	$-{0.682}_{-0.001}^{+0.005}$	${0.306}_{-0.008}^{+0.007}$	$-{2.899}_{-0.006}^{+0.004}$
$\mathrm{C17}$	$-{0.350}_{-0.029}^{+0.029}$	$-{0.716}_{-0.081}^{+0.077}$	${0.911}_{-0.307}^{+0.310}$	$-{2.399}_{-0.131}^{+0.131}$

Table 3. Table 3 Median values of the inferred parameters ( a , b , c , ln ⁡ α ) 𝑎 𝑏 𝑐 𝛼 (a,b,c,\ln\alpha) for the spectroscopic fit for both SC (343 stars) and LC data (2122 stars), with the physical parameter they relate to also indicated in brackets. Bayesian credible intervals of 68.3 % are included.

Model	$a\left(log\left(g\right)\right)$	$b\left(T_{\mathrm{eff}}\right)$	$c$ ([M/H])	$\ln \alpha$
$\mathrm{SC}$	$-{1.733}_{-0.050}^{+0.054}$	${2.321}_{-0.241}^{+0.212}$	${0.757}_{-0.0530}^{+0.0510}$	${3.904}_{-0.214}^{+0.203}$
$\mathrm{LC}$	$-{0.624}_{-0.014}^{+0.012}$	$-{5.056}_{-0.038}^{+0.092}$	$-{0.132}_{-0.012}^{+0.010}$	$-{0.557}_{-0.066}^{+0.034}$

Table 4. Table 4 Results of predicted Flicker ( F 8 subscript 𝐹 8 F_{8} ) and RV jitter for the full APOGEE sample. The full table is provided in the electronic version of the journal. A portion is shown here for guidance regarding form and content. a Stellar parameters from APOGEE DR14 catalog. b F 8 subscript 𝐹 8 F_{8} estimated from stellar spectroscopic parameters using new empirical relation (Sec.  3 ). d RV jitter amplitude estimated from empirical Flicker-jitter relation of Bastien et al. ( 2014 ) . Uncertainty includes error contributions from both F 8 subscript 𝐹 8 F_{8} and the Flicker-jitter relation. e RV jitter amplitude estimated from empirical Flicker-jitter relation of Oshagh et al. ( 2017 ) . Uncertainty includes error contributions from both F 8 subscript 𝐹 8 F_{8} and the Flicker-jitter relation. f RV jitter amplitude estimated from empirical Flicker-jitter relation of Cegla et al. ( 2014 ) . Uncertainty includes error contributions from both F 8 subscript 𝐹 8 F_{8} and the Flicker-jitter relation. g Light curve variability amplitude from KELT ( Oelkers et al. , 2018 ) . RV jitter estimates for stars with variability amplitudes ≳ greater-than-or-equivalent-to \gtrsim 10 mmag should be treated as lower limits; see the text.

2MASS ID	$T_{\mathrm{eff}}$aafootnotemark:	$\log g$aafootnotemark:	[M/H]aafootnotemark:	$F_8$bbfootnotemark:	[C/N]aafootnotemark:	RV1ddfootnotemark:	RV2eefootnotemark:	RV3fffootnotemark:	LC r.m.s.ggfootnotemark:
	(K)	(cgs)		(ppt)		(m s-1)	(m s-1)	(m s-1)	(mag)
00001977$-$2003393	5351	4.14	$-$0.223	0.026	string0.004	4.31 $\pm$ 1.69	5.71 $\pm$ 3.02	1.46 $\pm$ 0.54	$\cdots$
00002021$+$6302567	4933	2.52	$-$0.017	0.262	$-$0.586	11.85 $\pm$ 1.53	10.30 $\pm$ 3.90	5.71 $\pm$ 0.21	0.01
00002035$+$6250406	5049	2.56	string0.217	0.220	$-$0.098	10.50 $\pm$ 1.38	9.47 $\pm$ 3.59	4.95 $\pm$ 0.22	$\cdots$
00002038$-$1912052	6172	4.41	$-$0.010	0.027	string0.007	4.33 $\pm$ 1.67	5.72 $\pm$ 3.00	1.47 $\pm$ 0.54	$\cdots$
00002141$+$8606336	4856	3.19	string0.094	0.184	$-$0.213	9.37 $\pm$ 1.26	8.79 $\pm$ 3.34	4.31 $\pm$ 0.23	$\cdots$
00002142$-$1929009	5357	4.54	string0.069	0.016	string0.008	3.99 $\pm$ 2.45	5.51 $\pm$ 3.91	1.27 $\pm$ 0.77	$\cdots$
00002338$+$6141442	4987	2.52	$-$0.061	0.249	$-$0.445	11.43 $\pm$ 1.48	10.04 $\pm$ 3.80	5.47 $\pm$ 0.21	$\cdots$
00002388$+$6151472	4785	3.01	string0.199	0.219	$-$0.391	10.47 $\pm$ 1.37	9.46 $\pm$ 3.58	4.93 $\pm$ 0.22	0.017
00002561$+$5528511	5512	3.82	string0.010	0.059	string0.003	5.37 $\pm$ 1.11	6.35 $\pm$ 2.61	2.05 $\pm$ 0.34	$\cdots$
00002596$+$7433383	4952	3.09	$-$0.114	0.182	$-$0.276	9.30 $\pm$ 1.25	8.75 $\pm$ 3.33	4.27 $\pm$ 0.23	$\cdots$

Equations

$$F_8^{(\mathrm{seismic})}=\beta {\left(\frac{{\nu }_{max}}{{\nu }_{max,\odot }}\right)}^s{\left(\frac{M}{M_{\odot }}\right)}^t{e}^{u[M/H]}$$

$$F_8^{(\mathrm{spec})}=\alpha e^{a\log \mathrm{g}}{\left(\frac{T_{\mathrm{eff}}}{T_{\mathrm{eff},\odot }}\right)}^b{e}^{c[M/H]}$$

$$\ln F_8^{(\mathrm{spec})}=\ln \alpha +a\log g+b\ln \left(\frac{T_{\mathrm{eff}}}{T_{\mathrm{eff},\odot }}\right)+c[M/H].$$

$${\tilde{\sigma }}_{\mathrm{tot}}^2\left(a,b,c\right)={\tilde{\sigma }}_{F_8}^2+a^2{\tilde{\sigma }}_{\log \mathrm{g}}^2+b^2{\tilde{\sigma }}_{T_{\mathrm{eff}}}^2+c^2{\tilde{\sigma }}_{[M/H]}^2$$

$$\Lambda (\mathbf{\xi })={\Lambda }_0(\mathbf{\xi })-\frac{1}{2}\sum _{i=1}^N{\left[\frac{{\Delta }_i(\mathbf{\theta })}{{\tilde{\sigma }}_{\mathrm{tot},\mathrm{i}}(\mathbf{\theta })}\right]}^2,$$

$${\Lambda }_0(\mathbf{\xi })=-\sum _{i=1}^N\ln \sqrt{2\pi }{\tilde{\sigma }}_{\mathrm{tot},\mathrm{i}}(\mathbf{\xi }).$$

$${\Delta }_i(\mathbf{\xi })=\ln F_8^{\mathrm{obs}}-\ln F_8^{\mathrm{pred}}(\mathbf{\xi }).$$

$${\tilde{\sigma }}_{\mathrm{tot}}^2\left(s,t,u\right)={\tilde{\sigma }}_{F_8}^2+s^2{\tilde{\sigma }}_{{\nu }_{\max }}^2+t^2{\tilde{\sigma }}_M^2+u^2{\tilde{\sigma }}_{[M/H]}^2.$$

Full text

1. Introduction

One of the main challenges to measuring precise masses of extrasolar planets is that many stars exhibit Doppler noise with an amplitude comparable to or larger than the radial-velocity (RV) signal of the planet (e.g., Saar et al., 1998; Wright, 2005; Pont et al., 2011). This is especially the case for very low-mass planets and/or planets on wide orbits, where the RV amplitude of the stellar reflex motion can be on the order of $\sim$1 m s*-1* or smaller. If the star is highly magnetically active, such that the stellar surface has large spots, the RV noise can be tens of m s*-1* in amplitude (e.g., Dumusque et al., 2011), though it may also be sufficiently coherent that it can be modeled and removed, in particular if a simultaneous light curve is available (see, e.g. Aigrain et al., 2012; Dumusque et al., 2014).

RV noise can also arise from less coherent phenomena such that it is much more challenging to mitigate. For example, stellar plages represent a kind of chromospheric activity-related phenomenon in solar-type stars that can produce significant RV variations (so-called “RV jitter”) that may be less amenable to modeling from the light curve. Nonetheless, it may be possible for some dwarf stars to estimate the amplitude of the plage-driven RV noise if a measure of the stellar rotation period is available, such as from a light curve (see, e.g., Saar, 2003). More generally, attempts have been made to establish empirical relationships between chromospheric activity proxies (e.g., UV emission, Ca $R^{\prime}_{HK}$, etc) and RV jitter (e.g., Cegla et al., 2014).

However, RV jitter can also arise—at levels of up to $\sim$20 m s*-1*—in stars that are otherwise devoid of magnetic activity-driven variability (e.g., Wright, 2005), including very slowly rotating dwarfs, subgiants, and red giant stars. Indeed, RV planet surveys have been stymied in some cases from achieving their nominal RV precision of $\lesssim$1 m s*-1*, despite carefully selecting stars known to be chromospherically inactive (see, e.g., Wright, 2005; Isaacson & Fischer, 2010) and mitigating instrumental and astrophysical effects (Lovis & Fischer, 2010; Beatty & Gaudi, 2015). In the era of large-scale searches for small, transiting planets via precise light curves, such as CoRoT (Auvergne et al., 2009), MOST (Walker et al., 2003), Kepler (Borucki et al., 2010), and TESS (Ricker et al., 2015), it has become increasingly important to evaluate individual stars for the possibility of high RV jitter that may not be otherwise discernible from the light curve itself or from other activity proxies. For example, a key science goal for TESS is to identify at least 50 Earth-like transiting planets whose masses can be measured from follow-up RV observations (Ricker et al., 2015).

A likely driver of RV jitter that is independent of magnetic activity is convective motions at the stellar surface (i.e., granulation), as demonstrated by, e.g., Bastien et al. (2014). Indeed, Bastien et al. (2014) found that, among otherwise photometrically “quiet” stars (whose overall photometric variability is $\lesssim$3 parts per thousand; ppt), the observed RV jitter can range in amplitude from $<$4 m s*-1* to 20 m s*-1*, and that this RV jitter is most strongly correlated with the granulation “flicker”. The 8-hour Flicker ($F_{8}$) first defined by Bastien et al. (2013) refers to the low-level photometric variations arising from granulation in the integrated light of a star occurring on timescales shorter than 8 hr. The typical $F_{8}$ amplitudes in white light (such as in a Kepler light curve) range from $\approx$0.015 ppt to $\approx$0.4 ppt, depending most directly on the stellar surface gravity, $\log g$ (Bastien et al., 2016). Because the $F_{8}$ amplitude is so small, it generally can only be measured in an ultra-precise light curve such as from Kepler or TESS, but it provides access to predicting RV jitter amplitudes for otherwise photometrically “quiet” stars whose RV noise would be difficult to predict in other ways.

More recently, Oshagh et al. (2017) demonstrated that the predictive power of $F_{8}$ extends to stars with overall photometric activity amplitudes as large as $\sim$10 ppt and RV jitter as high as $\sim$100 m s*-1*. Thus, measuring or estimating $F_{8}$ for large numbers of stars to be targeted by TESS could serve to refine the RV followup strategy by focusing on those planet candidates whose stellar hosts are most likely to be sufficiently RV stable to enable precise measurement of planetary masses. At the same time, through its strong correlation with $\log g$, $F_{8}$ provides an opportunity to measure stellar masses and radii for stars that will be observed by TESS and Gaia (see, e.g., Stassun et al., 2018a). Including both TESS targets slated for 2-min cadence observations (Stassun et al., 2018b) as well as those that will be observed in the 30-min cadence full-frame images (e.g., Oelkers & Stassun, 2018), should enable precise, fundamental stellar masses and radii for tens of thousands of stars (see Stassun et al., 2018a).

Recent work has also demonstrated that while $F_{8}$ correlates most fundamentally with $\log g$, opacity effects cause it to also depend on the stellar metallicity. For example, Corsaro et al. (2017) used a small sample of red giant stars in open clusters observed by Kepler to show that a change in [M/H] of $\sim$0.5 dex can produce a change in $F_{8}$ of $\sim$50%. Accounting for the metallicity dependence of $F_{8}$ can therefore improve predictions of granulation-driven RV jitter and also improve estimates of stellar $\log g$. Indeed, Stassun et al. (2018a) found in a sample of Kepler stars having Gaia parallaxes and asteroseismically determined masses and radii (e.g., Huber et al., 2017), the precision on the stellar $\log g$ inferred from $F_{8}$ improved from $\sim$0.1 dex to $\sim$0.05 dex when accounting for the dependence of $F_{8}$ on [M/H]. Consequently, the precision with which individual stellar masses can be estimated with $F_{8}$ from a TESS light curve improves from $\sim$20% to $<$10% (see Stassun et al., 2018a). In turn, such precise stellar measurements will permit even more precise determinations of planet masses and radii (see, e.g., Johns et al., 2018; Berger et al., 2018; Stassun et al., 2017).

The purpose of this paper is twofold. First, we seek to use the large sample of stars with Kepler light curves for which accurate masses, radii, temperatures, and metallicities are now also available in order to refine the relationships between $F_{8}$, $\log g$, and [M/H]. Second, we aim to use these refined relationships to predict the granulation-driven RV jitter amplitudes for a large number of stars that will be observed by TESS.

In Sec. 2 we summarize the data that we use in this study, as well as the method that we use to robustly estimate the dependence of $F_{8}$ on $\log g$, [M/H], and other parameters. Sec. 3 presents the main results of this study, including our refined $F_{8}$ relationships, their application to a very large sample of likely TESS targets having spectroscopic parameters in order to infer their $F_{8}$, and finally our predicted RV jitter amplitudes for this large sample of stars. We discuss these results, caveats, and guidelines for their use in Sec. 4. Finally, we conclude with a summary in Sec. 5.

2. Data

In this work, we use two spectroscopically characterized samples of stars: a smaller calibration sample with asteroseismic measurements, and a broader sample with only spectroscopic information to which we apply the relationship we derive. We discuss the spectroscopic and asteroseismic characterization of the 2465 stars from the APOGEE-Kepler overlap sample used for calibration in Section 2.1.1 and the broader sample of 129,055 stars with only APOGEE spectra in Section 2.1.2. For the calibration sample, we also require $F_{8}$ measurements from Kepler, which are described in Section 2.2.1. For the larger sample, it is important to identify active stars likely to deviate from the flicker relation, and our method for doing so is discussed in Section 2.2.2.

2.1. Spectroscopy & Asteroseismology

2.1.1 The APOGEE-Kepler sample

In order to calibrate an accurate relationship between $F_{8}$, RV jitter, and spectroscopic observables, we choose to use the combined APOGEE-Kepler (APOKASC) sample of dwarfs, subgiants, and first ascent red giants. This sample is ideal for this sort of work because it has a large number of stars covering a wide range for parameter space. They have been precisely characterized using both asteroseismology, the study of stellar oscillations (e.g. Aerts et al. 2010), and high resolution spectroscopy. Because of the change in frequency of oscillations as a star moves across the HR diagram, the asteroseismic characterization of these stars is naturally broken up into a dwarf/subgiant sample, which oscillates more rapidly and requires short cadence data, and a red giant sample which can be characterized using the standard long cadence data from the Kepler mission.

Parameters for dwarfs and subgiants are taken from the analysis of Serenelli et al. (2017). These stars represent a subset of the Chaplin et al. (2011) sample of stars with detected seismic oscillations in the short cadence Kepler data. Seismic analysis was carried out using several pipelines, with the central values taken from the SYD pipeline (Huber et al., 2009), and improved via comparison with grids of stellar models. Spectroscopic parameters for these stars are taken from Data Release 14 (DR14, Abolfathi et al., 2017) of APOGEE-2 (Majewski et al., 2017), one of the components of the Sloan Digital Sky Survey IV (SDSS-IV, Blanton et al., 2017), which is using the 2.5m Sloan Digital Sky Survey telescope (Gunn et al., 2006) to take H-band spectra. The observations are normalized and compared to a grid of synthetic spectra by the automated ASPCAP pipeline (Nidever et al., 2015; García Pérez et al., 2016) and stellar parameters are determined by a global chi-squared minimization. For consistency with the red giant seismic analysis, we use the dwarf and subgiant results determined using the APOGEE spectroscopic temperatures.

We add to the dwarf and subgiant sample the sample of first ascent red giants analyzed in Pinsonneault et al. (2018) by five different seismic pipelines. The resulting seismic parameters were corrected for pipeline dependent systematics, averaged, calibrated to an absolute scale using the open clusters in the Kepler field, and corrected for deviations from homology (Serenelli et al., 2017). This analysis also uses temperatures from APOGEE DR14 (Abolfathi et al., 2017). We use only stars determined to be first ascent red giants by consensus of the APOKASC collaboration (Pinsonneault et al., 2018, see also Y. Elsworth et al. in prep) to avoid additional correlations in the expected flicker signal with evolutionary state (Bastien et al., 2016) and because most red clump stars fall outside of the validated gravity range of the flicker technique. We have also excluded KIC 9893440, which has a close companion contaminating the light curve, and thus an anomalous flicker measurement for its surface gravity. To prevent other unidentified binary stars from contaminating our results, we have also required that the stars in our sample have radial velocity variability of less than 100 m s*-1* in the APOGEE spectra.

Additionally, we restrict both our dwarf/subgiant and giant samples to stars where the flicker ($F_{8}$) is considered a reliable proxy of surface gravity via granulation, specifically, stars with log(g) between 4.6 and 2.5 dex (Bastien et al., 2016). We indicate the temperatures and gravities of the stars in our APOKASC sample in Figure 1. Basic data for all the APOKASC stars used in this analysis, including their evolutionary state, and listed in Table 1.

2.1.2 The APOGEE sample

Once we have fit a relation that predicts the flicker and the radial velocity jitter as a function of spectroscopic observables, we want to extend those predictions to a larger sample of stars. For this purpose, we take the stars from the full APOGEE survey, specifically Data Release 14 (Abolfathi et al., 2017). Where possible, we use corrected APOGEE parameters, otherwise we use the raw values from the ASPCAP pipeline.

2.2. Light Curves and Variability Measures

2.2.1 Flicker from Kepler

In order to develop our updated relationships between granulation Flicker ($F_{8}$) and other stellar parameters, we require a set of calibration stars for which precise $F_{8}$ measurements are available. The most extensive $F_{8}$ measurements published are those of Bastien et al. (2016), which provides $F_{8}$-based $\log g$ determinations for 27,628 Kepler stars brighter than 13.5 mag with 4500 K $<T_{\rm eff}<$ 7150 K, $2.5<\log g<4.6$, and overall photometric amplitudes of less than 10 ppt. The $F_{8}$ values in the catalog published by Bastien et al. (2016) were erroneous (see also Oshagh et al., 2017), and to our knowledge the corrected values have not been published. Therefore, we redetermined the $F_{8}$ values for the 2465 stars in our calibration sample using the same Kepler light curves and employing the same methodology as in Bastien et al. (2016).

2.2.2 Photometric Variability from KELT

Previous work examining the ability of $F_{8}$ to predict RV jitter has found that the $F_{8}$-jitter correlation holds for stars that are photometrically “quiet”, presumably because the RV jitter becomes increasingly dominated by magnetic activity effects (e.g., spots, plage) which manifest as large amplitude photometric variations. In particular, Bastien et al. (2014) estimated that the $F_{8}$-jitter relationship holds for stars with overall light curve amplitudes less than $\sim$3 ppt. Oshagh et al. (2017) found that the $F_{8}$-jitter relationship continues to hold for stars with overall light amplitudes as high as $\sim$10 ppt.

TESS will observe stars over nearly the entire sky, so to estimate the photometric variability amplitudes of as many stars as possible, we use the variability catalog of 4 million stars observed by the Kilodegree Extremely Little Telescope (KELT; Pepper et al., 2007, 2012) survey (Oelkers et al., 2018). In our analysis and results we flag stars whose photometric variability amplitudes have been determined to be less than the thresholds mentioned above.

3. Methods & Results

3.1. Recalibrating the Asteroseismic Flicker Relation

As discussed in Bastien et al. (2016), the flicker signal depends most strongly on surface gravity, and this is consistent with previous work that indicates that stellar granulation, for which flicker is a proxy, correlates with $\nu_{\rm max}$ (Hekker et al., 2012). More recent analysis by Corsaro et al. (2017) has shown that for red giants, stellar granulation also depends on the mass of the star and on its metallicity. However, the fit by Corsaro et al. (2017) (here after C17) was done using only 60 stars over the relatively small metallicity range, covered by the open clusters NGC 6811, NGC 6819 and NGC 6791, namely $-0.09<\mbox{[M/H]}<0.32$.

We show in Figure 2 that when the C17 relation is applied to the full metallicity and gravity range of our APOKASC sample, the resulting predictions are not consistent with the observed $F_{8}$ values. This is particularly true in the dwarf regime, a result that is not surprising since the C17 relationship was not calibrated in this range of surface gravities. In general, the C17 predictions for dwarfs are overestimated with a rather constant offset across the range of surface gravities. In addition, in the case of giant stars, the C17 relation implies a metallicity dependence that seems somewhat overestimated compared with observations.

Given that we have measurements of $F_{8}$ for our APOKASC sample, accounting for over an order of magnitude more stars than the original C17 paper, and the difference arising from the C17 predictions as highlighted in Figure 2, we perform a new fit using the same Bayesian framework as C17 in order to quantify the differences arising when using our new sample of stars. Specifically, we use the scaling law originally introduced by C17 (their Eq. 17)

[TABLE]

where $\nu_{\rm max}$ is the frequency of maximum oscillation power and directly proportional to the stellar surface gravity (Brown et al., 1991), $M$ is the stellar mass, and [M/H] the metallicity. This sort of scaling relation relies on the assumption of homology among stars, which is generally reasonable across our sample range. Estimating the exponents $(s,t,u)$ and the proportionality term $\beta$ of the scaling relation will allow us performing a direct comparison with the results obtained by Corsaro et al. (2017) using our sample of stars. Also, following an approach similar to that used by Corsaro et al. (2013), we decided to calibrate this relation separately for short cadence (SC, dwarf and subgiant) and long cadence (LC, giant) samples because, as already shown in Figure 2, the two regimes exhibit significantly different dependencies on the parameters considered.

The description of how the fits are obtained is given in Sect. 3.3. As shown in Table 2, using Eq. (1) we find strong evidence that the dependence of the fit on $\nu_{\rm max}$ is not the same for dwarf/subgiant sample and the giant sample, the latter being similar to the C17 result, while the former is almost twice as strong (See Figure 3). We find that the dependence on mass is similar for the dwarf/subgiant and giant samples, and that it is compatible within 1-$\sigma$ to the C17 fit. We also see the inverted trend or anti-correlation between the dependence on mass and the dependence on metallicity that was found by C17 in both LC and SC data.

However, while interestingly the metallicity dependence for the SC sample (u $\sim$ 0.64) is compatible within the quoted error bars with that found for the cluster data ($u\sim 0.9$), the metallicity dependence of the long cadence sample is a factor of about three times smaller ($u\sim 0.3$). Some of this difference can be explained by the change in the average metallicity measured for stars in NGC 6791 between APOGEE Data Release 13 (SDSS Collaboration et al., 2016), used by C17, and the DR14 data used here, but additional effects of $\alpha$-element abundances and [C/N] variations in stellar granulation that were not taken into account in this analysis may also play a role. We refer the interested reader to S. Mathur et al. (in prep) for a detailed analysis and discussion of other parameters besides those in our fit that could impact the relationship investigated here. The errors on the fitted parameters of the scaling relation are significantly smaller for the new fits, especially the LC sample, than those from C17 because of the much larger number of stars used in our analysis (about two orders of magnitude more than C17 for the LC sample and about one order of magnitude for the SC sample). For completeness, we also note that our results for the dwarf stars are consistent with the metallicity dependence found by Serenelli et al. (2017), although we prefer to adopt the C17 formalism because it additionally takes into account dependencies on mass and $\nu_{\rm max}$.

3.2. Calibrating the Spectroscopic Flicker Relation

Having calibrated the relationship between precise stellar properties of $\nu_{\rm max}$, mass, and flicker, we want to determine whether spectroscopic observables alone provide sufficient information to predict the flicker value. Using the insight from the seismic sample, we again divide our stars into a dwarf/subgiant and a giant sample, using a cut at $\log g=3.4$ dex, approximately equivalent to the cut used to divide the seismic samples. We also enforce a similar form for the flicker relation, searching for the exponents $(a,b,c)$ such that

[TABLE]

where $\log g$ is the stellar surface gravity, $T_{\mathrm{eff}}$ its temperature, and [M/H] is again the metallicity. $(a,b,c)$ are the corresponding exponents of the scaling relation, and $\alpha$ is a proportionality term; these are the parameters that need to be calibrated. Similarly to what done for Eq. (1), we apply this relation separately for SC and LC observations. We expect the differences between the SC and LC fits to be even more significant here since the parameter acting as the evolutionary coordinate is log(g) in the LC sample whereas for subgiants in the SC sample $T_{\mathrm{eff}}$ is serving that purpose.

We deliberately choose to adopt different letters for the exponents in the two scaling relations considered, because they represent different physical relationships even though the functional forms look similar. We therefore do not expect a priori that the exponents of the two scaling laws end up in having similar estimates, even for the case of the metallicity term, which formally appears in the same way in both relations of Eqs. (1) and (2). This is because different correlations among the observables $\nu_{\rm max}$, $M$, and [M/H] on one side and $\log g$, $T_{\mathrm{eff}}$, and [M/H] on the other could be present, thus changing the way each term in the scaling relation will contribute to the overall fit.

The resulting fit for the dwarf and giant samples are obtained following the same approach used for Eq. (1) and described in Sect. 3.3. The results are shown in Figure 4, and the exponents and their uncertainties are listed in Table 3. It is worth mentioning that the differences in the fitted parameters between SC and LC sample are even more prononced when using the spectroscopic quantities, in particular with an inverted trend in the exponents related to temperature and metallicity. While in the case of the SC sample the trend with metallicity is in agreement with that found by C17 and our new fit using the asteroseismic quantities, for the LC sample this trend has an opposite direction, although the exponent is only slightly different than zero. We motivate this change as the combined effect of an offset $\ln\alpha$ having an opposite sign from SC to LC sample, a strong correlation between the exponent $b$ and $\ln\alpha$ (about 0.8), and the correlation between $b$ and the exponent $c$ of the metallicity term (corresponding to 0.46).

The rms scatter of the spectroscopic fit was compared to the results from the seismic fit. This indicates that using only spectroscopic observables allows us to predict the flicker value to 6.6 % for the LC sample and 19.7 % for the SC sample, with an error likely dominated by the 0.02 dex uncertainty in surface gravity. This results in an overall precision arising from the fit to the spectroscopic relation with respect to the asteroseismic relation that is about 38 % and 47 % lower for the LC and SC samples, respectively.

3.3. Bayesian Inference Approach

For estimating the free parameters of the scaling laws given by Eq. (1) and Eq. (2), we adopt a Bayesian approach similar to that originally used by Corsaro et al. (2013), and subsequently by Bonanno et al. (2014) and by C17.

Considering Eq. (2) as a reference, the actual fitting model that we need to consider is that obtained by converting the scaling relation to its natural logarithm, yielding

[TABLE]

In this way, as shown by Corsaro et al. (2013), we can take into account the uncertainties on all the observed quantities, including the dependent variable $F_{8}$, by implementing an analytical, parameter-dependent relation for the total relative uncertainty on the flicker predictions, which is given as

[TABLE]

and includes the relative uncertainties on surface gravity, temperature and metallicity, on top of those of the measured flicker of stars. Similarly to what reported by C17, we note that in this case $\widetilde{\sigma}_{\mathrm{\log g}}$ and $\widetilde{\sigma}_{\mathrm{[M/H]}}$ correspond to the formal uncertainties in $\log g$ and [M/H], respectively, since the two terms are logarithmic quantities used in the original scaling relation, and their uncertainties are therefore already in relative units.

For the statistical inference on the flicker data we apply uniform priors on each free parameter of the scaling relations and consider a modified Gaussian log-likelihood of the form

[TABLE]

with $N$ the total number of flicker measurements, $\boldsymbol{\xi}$ the parameter vector of the free parameters (e.g. $(a,b,c)$, and $\Lambda_{0}(\boldsymbol{\xi})$ an additional term depending only on the relative uncertainties, that is

[TABLE]

Finally, as shown in C17, we note that the residuals between observed and predicted flicker values are defined as

[TABLE]

A perfectly analogous approach is used for the scaling law given by Eq. (1), in which the total relative uncertainty obtained is given as

[TABLE]

3.4. Application to the full APOGEE DR14 sample

Using our new empirical relations between $F_{8}$ and spectroscopic stellar parameters, in Table 3.4 we report $F_{8}$ for all 129,055 APOGEE DR14 stars having $T_{\rm eff}$, $\log g$, and [M/H] measurements, although we caution that flicker is only formally valid for 4500 K $<T_{\rm eff}<$ 7150 K and $2.5<\log g<4.6$, and in fact the flicker-gravity relation turns over at $\log g<2.5$ as the timescale of granulation shifts out of the frequency window measured by flicker. A Hertzsprung-Russell diagram for these stars is presented in Figure 5(a), and the resulting $F_{8}$ as a function of $\log g$ and [M/H] is shown in Figure 5(b).