SAG: Stars and Galaxies Vol. 3, id. 4, 2020 December 28© 2020. Center for Astronomy, University of Hyogo.

Estimation of the mass and luminosity of Polaris from resonance hypothesis
Toshihito I SHIDA
Nishi-Harima Astronomical Observatory, Center for Astronomy, University of Hyogo,
407–2 Nishigaichi, Sayo-cho, Hyogo 679–5313, Japanishida@nhao.jp(Received 2020 November 16; accepted 2020 December 24)

Abstract
The resonance hypothesis for the cause of the decreasing event in the pulsation amplitude of the Cepheid component of Polaris is proposed. If Polaris is at the center of the 2:1 resonance between the first and the fourth overtone
modes, we can estimate the mass and luminosity from the results of the linear nonadiabatic pulsation models using resonance conditions. The estimated stellar parameters from the resonance hypothesis are within the possible
parameter range.
Key words: stars: individual (Polaris) — stars: oscillations — stars: variables: Cepheids
1.

Introduction

Classical Cepheids are population I yellow supergiants pulsating radially. These stars are well known to be the standard candles for the extragalactic distance scale, thanks to theperiod-luminosity relation. Their fundamental properties are
explained by the stellar pulsation theory, however, it is pointed
out that there are some discrepancies between results from thestellar evolution theory and those from the pulsation theory.
Among those objects, Polaris (P = 3.9696 days, F8 I;
GCVS: Samus’ et al. 2017) is one of the most popular andwell-observed object. However, there is still some unsolved
problems concerning this object.
Feast and Catchpole (1997) investigated the periodluminosity relation of Cepheids with Hipparcos trigonometricparallax, and pointed out that the weight of Polaris on this relation is much greater than others. They also indicated that
Polaris is considered to be the first overtone pulsator. However,
Turner (2004) pointed out the existence of the overlooked cluster around Polaris and derived closer distance 94 ± 4 pc (seealso Turner et al. 2005), from which they concluded that Polaris
pulsates in the fundamental mode. On the contrary, revised+2.9
pc (van Leeuwen 2007). Selected
Hipparcos value is 132.6−1.9
recent distance estimations of Polaris including Gaia DR2 es+0.53pc) are summatimate for companion Polaris B (137.14−0.52
rized in Engle, Guinan, and Harmanec (2018).
Because interferometric angular diameter is estimated(Mérand et al. 2006), adapting longer distance means not only
brighter luminosity but also the larger radius and prefers thefirst overtone mode pulsation, and vise versa. From two different sets of luminosity and radius, we can estimate two different
mass to obtain the observed period. Therefore, there are stillsome uncertainty for the stellar parameters for Polaris.
The evolutionary status and crossing number of Polaris isalso one of the controversial subjects. The change of the pulsation period with the stellar evolution of the Polaris is summa-

rized in Turner et al. (2005). Saitou (1989) classified Polaris as
the first crossing objects, but mentioned that we need further
investigation to identify the pulsation mode. Using the shorter
distance (99 ± 2 pc), Turner et al. (2013) concluded that Polaris
is pulsating in the fundamental mode and crossing the instability strip for the first time. Neilson (2014) used new stellar evolution models and pointed out that Polaris is in its third crossing and pulsate in the first overtone mode. More recently, Bondet al. (2018) pointed out that isochrones for Polaris A (Cepheid)
and its wide companion B are different. In summary, the evolutionary status of Polaris is not clear yet.
The decrease in the pulsation amplitude of Polaris fromabout 1.5 mag in 1940s to 0.05 mag in 1980s was first reported
by Arellano Ferro (1983) using photometric observations andconfirmed by Kamper et al. (1984) by radial velocity observations. Fernie et al. (1993) pointed out that the decrease in theamplitude is exponential and deduced that the pulsation willstop in 1994. However, Bruntt et al. (2008) reported that pulsation amplitude increased again. More recent observations indicate the further increase (Mkrtichian et al. 2014) and the following possible decrease (Usenko et al. 2018). It is also worth
noting that more gradual brightening in longer time scale, sayover 2000 years is also pointed out (Engle et al. 2004).
Since the observed decrease in the pulsation amplitude isremarkable, some authors have proposed a possible cause ofthis behavior. Moskalik and Ogłoza (2000) speculated that theobserved decrease is a part of the periodic or quasi-periodic
amplitude modulation near the 2:1 resonance between the first
and the fourth overtone radial modes. Stothers (2009) pointed
out that if the amplitude modulation is cyclic, it resembles theBlazhko effect observed in RR Lyrae stars. He proposed a magnetoconvective cycle in the stellar envelope as a possible explanation. However, it is still unclear whether the decrease isa cyclic or intrinsically one-time event. Both of the above two
proposals are assuming cyclic variation, and we seem to haveno explanation if the event is one-time.

2

T. IshidaIn this paper, we would like to propose the resonance itself
as the possible cause of the decrease of the pulsation amplitude
which is in one-time only in their life as a classical Cepheid.
Furthermore, the mass and the luminosity under the resonancehypothesis are estimated from the linear nonadiabatic pulsation calculations and compared with other estimations. In section 2, the resonance hypothesis for the decrease in the pulsation amplitude is explained. Models and their results of the linear nonadiabatic pulsation calculations are shown in section 3.
The discussions are presented in section 4. We conclude insection 5.
2.

Resonance hypothesis

2:1 resonance between two radial pulsation modes is theoretically investigated in relation to the bump Cepheid phenomena between the fundamental and the second overtone modes(Takeuti and Aikawa 1981; Buchler and Kovacs 1986: see also
Aikawa 1984). The apparent phenomena between the bump
Cepheids and Polaris amplitude differ largely. The decrease of
pulsation amplitude of Polaris is observed in one star withina hundred years, while the bump Cepheid phenomena are observed in many stars with period around 10 days. However, weneed to consider that the coupling coefficients between relatedmodes have only calculated for the first three modes, namely
fundamental, first overtone, and second overtone. It is known
that the higher overtones usually have smaller amplitude if theamount of oscillating energy is the same, therefore, it seems
probable that the decrease in amplitude occurs in the narrower
period range and is deeper around the period for the exact resonance for the fourth overtone case.
From the observational data, the center of the 2:1 resonancefor the first and fourth overtone for the light curve data is estimated around 3.2 days from the jump of the Fourier phases φ21 .
However, for the radial velocity data, it is pointed out that there
is no jump in the Fourier phases around the above period andchanges smoothly for all the period range. Because the centerof the resonance need to occur at the same period for both ofthe light curve and radial velocity data, Kienzle et al. (1999)
estimated the center to occur at the longer period, 4.58 days in
our Galaxy. Furthermore, Feuchtinger, Buchler, and Kolláth(2000) surveyed their results obtained by hydrodynamic models, and reproduced the observed change of the Fourier parameters for both the light curve and the radial velocity by theirmodels. They estimated the center of the resonance is near
4.2 days in the case of their models.
For Polaris itself, Moskalik and Ogłoza (2000) analyzed radial velocity data to derive its vicinity to the resonance center. They estimated that the ratio between the fourth and thefirst overtone periods for Polaris is 0.51. However, consideringmuch larger uncertainty of Polaris than other stars for the estimated Fourier parameter φ21 into account, it seems possiblethat Polaris is nearer to the resonance center than their estimate.
If the Polaris is at the center of the resonance, we can confine
the possible region of its properties from the resonance condition and the observed value of the period. The first overtone
also needs to be pulsationally unstable, and the fourth overtone
to be stable. In the next section, we would like to present linear
nonadiabatic pulsation calculation and its results.

3.
3.1.

Vol. 3
Linear non-adiabatic pulsation calculations
ModelsWe used the same code used in Ishida (1995, 2017),
namely, Castor (1971) type procedure with OPAL opacity tables (Iglesias et al. 1992; Rogers & Iglesias 1992) for calculation of the linear nonadiabatic periods. As the purpose ofthis paper is to examine the possibility that the decrease of theamplitude in Polaris may be explained by the resonance, theenvelope model is purely radiative i.e. the effect of the convective energy transport is not included in all of the models. Thiseffect is discussed in section 4.
Results of effective temperature estimations are summarizedin Usenko et al. (2005). The mean effective temperatures during the last 60 yrs spread within 5800–6200 K. Therefore, we
set the parameter range for effective temperature from 5800 to
6200 K with intervals of 100 K.
Recently, the close companion of the Polaris is resolved by
Hubble Space Telescope observation and reported that the dynamical mass of the Cepheid is estimated to be 3.45±0.75 M⊙(Evans et al. 2018). Considering Anderson (2018)’s estimationof 7 M⊙ into account, we set also the parameter range widelyfrom 4.0 to 7.5 M⊙ with intervals of 0.1 M⊙ .
We can derive luminosity estimation in several ways. We
can estimate it from Fouqué et al. (2007)’s period-luminosity
relation using the observed period of Polaris, while we needtake the possibility into account that the observed period
3.9696 days is for the first overtone. The derived luminosity
by this method is about 1150 L⊙ if the observed period corresponds to the fundamental mode while about 1660 L⊙ if thefirst overtone is assumed. We can also estimate it from theobserved mean visual magnitude (< V > = 1.982), reddening(E(B − V ) = 0.01 ± 0.01), and parallax with Torres (2010)’s
polynomial fit for Flower (1996)’s bolometric correction. We
can derive several values depending on which parallax we use.
For example, we can estimate about 2530 L⊙ for the Gaia DR2
parallax of Polaris B, namely, 7.292 mas. For the preliminary
calculations, we set the parameter range from 1000 to 4000 L⊙ .
Although it is reported that Polaris has a small supersolar
abundance in the majority of elements (Andrievsky et al. 1994),
we assume Z = 0.02 as the reference composition. Later wewould like to discuss the effect of chemical composition.
From the preliminary calculations, we can recognize thatthe 2:1 resonance seems to be possible only for the first andfourth overtones as the Polaris model, we will show the results
only related to these two radial modes. We set the parameter
range for the luminosity from 1500 to 4000 L⊙ with intervalsof 100 L⊙ hereafter.
The final selected parameter range is summarized in table 1.
3.2.

Results

As a typical example of the results, the contour plot for theeffective temperature 6000 K is presented in figure 1. The exact resonance with the observed period of Polaris is realizedaround M = 5.40 M⊙ and L = 2470 L⊙ .
In the same manner, we can estimate the masses and luminosities for the exact resonance between the first and the fourth
overtones with the observed period of Polaris for every Teffid. 4

Estimation of the mass and luminosity of Polaris from resonance hypothesis

3

Table 1. Summary of the model parameters

Polaris Resonance Model Summary

3000

6

6200–5800

Mass (in M⊙ )4.0–7.5

X

0.70

Z

0.01, 0.02, 0.03

Luminosity (in L⊙ )1500–4000

2800
2600

5
2400

ble

sta

ys

50

5da

0.
1=

9

/P

0.4

s

P(P

P4

)

s
olari

P1=

○
.51

/P

0
1=

P4

P4

3

0.5

1=

/P

.52

0
1=

/P

P4

days

P1=3.0Fig. 1. Contour plot for the effective temperature 6000 K. Abscissa isthe model mass and ordinate is the model luminosity both in the solar
unit. Dashed lines are period ratio contours with corresponding period
ratio values of the fourth overtone to the first overtone. Solid lines are
the result of the mass-luminosity relations for the first overtone modes
corresponding to a given period value. Observed period of the Polaris3.9696 days is also indicated. The thick line near the top left corneris the instability edge of the first overtone. Below this line, the first
overtone is pulsationally unstable. If the exact resonance is realized in
the Polaris, the circle point will satisfy the necessary conditions.

values. Thus, we can summarize the whole of the results asshown in figure 2. The derived mass gradually increases as we
use lower effective temperature, while the derived luminosity
mostly decreases.
4.

Discussions

As described in the previous section, we can derive the massof Polaris as 5.40 M⊙ . As mentioned in the previous section, the dynamical mass of the Cepheid component of the
Polaris system is estimated to be 3.45 ± 0.75 M⊙ (Evans et al.
2018). However, Engle, Guinan, and Harmanec (2018) reportas their results for preliminary fit given by evolutionary tracks
including rotation effects as M ∼ 6.2–6.7 M⊙ . The mass for
Teff = 6200 K models are marginally matched to the dynamical
mass estimation. If we adopt Teff = 6000 K models, our estimate mass is larger than the dynamical mass, but smaller thanthe evolutionary mass. The derived mass 5.40 M⊙ seems to bewithin the possible parameter range.

6200

6100

6000
Teff

5900

5800

4
5700

Fig. 2. The masses and luminosities derived from the following conditions are plotted for a given Teff value. Period ratio of the fourth tothe first overtones is 0.5, namely, the exact 2:1 resonance, the periodof the first overtone is the same as the observed period of Polaris, thefirst overtone is pulsationally unstable, and the forth overtone is pulsationally stable. Filled circles with thick lines are derived luminosities
for the Z = 0.02 models. Open squares with broken lines are derived
masses for the Z = 0.02 models. Luminosity scale is given on theleft ordinate and mass scale is given on the right ordinate. Mass andluminosity derived for the Z = 0.01 models are filled and open upper peaked triangles, respectively. Those for the Z = 0.03 models arebottom peaked triangles with the same way.

.0day

P1=4

/P

1

=

P1=

P4

un

4.5

2200
2000
6300

1O

5.5

Mass (in solar unit)

Teff

Luminosity (in solar unit)

Parameter

If we adopt the Gaia DR2 distance to Polaris B as the parallax to the Cepheid component of Polaris, we can estimate theluminosity of the Polaris as 2530 L⊙ . It is worth noting that
our estimate for the Z = 0.02 models with Teff = 6000 K is asnear to the value from Gaia DR2 as about 2470 L⊙ .
Z = 0.01, and 0.03 models for Teff = 6000 K are also calculated to check the possible effect of Z. The derived parameters
are also plotted in figure 2. Open marks are for mass estimation and filled marks are for luminosity estimation. We can seethat all of the mass estimations are within 0.21 M⊙ width, andluminosity estimations are within 130 L⊙ width. The effect of
Z seems to be not so large.
For evaluating the effect of convection, we tried to use
Radial Stellar Pulsation (RSP) feature of the Modules forExperiments in Stellar Astrophysics (MESA) project, which
is described in MESA paper 5 (Paxton et al. 2019). We usedalmost default values but grid points are increased to be nearly
500, as our no convection model also uses nearly 500 zones.
We picked up only the linear results. The most significant difference is that the first overtone modes are pulsationally stable
in a wider parameter range than those for no convection models. The stability edge for the first overtone exists at a much
shorter than the observed Polaris period. We seem to need moredetailed considerations concerning the inclusion of convection.
5.

Concluding Remarks

We proposed that the 2:1 resonance between the first and thefourth overtone modes itself as the possible origin of the observed pulsation amplitude decrease in the Cepheid componentof Polaris. We can estimate the stellar parameters of Polaris if
Polaris is indeed in the resonance. From the result of the linear

4

T. Ishida

nonadiabatic pulsation calculations, we have shown that the parameters derived from the resonance hypothesis are within thepossible parameter range.
Coupling coefficients between some radial pulsation modes
are evaluated in some papers (e.g., Takeuti and Aikawa 1981;
Klapp et al. 1985). However, their studies are limited to thebump Cepheid phenomena between fundamental and the second overtone modes. We seem to need to evaluate coupling
coefficients between the first and fourth overtone modes.
One need to remark that the proposed origin will induce the
one-time event during life as a Cepheid. If the amplitude shows
clear cyclic change, the modulation proposed by Moskalik andOgłoza (2000) may be the origin. We need more observations
to confirm the possible increase/decrease of the pulsation amplitude.
I would like to thank my colleagues for their useful comments during this investigation. I also would like to thank thereferee for his/her valuable comments. Computations are carried out mainly on the workstation provided by a research grantfrom University of Hyogo.
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