A Measurement of Stellar Surface Gravity Hidden in Radial Velocity Differences of Co-moving Stars
Matthew Moschella, Oren Slone, Jeff A. Dror, Matteo Cantiello, Hagai B. Perets
DDraft version February 3, 2021
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A Measurement of Stellar Surface Gravity Hidden in Radial Velocity Differences of Co-moving Stars
Matthew Moschella, Oren Slone,
1, 2
Jeff A. Dror,
3, 4, 5
Matteo Cantiello
6, 7
And Hagai B. Perets — Department of Physics, Princeton University, Princeton, NJ 08544, USA Center for Cosmology and Particle Physics, Department of Physics, New York University, New York, NY 10003, USA Department of Physics and Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, CA 95064, USA Theory Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Berkeley Center for Theoretical Physics, University of California, Berkeley, CA 94720, USA Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA Faculty of Physics, Technion – Israel Institute of Technology, Haifa, 3200003, Israel
Submitted to AJABSTRACTThe gravitational redshift induced by stellar surface gravity is notoriously difficult to measurefor non-degenerate stars, since its amplitude is small in comparison with the typical Doppler shiftinduced by stellar radial velocity. In this study, we make use of the large observational data set ofthe
Gaia mission to achieve a significant reduction of noise caused by these random stellar motions.By measuring the differences in velocities between the components of pairs of co-moving stars andwide binaries, we are able to statistically measure gravitational redshift and nullify the effect of thepeculiar motions of the stars. For the subset of stars considered in this study, we find a positivecorrelation between the observed differences in
Gaia radial velocities and the differences in surfacegravity inferred from effective temperature and luminosity measurements. This corresponds to thefirst ever measurement of extra-Solar surface gravity induced gravitational redshift in non-degeneratestars. Additionally, we study the sub-dominant effects of convective blueshifting of emission lines,effects of binary motion, and possible systematic errors in radial velocity measurements within
Gaia .Results from the technique presented in this study are expected to improve significantly with datafrom the next
Gaia data release. Such improvements could be used to constrain the mass-luminosityrelation and stellar models which predict the magnitude of convective blueshift. INTRODUCTIONThe advent of the
Gaia space telescope has given riseto a new era of precision astrometry with a current cat-alog that includes over one billion stars, of which sev-eral millions have radial velocity (RV) measurements.Such a large data set offers new opportunities to statis-tically measure stellar properties. In this Letter, we useRV measurements from
Gaia to measure the gravita-tional redshift (GR) due to stellar surface gravity (SG),
Corresponding author: Matthew [email protected] as well as the subdominant effect of convective outflowsand downflows at the stellar surface.Understanding these effects is important since theygive rise to systematic noise in RV measurements, and inparticular raise difficulties in RV detection of exoplan-ets (see e.g. Wright 2018, for a review). Additionally,measurement of convective effects could shed light onphysical processes occurring within stars and constraincurrent and future modeling of these processes.The
Gaia mission performs RV measurements by mea-suring the Doppler shift of a few common emissionlines using the Radial Velocity Spectrometer (RVS). Al-though the frequency shift observed by the RVS is re- a r X i v : . [ a s t r o - ph . S R ] F e b Figure 1.
The observed correlation between differences in radial velocity measurements between co-moving stellar pairs and theexpected difference due to gravitational redshift and the smaller effect of convective blueshift.
Left panel:
Stars that passedthe selection criteria but do not have precise spectroscopic mass measurements (primary data set). Since mass measurementsare imprecise for this data set, convective blueshift modeling is challenging.
Right panel:
The subset of co-moving pairs whichhave precise mass measurements (secondary data set). For each panel, a model which perfectly fits the data would correspondto a linear fit of the form f ( x ) = x , with a slope of unity. The observed best fit linear curves are given as insets in each panel.The result for the primary (secondary) data set is consistent with a non zero slope at the ∼ σ ( ∼ . σ ) level. The large errorfor the secondary data set is mainly due to low statistics. ported as a relative velocity between the emitting starand the Gaia satellite, there are many additional effectsthat can contribute (see e.g. , Lindegren & Dravins 2003,for an overview). In particular, the SG of the emittingstar results in a redshift of line frequencies measured bythe RVS, biasing RV measurements to be slightly morepositive than their true values. While the true velocitiesare typically on the order of the galactic virial velocity, v gal ∼
220 km/sec, typical values from GR are, v GR ≈ . (cid:18) M M (cid:12) (cid:19) (cid:18) R (cid:12) R (cid:19) kmsec . (1)Thus, GR in non-degenerate stars is usually a small con-tribution to the RV.An additional effect on RV measurements arises fromconvective motion on the stellar surface. Hot and lu-minous outflows typically result in a net blueshift ofemitted photons and measured RVs appear, on average,smaller (more negative) than their true values. This ef-fect is known as convective blueshift (CB). The size ofthe effect depends on stellar type and can induce effec-tive RV measurements of order, v CB ∼ O (0 . v gal . This difficulty has been overcome in only afew types of systems. Original measurements of GR were performed on white dwarfs in binary systems, most no-tably the nearby Sirius B (Greenstein et al. 1971). Thesecan have GRs as large as O (100 km / sec) (Greenstein &Trimble 1967), making detection relatively straightfor-ward. Additionally, there have been long-standing ef-forts to extract the GR of the Sun, which is expectedto be 636 . / sec (Lindegren & Dravins 2003). Thisis challenged by a CB of comparable size, which itselfdepends sensitively on the spectral line and its angularposition. A recent clean measurement and some discus-sion of its history can be found in Gonz´alez Hern´andezet al. (2020).The difficulty of disentangling GR from physical mo-tion in stars can be reduced with statistics of large datasets and an appropriate choice of stellar system. Aprevious attempt to statistically measure GR in ordi-nary stars was performed using the M67 open clustersin Pasquini et al. (2011). However, that study was un-able to extract a signal above the background, which theauthors hypothesized as due to unsubtracted CB effects.More recently, Dai et al. (2019) attempted a measure-ment of stellar GR by averaging Gaia
RV measurementsover stars of different types. Their results show some ev-idence for a combination of GR and CB, however theydid not attempt to distinguish between or characterizethe different contributions.In this Letter, we search for GR in co-moving pairs ofstars, thousands of which have been identified in
Gaia
DR2. If these co-moving pairs are wide binaries, theirorbital velocities should be small, providing a clean dataset with which to study small contributions to RVs. Inparticular, the difference of the RVs of two stars in awide binary can be dominated by the difference in v GR ,especially for pairs of stars with sizeable differences inmass and/or radius. This opens the possibility of usingwide binaries to directly probe GR and indirectly gaininformation regarding stellar structure and dynamics.The main results of this study are presented in Fig. 1,which shows the correlation between the differences ofthe observed RVs of stars in co-moving pairs and the ex-pectation from GR and CB. The left panel correspondsto a data set with higher statistics but also high uncer-tainty on stellar mass, while the right panel correspondsto a smaller sample with high quality mass measure-ments. A positive correlation is observed, correspondingto an agreement between model and data within mea-surement errors, with a non-zero slope at the ∼ . σ − σ level. This marks the first conclusive measurement ofGR for non-degenerate stars (other than the Sun).The Letter is structured as follows. First, we presentdetails of the data selection used for this study. We fol-low with a discussion of GR- and CB-induced changes inmeasured RVs. Next, we present our analysis techniqueand discuss the dispersion observed in the data, con-cluding with a discussion of our main results, possibleissues, and future prospects. DATA SELECTIONComoving pairs in physical (3-dimensional) velocitywere first compiled using
Gaia
DR1 by Oh et al. (2017).Such stars are typically wide binaries with separationdistances (cid:38)
25 AU Allen et al. (2000), and are distinctfrom typical binaries in that they are likely to have beenborn at the same time and share a similar chemical com-position. This has led to considerable interest in com-piling lists of wide binaries with 3741 pairs identifiedin DR2 Jim´enez-Esteban et al. (2019), and statisticalanalyses performed by Zavada & P´ıˇska (2018). Addi-tionally, Hartman & Lepine (2020) created the SUPER-WIDE catalog of wide binaries. This catalog is selectedfrom
Gaia
DR2 and the SUPERBLINK high proper mo-tion catalog (L´epine 2005; L´epine & Gaidos 2011). Im-portantly, selection of these wide binary pairs is com-pletely independent of RV measurements.We analyze the subset of SUPERWIDE pairs thatalso have RV measurements in
Gaia
DR2. Addition-ally, we require that the difference in the RVs of eachstar in a pair, ∆RV, has a small measurement error, (cid:113) σ , + σ , < / sec, and we remove outlierpairs with | ∆RV | > / sec. To reduce contaminationfrom the orbital velocities of wide binaries, we require Figure 2.
The Hertzsprung-Russell Diagram for the datasets used in this study. Grey points correspond to all SU-PERWIDE stars that passed the basic quality cuts. Of these,stars in the region between the solid red curves are classifiedas main sequence and stars above the dashed red curve areclassified as giants. Pairs for which either star is outside theseregions are removed from the primary data set. Blue pointscorrespond to stars in pairs for which both have spectroscop-ically determined mass estimates (secondary data set). that the transverse separation of the stars in a pair, d , begreater than 10 − pc. Finally, we require that all starshave estimated values for effective temperature, T eff , lu-minosity, L , and radius, R , from Gaia
DR2. A total of1,682 pairs pass these selection criteria.Since we are interested in estimating the masses andradii of stars in our data set with high accuracy, it isbeneficial to categorize stars as either main sequence(MS) stars or as giants. The selection criteria used inthis study are marked in red in Fig. 2. MS stars aredefined to lie within the solid curves. Giants are de-fined to lie above the dashed curve, which correspondsto
R > (cid:12) . A total of 1,139 pairs pass these additionalselection criteria in our primary data set, of which 1,084contain two MS stars, 53 contain one MS star and onegiant, and 2 contain two giant stars. These curves are a multiplicative factor of 2 from an L ( T eff ) curvewhich corresponds to the luminosity at which the highest numberdensity is obtained across all stars in Gaia
DR2 that pass the se-lection criteria suggested in Gaia Collaboration et al. (2018). Thecurve is restricted to 4000 K < T eff < L > − . L (cid:12) to ensure that the mass-luminosity relation from Malkov (2007)is valid. These criteria also assist in excluding wide binaries thatare in fact triple systems containing an unresolved inner binary,which otherwise might contaminate the sample. We estimate the masses of MS stars using the mass-luminosity relation,log (cid:20) M M (cid:12) (cid:21) ≈ α + α log (cid:20) L L (cid:12) (cid:21) + α log (cid:20) L L (cid:12) (cid:21) , (2)where α = 0 . α = 0 . α = 0 . . (cid:12) inthe masses estimated using Eq. (2).For giant stars, the stellar mass can in principle beinferred directly from photometric observations via as-teroseismology. However, this requires long photomet-ric observation times and cannot be done with Gaia photometry. In order to preserve a sizeable statisticalsample of giants, we uniformly set the masses of all se-lected giants to the average asteroseismologically deter-mined mass from the
Kepler mission, 1 . (cid:12) (Yu et al.2018). We estimate the uncertainty on the fixed giantmass to be 0 . (cid:12) . This ad hoc estimation of the giantmasses has a negligible effect on the GR signal in a MS- giant comoving pair, since the radius selection cut of R > (cid:12) ensures that the GR contribution from giantsis (cid:46)
20% of that from MS stars.In addition to the primary data set discussed above,we also identify a distinct subset of comoving pairs forwhich both stars have high quality spectroscopic massand radius measurements from Sanders & Das (2018).Stars from this set of 114 pairs are marked in blue inFig. 2. With these two data sets at hand, we proceed toestimate the effects of differences in GR and CB betweenstars in each pair. GRAVITATIONAL REDSHIFTThe difference in measured RV between two stars (la-belled 1 and 2), induced by GR alone, is given by∆ v GR = G (cid:18) M R − M R (cid:19) . (3)Since giants have considerably larger radii, they tendto contribute negligibly to GR. Additionally, the GR oftwo MS stars tends to partially cancel. Thus, the largestsignal comes from MS - giant pairs.The masses and radii in Eq. (3) are estimated as de-scribed above (depending on the data set being used).For each co-moving pair, a distinct value of ∆ v GR can becalculated and compared to the measured ∆RV of thepair. This comparison is complicated by the presence ofsub-dominant effects such as CB. CONVECTIVE BLUESHIFT The surface of a star is characterized by the presence ofconvective cells where hot gas rises outwards, and coolergas sinks through intergranular lanes. Since the hotterrising gas is brighter than the cooler sinking gas, a netblueshift is produced in the disk-integrated light emittedby the star. This phenomenon is usually referred to asCB (e.g. Beckers & Nelson 1978; Dravins 1982).Since CB depends on complex details of turbulentconvection, it is notoriously difficult to predict. Thispresents a major challenge in, for example, exoplanetsearches looking for modulations in spectral lines due toplanetary motion around distant stars (see e.g. Wright2018, for a review). Attempts to estimate the amplitudeof this effect for different stellar types rely on hydrody-namic simulations of surface convection. Such calcula-tions were performed in Allende Prieto et al. (2013) fortypical stellar types in the
Gaia catalog, finding am-plitudes of CBs in the range v CB ≈ . − . / sec,depending on the mass and temperature of the star.While magneto-hydrodynamic calculations might pro-vide a more complete picture, recent measurements showthat magnetic activity is only mildly correlated withCB (Meunier, N. et al. 2017; Meunier et al. 2017).To parametrize the effect of CB we define the differ-ence between stars 1 and 2 as ∆ v CB ≡ v CB , − v CB , (note that v CB ,i are typically negative) and apply thetheoretical fitting formula provided in Allende Prietoet al. (2013). This fit requires estimates of both metallic-ity and SG, which are not reported in Gaia spectroscopicdata. For our primary data set, we estimate metallic-ity using the
Gaia
RV template metallicity, and SG us-ing the mass and radius estimates described above. Forour secondary data set, we use metallicity and SG mea-surements reported in Sanders & Das (2018). Althoughthe dependence of CB on metallicity is relatively weak,∆ v CB is highly sensitive to SG. Therefore, the use ofEq. (2) may introduce sizeable errors in ∆ v CB for theprimary data set. ANALYSISFor each pair in our data sets we calculate ∆ v GR ac-cording to Eq. (3) and ∆ v CB using the fitting formulaof Allende Prieto et al. (2013). For stars in our pri-mary data set, we estimate stellar masses and radii asdescribed above; for our secondary data set we use themass and radius estimates provided in Sanders & Das(2018).In the absence of background effects, one expects ∆RVto be directly correlated with ∆ v GR + ∆ v CB . This rela-tionship is shown in Fig. 1. The left panel correspondsto the primary data set, for which the lack of high qual-ity mass estimates introduces large errors in ∆ v GR and Figure 3. Left panel:
The dispersion in ∆RV as a function of projected separation for the data used in this study comparedwith Monte Carlo mock data of wide binaries.
Right panels:
Distributions of | ∆RV | , | ∆TV | and d . For the mock data,the distribution of eccentricities is modeled either as a power law of the form f e ∝ e η or as circular orbits for all binaries.Differences resulting from the choice of eccentricity distribution are negligible and the consistency of the real data and mockdata distributions in the right panels validate the Monte Carlo procedure. We find that the dispersion in the data is likelydominated by binary motion at low d , measurement errors at large d , and some unknown source at intermediate d . ∆ v CB . The right panel corresponds to the secondarydata set for which ∆ v GR + ∆ v CB is more precise butstatistics are much lower. Grey points represent thepairs in each data set along with their correspondinguncertainties. The ordering of stars in each pair is cho-sen such that the value on the horizontal axis is positive.Orange points correspond to the average value of ∆RVin bins of ∆ v GR + ∆ v CB , and are shown for illustrativepurposes only.In each case we test how well the model (horizontalaxis, denoted x ) fits the data (vertical axis, denoted y ).If all background effects averaged to zero, one would ex-pect a linear fit of the form y = f ( x ) with f ( x ) = ax and a = 1. We find that this linear model is a poor fit toboth data sets, mainly due to a dispersion of the mea-sured ∆RV values which is larger than that expectedfrom measurement errors alone. That is, we find thatthere is an intrinsic source of dispersion in our ∆RVmeasurements. We discuss possible sources of this dis-persion below and account for it by introducing an ad-ditional model parameter, σ . We then analyze the datausing the Gaussian likelihood, L ( a, σ | x , y ) ∝ (cid:112)(cid:81) i ˜ σ i exp (cid:34) − (cid:88) i ( y i − ax i ) ˜ σ i (cid:35) , (4)where ˜ σ i = a σ x i + σ y i + σ is the effective uncertaintyof each data point, and the index i runs over all pairs ina given data set. This likelihood is maximized and usedto compute confidence intervals for the slope parameter a , with the dispersion σ treated as a nuisance param- eter. The dashed black lines in Fig. 1 represent thefunction f ( x ) = ax , where a is the maximum likelihoodvalue of the corresponding data set, and the blue bandscorrespond to the 1 σ (68%) confidence interval aroundthis slope. For our primary (secondary) data set, wefind that the maximum-likelihood excess dispersion is σ = 0 . ± .
09 (0 . ± .
3) km/sec. EXCESS DISPERSION OF THE DATARadial orbital motions of binary systems should con-tribute a dispersion to ∆RV measurements and could bethe source of σ . To check this assumption, we have per-formed a Monte Carlo (MC) analysis by simulating 5 · binary systems with random orientations, including theeffects of ∆RV measurement errors in Gaia . The massdistribution of stars within the simulation and ∆RVmeasurement errors are taken from the data. The dis-tribution of semi-major axes is estimated by performinga de-convolution (Lucy 1974) of the projected 2D sep-aration of stars, d , as measured in the data, under theapproximation of circular orbits for all binaries. The dis-tribution of eccentricities is modeled either as a powerlaw (Moe & Di Stefano 2017), such that the distributionscales as f e ∝ e η with η = 0 or 1, or taken to be zerofor all binaries (circular orbits).The left panel of Fig. 3 shows the dispersion of mea-sured ∆RV as a function of d for the primary data set,compared with the MC results. The MC curves are dom-inated by binary motion at small values of d and by∆RV measurement errors at large d . The data is a goodfit to the MC curve, except at intermediate values of systematic checks slopeprimary 1 . ± . . ± . . ± . R giant >
10 R (cid:12) . ± . M giant = 2 M (cid:12) . ± . M giant = 0 M (cid:12) . ± . | ∆RV | < . ± . | ∆RV | <
10 km/sec 1 . ± . d > . ± . d > − . pc 1 . ± . (cid:113) σ , + σ , < ∞ . ± . Table 1.
Table indicating the systematic variations to ourprimary data selection and modelling ( left ) and the result-ing slope of the best-fit linear relation ( right ). The “nar-row main sequence” variation corresponds to making themain sequence selection bands in Fig. 2 three times nar-rower. The “observational main sequence” variation corre-sponds to selecting main sequence stars within ± . R giant >
10 R (cid:12) ” variation corre-sponds to adjusting the radius selection cut for giant starsto be
R >
10 R (cid:12) . The “ M giant ” variations correspond tovarying the fixed value used for the masses of all giant starsas indicated. The remaining variations are in the maximumdifference in radial velocity, | ∆RV | , the minimum distancebetween the transverse separation between the stars, d , andthe maximum error in the radial velocity measurement. d ≈ − − − pc, hinting that binary motion is a sig-nificant source of excess dispersion at small separations.However, since the MC underestimates the dispersion atintermediate separations, there is likely an unidentifiedadditional contribution.One possibility is that the RV errors quoted in the Gaia catalog are underestimated. The excess dispersionfound in this study can be explained if there are exper-imental uncertainties of O (1 km / sec) present in Gaia
RVS measurements. DISCUSSIONThe main results of this study are summarized inFig. 1. We find a significant correlation between ∆RVand ∆ v GR + ∆ v CB , corresponding to a linear fit of theform f ( x ) = ax , with a slope of order unity. Specifically,for both data sets considered in this study, the best fitslope is consistent with a = 1 to within measurementerrors and inconsistent with zero at the ∼ σ ( ∼ . σ )level for the primary (secondary) data set. In the sec-ondary data set, even though the data is more precise,there is a large error in a due to low statistics. In addition to the results shown in Fig. 1, we have per-formed our analysis with slight variations to the data se-lection and parameter estimation described above. De-tails of these variations and the resulting values for a are summarized in Table 1. We find that these modifi-cations to the analysis do not qualitatively change ourresults.We find an excess dispersion in the data of order ∼ . Gaia are underestimated, however we areunable to pinpoint the origin of the excess dispersionwith a high level of certainty.Lastly, we note that
Gaia
RVS measurements mayexhibit a systematic bias toward high RVs for dimstars (Katz et al. 2019) with corrections up to 0 . / secfor apparent magnitudes of G ≈
12, and negligible biasfor stars brighter than G ≈
4. Since apparent mag-nitude is correlated with stellar SG, such a systematicbias would serve to enhance, or possibly mimic a GRsignal, and might explain the fact that we consistentlyfind a >
G <
G >
4, with aslope of (0 . / sec) / (8 mag), reduces the slope in theleft panel of Fig. 1 to a = 0 . ± .
23. One potentialdiscriminator between such a systematic error and a GRsignal is that a dependence on G need not manifest as alinear relation in Fig. 1 (due primarily to the non-trivialrelation between apparent magnitude and SG). With fu-ture data, one could robustly test whether this effect issubstantial.In principle, modifications of the technique developedin this study could be used to test relations such asEq. (2) and the v CB fitting formula provided in AllendePrieto et al. (2013). Ultimately, this would be a novelprobe of stellar structure and could, for example, assistin an improved understanding of systematics currentlyprohibiting better measurements of exoplanets and theirproperties. With the current data, low statistics limit ameaningful test of such models; however, the next Gaia data release (DR3) is expected to include many morestars with RV measurements and may allow for such ananalysis to be performed.In the current study we have undertaken the task ofshowing that a significant signal can be extracted from
Gaia data. This corresponds to the first ever measure-ment of extra-Solar GR in non-degenerate stars.ACKNOWLEDGMENTSWe thank David Hogg for advise and guidance in theearly stages of this study. We also thank David Spergel, Jason Hunt and George Seabroke for useful conversa-tions. JD was supported in part by the DOE undercontract DE-AC02-05CH11231 and in part by the NSFCAREER grant PHY-1915852.REFERENCESdata. This corresponds to the first ever measure-ment of extra-Solar GR in non-degenerate stars.ACKNOWLEDGMENTSWe thank David Hogg for advise and guidance in theearly stages of this study. We also thank David Spergel, Jason Hunt and George Seabroke for useful conversa-tions. JD was supported in part by the DOE undercontract DE-AC02-05CH11231 and in part by the NSFCAREER grant PHY-1915852.REFERENCES