Homing in on Polaris: A 7 M ⊙ first-overtone Cepheid entering the instability strip for the first time
AAstronomy & Astrophysics manuscript no. HomingInOnPolaris c (cid:13)
ESO 2018March 21, 2018
Letter to the Editor
Homing in on Polaris: A 7 M (cid:12) first-overtone Cepheidentering the instability strip for the first time
Richard I. Anderson (cid:63)
European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748 Garching b. München, GermanyReceived 3 January 2018 / Accepted 8 March 2018
ABSTRACT
A recently presented
HST/FGS parallax measurement of the Polaris system has been interpreted as evidence for theCepheid Polaris Aa to be pulsating in the second overtone. An age discrepancy between components A and B hasbeen noted and discussed in terms of a stellar merger. Here I show that the new parallax of Polaris is consistentwith a simpler interpretation of Polaris as a (cid:12) , first-overtone, classical Cepheid near the hot boundary of the firstinstability strip crossing. This picture is anchored to rates of period change, the period-luminosity relation, the locationin color-magnitude space, the interferometrically determined radius, spectroscopic N/C and N/O enhancements, and adynamical mass measurement. The detailed agreement between models and data corroborates the physical associationbetween the Cepheid and its visual companion as well as the accuracy of the HST parallax. The age discrepancybetween components A and B is confirmed and requires further analysis, for example to investigate the possibility ofstellar mergers in an evaporating birth cluster of which the Polaris triple system would be the remaining core.
Key words.
Stars: Individual: Polaris = North Star = α UMi = HD 8890 – Stars: variables: Cepheids – binaries: visual– Stars: rotation – Stars: evolution – Stars: oscillations
1. Introduction
Polaris is the closest “classical” Cepheid to the Sun. Giventhe importance of Cepheids for both stellar astrophysicsand cosmology, it is unsurprising that Polaris has beenstudied extensively for more than a century. Polaris fea-tures several peculiarities, such as a very low and changinglight amplitude, and abrupt variations in its fast changingperiod. A particular point of contention in the literature hasbeen its pulsation mode and parallax, since these two areintricately linked via the period-luminosity relation (PLR).Polaris has at least two companions that are discussedin the literature; see Kamper (1996) for a review. One isa visual companion at a separation of approximately (Fernie 1966; Evans et al. 2008, Polaris B), and the otheris a much more nearby companion that was first discoveredspectroscopically (Moore 1929) and later spatially resolvedapproximately . from the Cepheid (Evans et al. 2008,Polaris Ab). Although the separations are numerically dif-ferent, the Polaris system thus shares an important simi-larity with the prototype δ Cephei (Anderson et al. 2015).Recently, Bond et al. (2018, henceforth:B18) providedan independent parallax measurement of Polaris B, the vi-sual companion generally thought to be physically associ-ated with the Cepheid. Intriguingly, this new parallax ismuch smaller than the
Hipparcos parallax, that is, . ± . mas (B18) versus . ± . mas (van Leeuwen 2007),respectively. Another parallax estimate of . ± . mas (cid:63) ESO fellow, e-mail: [email protected] “Polaris” here denotes the Cepheid component, Polaris Aa,unless explicitly stated otherwise. (Turner et al. 2013) based on putative cluster membershiphas been vigorously disputed (van Leeuwen 2013).The B18 parallax of Polaris B should be applicable forthe Cepheid, if Polaris A and B are physically associated.This assumption appears warranted based on several mem-bership indicators, such as photometry, proper motion, andradial velocities; see B18 and references therein. Given theB18 parallax, the physical separation between A and B is ∼ au, which is within the range of physically associ-ated wide binaries among Cepheids (Evans et al. 2016b,a, a rel (cid:46) au). However, this poses a problem, since Po-laris B (age > . Gyr) is much older than the Cepheid(B18). Moreover, reconciling the B18 and
Hipparcos paral-laxes seems impossible, since the two values differ by . σ .B18 argue that Polaris should be pulsating in the sec-ond overtone based on its absolute V − band magnitude andpulsation period, P . However, B18 also note that the 2Opulsation mode for Polaris is unlikely, since 2O pulsatorstend to exhibit more than a single pulsation mode and sincePolaris would have an unusually long period for a Galac-tic 2O pulsator. To explain the age discrepancy betweencomponents A and B, B18 discuss the possibility of merg-ers among system components, although no final conclusioncould be reached.This letter is structured as follows. §2 provides period-age and period-radius relations of first overtone Cepheidsthat cross the classical instability strip (IS) for the firsttime based on Geneva stellar evolution models that incor-porate the effects of rotation. §3 compares the observedproperties of Polaris—adopting the B18 parallax for theCepheid—with these model predictions, and draws a con-sistent picture of its evolutionary status. §4 discusses how Article number, page 1 of 6 a r X i v : . [ a s t r o - ph . S R ] M a r &A proofs: manuscript no. HomingInOnPolaris this picture relates to a dynamical mass measurement andthe age discrepancy. §5 summarizes this work.
2. Model predictions
Anderson et al. (2016, henceforth: A16) recently provideda detailed linear non-adiabatic radial pulsation analysis ofGeneva stellar evolution models that incorporate the effectsof rotation on stellar evolution. Rotation is a key ingredientin stellar models that remains under-discussed and whoseinfluence on the evolution of Cepheids has been shown tobe very significant. For instance, Geneva stellar evolutionmodels (Ekström et al. 2012; Georgy et al. 2013) do not ex-hibit a mass discrepancy compared to other mass estimates(Anderson et al. 2014, 2017, A16).A16 published predictions for the position of the clas-sical IS as well as period-age, period-radius, and of courseperiod-luminosity relations, among others, for Cepheids ofdifferent metallicities (Solar, LMC, SMC) and initial ro-tation rates (no, average, and fast rotation) that pulsatein the first overtone (FO) or the fundamental (FU) mode.Moreover, A16 distinguished between Cepheids crossing theIS for the first, second, and third time, and provided ana-lytic expressions for different relations with P ; mainly forsecond and third crossings as these are the most commonamong classical Cepheids.To evaluate whether the observed properties of Po-laris, which is arguably not representative of the averageCepheid, are consistent with the Geneva models, the fol-lowing provides analytic expressions for period-age (§2.1)and period-radius (§2.2) relations of FO Cepheids crossingthe IS for the first time. These relations have been obtainedby fitting the published results from A16. Table A.1 provides analytic period-age relations for FOCepheids on the first IS crossing with average initial ro-tation rate ω = Ω / Ω crit = 0 . . For Polaris, assuming So-lar metallicity ( Z (cid:12) ), average initial rotation, FO pulsa-tion and a first IS crossing very near the hot IS bound-ary (cf. below), these relations yield an age of Myr, thatis, log ( a [yr]) = 7 . . For comparison, period-age relationsbased on models with no and fast ( ω = 0 . ) rotation wouldimply ages of and Myr, respectively.
Table A.2 provides analytic period-radius relations for FOCepheids on the first IS crossing with average initial ro-tation rate ω = Ω / Ω crit = 0 . . For Polaris, assuming Z (cid:12) ,average initial rotation, FO pulsation and a first IS crossingvery near the hot IS boundary (cf. below), these relationsyield a radius of . (cid:12) . We note, however, that the fit-ted linear relation predicts a slightly smaller radius thanthe closest computed model; see Sec. 3.3. Period-radius rela-tions for ω = 0 . and . predict values of . and . R (cid:12) .
3. A consistent picture for a peculiar Cepheid
This section compares observed properties of Polaris withpredictions from Geneva stellar evolution models adoptingthe B18 parallax for the Cepheid Polaris.
Rates of changing pulsation periods are commonly inter-preted as evidence for secular evolution, i.e., the large-scaleevolution of a Cepheid’s radius as it passes through theIS. Although Cepheids exhibit a diverse phenomenologyof observed period changes on short time scales (years toa decade) (e.g., Poleski 2008; Süveges & Anderson 2017),stellar evolution models generally provide a good match torates of period change determined using temporal baselinesof multiple decades (e.g., A16). Positive ˙ P implies a right-ward motion in the Hertzsprung-Russell diagram, that is,an increase in stellar radius. Since the first IS crossing hasa particularly short lifetime, ˙ P is expected to be . − orders of magnitude faster on the first crossing than on thethird.As noted previously in the literature, the rate at whichPolaris’ period changes is comparatively fast, on the orderof . to . − (based on data spanning ∼ years)(e.g., Evans et al. 2002; Turner et al. 2005; Spreckley &Stevens 2008; Bruntt et al. 2008, and references therein). Ofcourse, Polaris has previously been discussed as a Cepheidon the first IS crossing (e.g., Turner et al. 2013; Fadeyev2015, A16). A peculiar aspect of Polaris’ period change isthe observed discontinuity in the observed O-C diagramparabola, which implies a sudden change in pulsation periodto have occurred around 1963. Rapid monitoring using the SMEI instrument on board the
Coriolis spacecraft suggeststhat another such break may have occurred more recently(Spreckley & Stevens 2008; Bruntt et al. 2008). Polaris isfurthermore famous for its changing light amplitude, whichfor a while was thought to be disappearing.Figure 1 shows model predictions for Z (cid:12) FO Cepheidsof different initial (ZAMS) rotation rates ω on the first(faster ˙ P ) and third IS crossings. Predicted values of ˙ P for FU Cepheids are very similar, if slightly lower and arenot shown here (cf. A16). The observed ˙ P values of fourCepheids, one of which is Polaris, by far exceed the bulkof objects consisting of both FO and FU Cepheids. Thepredicted ˙ P for the first crossing (cid:12) model is consider-ably faster than the observed value, and a similar averagetendency is seen among third-crossing models. Hence, itappears that models systematically overestimate ˙ P , thatis, they underestimate evolutionary timescales. The sameeffect is seen also for other models (Fadeyev 2013, 2014,cf. A16). Despite the mismatch in absolute terms, the clearseparation into two groups likely indicates that Polaris iscrossing the IS for the first time. The pulsation mode of Polaris has been a matter of intensediscussion, in particular with regards to the calibration ofthe PLR. Adopting the
HST parallax, B18 noted the dif-ficulty of rendering the absolute magnitude of Polaris con-sistent with the PLR of either FU or FO Cepheids, whileimplicitly assuming that Polaris crosses the IS for the thirdtime. Figure 2 shows the position of Polaris in the period-luminosity relation using the optical reddening-free Wesen-heit magnitude W VI = I − . · ( V − I ) (Madore 1982;Soszynski et al. 2008). As the figure shows, the position ofPolaris agrees with the predictions for a FO Cepheid onthe first IS crossing very near the hot IS boundary. The Article number, page 2 of 6ichard I. Anderson: Homing in on Polaris . . . P [d]) − l og ( ˙ P [ s / y r ] ) (cid:12) ,
1X 7 M (cid:12) ,
1X 9 M (cid:12) , (cid:12) ,
3X 7 M (cid:12) , Fig. 1.
Predicted rates of period change for Z (cid:12) FO Cepheids(lines) and observed values for Galactic Cepheids (Turner et al.2006, markers include both FO and FU Cepheids). Adapted fromFigure 13 in A16 to illustrate predictions for FO Cepheids. FirstIS crossings are labeled ‘1X’, third IS crossings ‘3X’. Polaris(green filled circle) and three other Cepheids (DX Gem, BYCas, and HD 344787; yellow markers) exhibit much faster ˙ P than the bulk of Cepheids and are likely crossing the IS forthe first time. Line styles distinguish initial rotation rates of themodels: dotted red = no rotation, solid black = average rotation( ω = 0 . ), dashed blue = fast rotation ( ω = 0 . ). Groups oflines correspond to different initial mass or IS crossing. Eachline segment represents an IS crossing (left is hot IS boundary). much shorter-period FO Cepheid SU Cas (on the second IScrossing) lies below the period range where Z (cid:12) models ex-hibit blue loops sufficiently extended for predicting blue ISboundaries; see Anderson et al. (2017) for a discussion.Figure 3 shows isochrones computed using the Genevastellar isochrone online interpolation tool (Ekström et al.2012; Georgy et al. 2014) assuming average initial rotationrates together with the M V and ( B − V ) values from B18.These isochrones were computed for a range of ages near log a = 7 . ; see §2.1 and Table A.1. The right panel showsa closeup of the isochrones that pass through the uncer-tainties of Polaris Aa. Instability strip boundaries weredetermined by fitting predictions for all rotation rates ofFO Cepheids on the first crossing based on Geneva evo-lution models (A16), and are thus self-consistent with theisochrones. The isochrone suggests a slightly younger ageof log a ∈ [7 . , . than the P − a relation. As noted byB18, the position of Polaris B in the CMD does not matchthe isochrone for the Cepheid, regardless of the assumedage or IS crossing. The fitted period-radius relation for Z (cid:12) in Table A.2 pre-dicts R = 50 . (cid:12) for P = 3 . d near the hot IS edge.However, the predicted radius of a computed (cid:12) FO FO Cepheid on first crossing hot boundary: M V = 3 . − . · ( B − V ) ; cool boundary: M V = 6 . − . · ( B − V ) + 6 . · ( B − V ) . . . . . . . P [d]) − . − . − . − . − . W V I = I − . · ( V − I ) [ m ag ] PolarisSU Cas
1X FO avg rot1X FO no rot1X FO fast rot2X FO avg rot3X FO avg rot
Fig. 2.
The position of Polaris on the predicted reddening-freePeriod-Wesenheit relation for FO Cepheids on the first crossing;relations for models on the blue and red IS edges are shown inthese colors. Solid lines are model predictions featuring averagerotation (half of critical angular rotation rate on ZAMS). Mod-els without rotation or very fast rotation are shown by dottedand dash-dotted lines and are very similar near the blue edge.SU Cas is another FO Cepheid with positive ˙ P (likely on secondcrossing) and available parallax (van Leeuwen 2007, Hipparcos) .PLRs near red IS boundaries for second and third crossing FOCepheids are shown in cyan and yellow. Blue boundaries forsecond or third crossings could not be established at such shortperiods, since their blue loops are not extended to sufficientlyhigh temperatures (cf. Anderson et al. 2017). model (with average initial rotation) at the first crossing’shot IS boundary is R = 51 . (cid:12) (Fig. 4), and matchesthe interferometrically measured radius assuming the B18parallax to within the uncertainty (Mérand et al. 2006, θ LD = 3 . ± . mas yields R obs = 53 . ± . (cid:12) ). Theperiod of the same model and the Cepheid differ by less than (predicted P = 3 . d). Period-radius relations for noor very fast rotation match the observed radius even better(see Sec. 2.2) and would imply higher and lower mass, re-spectively. However, the observed rotational enhancementof N/C and N/O (Sec. 3.4) strongly favors a typical rate ofinitial angular rotation and the (computed) M (cid:12) solution. Pre-dredge-up enhancement of nitrogen relative to carbonand oxygen provides “smoking gun” evidence for the pres-ence of rotational mixing, since N has the slowest de-struction rate in the CNO cycle (Maeder 1985; Maeder& Meynet 2000). Estimating this enhancement using mea-sured CNO element abundances (Usenko et al. 2005) andassuming a Solar ZAMS composition (since [Fe/H] = 0.07)yields ∆[N / C] = 0 . ± . and ∆[N / O] = 0 . ± . .These numbers are in excellent agreement with the rota-tional enhancement predicted by the log a = 7 . isochrone( ∆[N / C] = 0 . , ∆[N / O] = 0 . ) near the first crossingof the instability strip, thus contradicting previous results(Neilson 2014). §3.3 and §3.1 suggested the same value for For an illustration of the dependence of these numbers on ω for a (cid:12) Z (cid:12) Cepheid, see Anderson et al. (2014, Fig. 8).Article number, page 3 of 6 &A proofs: manuscript no. HomingInOnPolaris . . . . B − V ) [mag] − − − M V [ m ag ] . . . . . B − V ) [mag] − . − . − . − . − . − . − . M V [ m ag ] Fig. 3.
Color-magnitude diagram for Polaris with predicted Z (cid:12) isochrones of different log a as labeled. The position of Polaris Bis clearly discrepant with the isochrone that matches Polaris Aa, indicating a significant difference in age (best fit isochrone for log a = 9 . ). M V and ( B − V ) values for Polaris Aa and B taken from B18. . . . . . P [d])1 . . . . . . . . l og ( R / R (cid:12) ) Polaris7 M (cid:12) model
FO 1XFO 3XFU 1XFU 3X
Fig. 4.
Predicted Z (cid:12) period-radius relations and Polaris’ in-terferometrically measured radius assuming the B18 parallax.Blue and red lines drawn are linear fits to the discretely com-puted models near the hot and cool IS edges; blue open squaresmark computed models along the hot boundary of the first cross-ing. 1X denotes first crossing models, 3X third crossing mod-els. P − R relations for FU Cepheids are shown for reference.The period and radius of a (cid:12)
1X model ( log P = 0 . , log R/R (cid:12) = 1 . ) on the first crossing nearly coincides withthe Polaris measurement ( . , . ). mass. For comparison, an isochrone for a Myr star en-tering the IS for the first time computed using fast rotation( ω = 0 . ) predicts ∆[N / C] = 1 . and ∆[N / O] = 0 . , farhigher than the spectroscopically measured values.
4. Discussion
The mass of Polaris implied by this work is very near (cid:12) .This is slightly above the σ range of the . +2 . − . M (cid:12) liter-ature value measured using a spectroscopic orbit, propermotions, and assuming (cid:36) = 7 . ± . mas (Evans et al. 2008, E08). Adopting the B18 parallax increases the totalmass of the Polaris system by nearly a factor of two, that is,from . to . (cid:12) (E08, Joint Fit) or from . to . (cid:12) (E08, HST
Only). Since the B18 parallax also implies agreater luminosity for Polaris B, its previous mass estimateof .
35 M (cid:12) also requires upward correction. Considering the(large) uncertainties of Polaris’ mass measurement and itsneed for upward revision, the predicted mass appears to bein very good agreement with observations.
B18 noted an age difference between Polaris Aa ( ∼ Myr)and B ( ∼ . Gyr) and discussed different possibilities ofexplaining this discrepancy via stellar mergers. The presentwork (Fig. 3) confirms that age discrepancy using Genevastellar evolution models.The Polaris system consists of a ∼ (cid:12) Cepheid (thiswork) and two ∼ . (cid:12) companions (E08): Polaris Ab(F6V) and Polaris B (F3V). Given the matching spectraltypes and similar masses of Polaris Ab and B, it seemsmost likely that the Cepheid would be the product of amerger. However, a merger scenario involving two stars of ∼ − (cid:12) does not resolve the age discrepancy, since suchstars have a lifetime of only about Myr, leaving theincreasingly unlikely scenario of multiple mergers.Interpreting the Polaris system ( M tot ≈
11 M (cid:12) , cf. §4.1)as the remaining core of a dispersed birth cluster would im-ply an original cluster mass M cl ≈ −
40 M (cid:12) and thus amaximum stellar mass in the order of − M (cid:12) (Kroupaet al. 2013, Fig. 4-5). Plotting all three components (Aa,Ab, and B) in a single CMD would help establish whethercomponent Ab is co-eval with the Cepheid or Polaris B.The high eccentricity of the inner binary (Aa-Ab) togetherwith the presence of a far outward component B providesevidence of Kozai-Lidov interactions that could have facil-itated a merger in the inner system. Stimulated evolution(Kroupa 1995) could have also been relevant for this sys-tem. The presence of a significant ( . ± . K − band fluxcontribution) circumstellar environment around Polaris Aa(Mérand et al. 2006, located at 2.4 times the Cepheid’spresent radius) may further point to a merger event, which Article number, page 4 of 6ichard I. Anderson: Homing in on Polaris would have taken place within the last ∼ Myr. Dynami-cal modeling of low-mass clusters would provide importantinsights into such systems.In this context, two things are especially worth not-ing. First, there does not seem to be a need to invoke anypeculiar evolutionary scenarios from the point of view ofthe Cepheid alone, given the above excellent agreement be-tween observations and predictions. Thus, if a merger oc-curred in the past, then single star evolution models (of adiscrepant age) are sufficient to explain the Cepheid’s cur-rent properties. Second, there are striking similarities be-tween the Polaris system and δ Cephei, which is also a widetriple system with an eccentric inner binary (Anderson et al.2015) and a nearby overdensity of dispersed young stars (deZeeuw et al. 1999; Gaia Collaboration et al. 2017). More-over, an important CSE around δ Cephei has been detectedusing different techniques (Mérand et al. 2006; Marengoet al. 2010; Matthews et al. 2012; Nardetto et al. 2016).
5. Conclusions
Model predictions for a (cid:12) Z (cid:12) FO Cepheid with typicalinitial rotation near the hot edge of the first IS crossing pro-vide a consistent picture for the observed properties of theenigmatic classical Cepheid Polaris when assuming that therecently measured
HST/FGS parallax of its visual compan-ion Polaris B also applies to the Cepheid. This consistentpicture is anchored to rates of period change, the period-luminosity-relation, the location in color-magnitude space,the interferometrically measured radius, spectroscopic N/Cand N/O enhancements, and the dynamical mass measure-ment from the literature.The strong level of agreement between models and datacorroborates independent evidence supporting the physicalassociation between Polaris A and B and the accuracy of theB18 parallax. This represents a success of stellar evolutionmodels that are particularly sensitive to Cepheid proper-ties. As a star that has only recently turned off from theMain Sequence, Polaris is sure to provide important con-straints on rotational mixing by comparing its propertiesto Cepheids on later IS crossings. The age discrepancy be-tween components Aa and B points to dynamical effectsthat require further study, via N-body simulations, for ex-ample.
Acknowledgements.
The author thanks the anonymous referee for aconstructive report and Nancy R. Evans for drawing his attentionto the recent B18 parallax result and the age discrepancy betweenPolaris Aa and B. Useful discussions with Pavel Kroupa, HideyukiSaio, Antoine Mérand, Martino Romaniello, and Henri Boffin are ac-knowledged. This research has made use of NASA’s Astrophysics DataSystem.
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Article number, page 5 of 6 &A proofs: manuscript no. HomingInOnPolaris
Appendix A: Tables
Appendix A.1: Period-age relation for FO Cepheids on thefirst IS crossing
Metallicity B b A b B r A r Solar (0.014) -0.890 8.267 -0.867 8.422LMC (0.006) -0.944 8.264 -0.923 8.432SMC (0.002) -0.993 8.265 -0.974 8.446
Table A.1.
Period-age relations of the form log ( a [yr]) = A + B · log ( P [d]) for FO Cepheids on the first IS crossing based onfit of predictions provided by A16. Subscripts b and r denoterelations valid at the hot (blue) and cool (red) IS boundary,respectively. Appendix A.2: Period-radius relations for FO Cepheids on thefirst IS crossing
Metallicity B b A b B r A r Solar (0.014) 0.748 1.256 0.737 1.232LMC (0.006) 0.760 1.250 0.751 1.223SMC (0.002) 0.764 1.240 0.756 1.212
Table A.2.
Period-radius relations of the form log ( R/ R (cid:12) ) = A + B · log ( P [d]) for FO Cepheids on the first IS crossing basedon fit of predictions provided by A16. Subscripts b and rr