Issue 
A&A
Volume 560, December 2013



Article Number  A2  
Number of page(s)  7  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/201321333  
Published online  28 November 2013 
On the use of the ratio of small to large separations in asteroseismic model fitting
Astronomy Unit, Queen Mary University of London Mile End Road London E1 4NS UK
email: I.W.Roxburgh@qmul.ac.uk
Received: 21 February 2013
Accepted: 1 November 2013
Aims. We aim to show that model fitting by searching for a best fit of observed and model separation ratios at the same radial orders n is in principle incorrect, and to show that a correct procedure is to compare the model ratios interpolated to the observed frequencies.
Methods. We compare models with different interior structures and outer layers, relate the separation ratios to phase shift differences, conduct model fitting experiments using separation ratios, and relate phase shift differences to internal phase shifts.
Results. We show that the separation ratios of stellar models with the same interior structure, but different outer layers, are not the same when compared at the same radial order n, but are the same when evaluated at the same frequencies by interpolation. The separation ratios trace the phase shift differences as a function of frequency, not of n, and it is the phase shift differences which are determined by the interior structure. We give examples from model fitting where the ratios at the same n values agree but the models have different interior structure, and where the ratios agree when interpolated to the same frequencies and the models have the same interior structure. The correct procedure is to compare observed ratios with model values interpolated to the observed frequencies.
Key words: stars: oscillations / asteroseismology / stars: interiors / methods: analytical / methods: numerical / stars: solartype
© ESO, 2013
1. Introduction
The ratio of small to large separations of stellar pmodes as a diagnostic of the internal structure of stars was proposed in Roxburgh & Vorontsov (2003) (see also Roxburgh 2004, 2005; Otí Floranes et al. 2005) and is increasingly being used in model fitting, that is finding that model (or models), out of a set models, whose separation ratios best fit the ratios of the observed frequencies. These ratios, defined in terms of the frequencies ν_{nℓ} as (Roxburgh & Vorontsov 2003) (1)(and similarly defined ratios ) subtract off the major contribution of the outer layers of a star and so constitute a diagnostic of the stellar interior that is almost independent of the structure of the outer layers of a star which are subject to considerable uncertainties in modelling.
This validity of this procedure rests on the result that that the contribution of the outer layers of a star to the oscillation frequencies ν_{nℓ} is very nearly independent of the degree ℓ; the separation ratios are combinations of frequencies that seek to subtract off this ℓ independent contribution.
We here point out that seeking a best fit model by direct comparison of observational and model ratios at the same {n,ℓ} values, e.g. searching for minimum in the reduced where (2)is, in principle, incorrect, and can give misleading, indeed erroneous results. Here are the error estimates on the ratios derived from σ_{nℓ}, the error estimates on the frequencies ν_{nℓ}, and N is the number of ratios. We show below that the correct procedure is to compare the observed ratios with the model ratios interpolated to the observed frequencies.
2. Examples
To illustrate this we take a model “observed” star of mass 1.2 M_{⊙}, radius 1.34 R_{⊙}, luminosity 2.45 L_{⊙}, effective temperature T_{eff} = 6242° K, central hydrogen abundance X_{c} = 0.30 and age of 2.90 × 10^{9} ys. We assume a total of 33 “observed” frequencies (11 n values, ℓ = 0,1,2) in the range 1534 − 2550 μHz which have a mean large separation Δ = ⟨ν_{n,0} − ν_{n − 1,0}⟩ = 96.2 μHz. For illustrative purposes, we assume a constant error on the frequencies of σ_{ν} = 0.2 μHz since, for the “best” observed stars from CoRoT and Kepler (eg HD 43587, 16CygA) the errors typically lie in the range 0.1 − 0.3 μHz (cf. Appourchaux et al. 2012).
Fig. 1 Internal structure of the “observed” and model stars. 

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Fig. 2 Top panel: the separation ratios r_{01}(n) for model99 (triangles) and scaled model (squares) superimposed on those for the “observed” star; bottom panel: the same but for the r_{02} ratios. 

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We then constructed two extreme comparison models:
 1.
A “scaled model” with the same dimensionless interior structure: density ρ(r)/(M/R^{3}), pressure P(r)/(GM^{2}/R^{4}) and adiabatic exponent Γ_{1}(r) as functions of the dimensionless radius x = r/R as the “observed’ star, but with a mass of 1.35 M_{⊙}, and radius of 1.25 R_{⊙}. This model has a mean large separation of ~ 114 μHz;
 2.
“model99” which has the same interior structure as our “observed” star ρ(r),P(r),Γ_{1}(r) but is truncated at a radius 0.99 R_{o} where R_{o} is the radius of the “observed” star. This model has a mean large separation Δ ~ 112 μHz.
Fig. 3 Top panel: the separation ratios r_{01}(ν) for model99 (triangles) and scaled model (squares) superimposed on those for the “observed” star; bottom panel: the same but for the r_{02} ratios. 

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3. Separation ratios and phase shifts ϵ(ν)
The eigenfrequencies of a slowly rotating star can always be expressed in the form (3)where is some estimate of the mean large separation, and the phase shifts ϵ_{ℓ}(ν_{nℓ}) are determined at the eigenfrequencies from this equation. The ϵ_{ℓ}(ν_{nℓ}) contain all the information on the departure of the structure of the star from a uniform sphere whose frequencies are the roots of the spherical Bessel functions which rapidly approach as n increases (Rayleigh 1894).
As shown in Roxburgh & Vorontsov (2000, 2003) (see Sect. 6 below) the ϵ_{ℓ}(ν) can be effectively split into an ℓ dependent inner contribution determined by the structure of the interior of the star, and an (almost) ℓ independent outer contribution determined by the structure of the outer layers. Interpolating in ϵ_{ℓ}(ν), which is known at the frequencies ν_{nℓ}, to determine the value at any ν, and subtracting values for different ℓ, cancels out both the ℓ independent contribution from the outer layers and the linear term ; the differences ϵ_{0}(ν) − ϵ_{ℓ}(ν) as a function of ν being determined solely by the structure of the inner layers. The separation ratios r_{0ℓ} are approximations to values of ϵ_{0}(ν) − ϵ_{ℓ}(ν) at the frequencies ν_{n0}.
Fig. 4 Top panel: phase shifts ϵ_{ℓ}(ν_{nℓ}), and interpolated curves ϵ_{ℓ}(ν); middle panel: phase shift differences ϵ_{0}(ν) − ϵ_{1}(ν) and the 9 separation ratios r_{01}: bottom panel as above but for the 11 r_{02}. 

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The top panel in Fig. 4 shows the phase shifts ϵ_{ℓ}(ν_{nℓ}) at the “observed” frequencies for our “observed” star, and the cubic interpolated curves ϵ_{ℓ}(ν), the bottom panels show the phase shift differences ϵ_{0}(ν) − ϵ_{ℓ}(ν) together with the separation ratios r_{01},r_{02} which lie on or close to the curves ϵ_{0}(ν) − ϵ_{ℓ}(ν).
If we now compare the separation ratios of a star with observed frequencies to those of a model with different frequencies (as in Eq. (2)) then we are comparing the values of the phase shift differences ϵ_{ℓ}(ν) − ϵ_{0}(ν) at different frequencies so, even if the star and model have the same interior structure, that is the same ϵ_{ℓ}(ν) − ϵ_{0}(ν) as a function of ν, they will not have the same values of the separation ratios at their respective {n,ℓ} values. This explains why the ratios for model99 in Fig. 3 are offset from the “observed” values. The scaled model on the other hand has a different interior structure and therefore a different function for its phase shift differences.
In summary the separation ratios of model99 with the same interior structure as the “observed” star lie on the same ϵ_{0}(ν) − ϵ_{ℓ}(ν) curves as those of the “observed” star when considered as a function of frequency, but disagree as a function of radial order n. Conversely the ratios for the scaled model which has a different interior structure from that of the “observed” star agree with the observations when considered as a function of n, but do not lie on the same ϵ_{0}(ν) − ϵ_{ℓ}(ν) curves as those of the “observed” star when considered as a function of frequency.
This demonstrates that seeking a best fit model by comparing separation ratios at the same n values can give erroneous results.
This is not surprising, the radial order n of an eigenfrequency depends on the structure of the outer layers of a star as well as on the interior structure, being determined approximately by the integral number of wavelengths that fit into the acoustic radius of the star T = ∫dr/c where c is the sound speed. Since c is small in the outer layers these layers can make a significant contribution to the acoustic radius. So the n values are not independent of the structure of the outer layers; comparing observed and model separation ratios at the same n does not subtract off the effect of the outer layers.
Fig. 5 Comparison of “observed” with model99 ratios interpolated to the “observed” frequencies. 

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4. Correcting the model fitting
This comparison procedure can be corrected by interpolating the values of the model separation ratios , which can be calculated for any { n,ℓ } values, to derive values at the observed frequencies, and then comparing the interpolated model ratios to the observed ratios. The results using local cubic interpolation on model99 ratios are shown in Fig. 5 along with the separate for the r_{0ℓ} separations. The total defined as (4)is 0.006 (with σ_{ν} = 0.2 μHz). The “observed” and interpolated model values are in excellent agreement. This is the procedure to be used in comparing observed and model separation ratios; it is independent of the n values the astronomer assigns to the frequencies.
Fig. 6 “Observed” and model ratios r_{01},r_{02} as functions of radial order n for 4 models within the L,T_{eff} error box in the HR diagram and their separate values. 

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5. Examples from model fitting
The above example is somewhat extreme, although it illustrates the point we are making. We therefore undertook a study which closely follows a realistic model fitting procedure, taking our “observed” star and seeking best fit models out of a large data base of models. The data base we use was kindly supplied by Miglio (priv. comm.) and contains over 200 000 models, computed using the CLES and OSC codes (Scuflaire et al. 2008a,b). The model set has masses in the range 1.04 − 1.5 M_{⊙}, 4 values of the initial Hydrogen abundance XH = 0.684,0.694,0,704,0.714, 4 values of heavy element abundance Z = 0.015,0.02,0.025,0.03, 2 values of the mixing length parameter α_{con} = 1.705,1.905, 2 values of a chemical overshoot parameter α_{ov} = 0,0.2H_{p} and with and without microscopic diffusion (details of the physics are given in Scuflaire et al. 2008a). The models are evolved from the premain sequence to the subgiant phase. Our “observed” star has similar physics with XH = 0.7,Z = 0.02,α_{con} = 1.6,α_{ov} = 0 and no diffusion (cf. Roxburgh 2008a).
We searched for best fit models that lie within an error box in the HRdiagram around the “observed” star defined by T_{eff} = 6242 ± 70°K, L/L_{⊙} = 2.45 ± 0.245, these being realistic error estimates. Some 1,324 models lie within the error box in the HR diagram of which 53 have both and ; 38 have and ; and 17 have and . The fact that a large proportion of the models which have also have is not a great surprise; if the ϵ_{0}(ν) − ϵ_{ℓ}(ν) are slowly varying functions of ν, their difference, , is given by (5)so if d(ϵ_{ℓ} − ϵ_{0})/dν is small, and the frequency offset is not too large, this difference is small.
Fig. 7 “Observed” and model ratios r_{01},r_{02} for the 4 models within the L,T_{eff} error box in the HR diagram interpolated to the “observed” frequencies, and their separate values. 

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To illustrate the point that matching observed and model ratios at the same n values can lead to erroneous inferences we show in Fig. 6 the ratios for 4 models that lie within the L,T_{eff} error box in the HR diagram and have a combined of 0.32,0.42,0.53,0.66; apparently good fits to the “observed” data. Details of the models are given in Table 1, they all include diffusion, have the same heavy element abundance Z and similar ages (age9 is the age in units of 10^{9} years). However if we interpolate for the values of the ratios at the “observed” frequencies the fit is not so good – this is shown in Fig. 7; the combined for the 4 models are 2.06,2.45,2.61,2.10 which for for the best case has has a probability of being due to chance of 0.22%.
Properties of the 4 models with .
Fig. 8 Internal structure of the “observed” star and that of the 4 models obtained by fitting at the same n values. 

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Properties of the 10 models with the lowest .
In Fig. 8 we compare the sound speed and density of the 4 models in the inner layers with that of the “observed” star; the models have a higher central temperature, lower central density and larger convective core (the core boundary being at the point of increase in c^{2}). This demonstrates that models that fit the observed ratios when compared at the same n values (within reduced for σ_{ν} = 0.2 μHz) may not fit the observed ratios when compared at the observed frequencies and can have an interior structure that is not that of the observed star.
Fig. 9 Internal structure of the observed star and that of the best 10 models obtained by fitting at the same n values. 

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Fig. 10 Comparison of “observed” ratios with those of the best 10 models interpolated to the observed frequencies. 

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The above examples were chosen to illustrate the argument advanced in this paper – but they are not the models with the lowest reduced . The 10 best fit models when compared at the same n values have a combined reduced ; details of these models are given in Table 2 and their inner structure is shown in Fig. 9. 4 of the 10 models (numbers 2,4,7,10) fit the structure very well; but all 4 also have a very small reduced (also shown in Table 2) and are amongst the 10 best fit models obtained by comparing at the “observed” frequencies (see Table 3 below). As pointed out above the difference between ν and n comparison (Eq. (5)) can be small. The other 6 models, which includes the model with the lowest , are not such good fits, having larger convective cores, lower central densities and higher central temperatures.
Were one to seek a best fit model by comparing ratios at the same n values, the model with the lowest would not necessarily reproduce the internal structure of the observed star. Matching ratios at the same n values can therefore give erroneous results on the interior structure of an observed star, and on the physical processes that govern stellar evolution.
Fig. 11 Comparison of the internal structure of the observed star with those of the best 10 models. 

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Properties of the 10 models with the lowest .
The 10 best fit models when comparing ratios interpolated to the “observed” frequencies are illustrated in Fig. 10 and details of the models are given in Table 3. The fits are very good with 9 of the 10 models having a . The internal structure of these models is shown in Fig. 11; all the models satisfactorily reproduce the internal structure of the “observed” star. Note that models 1,4,5,7 in Table 3 are the same as models 7,4,10,2 in Table 2 and are the 4 models that fit the internal structure of the “observed” star in Fig. 9.
The properties of the best fit model with are very close to those of the “observed” star and all 10 models have very similar central hydrogen abundances and ages. But the masses range from 1.16 − 1.22 M_{⊙}, the smaller masses having lowest initial hydrogen content, some models incorporate diffusion and some do not, and all models have a large separation Δ greater than that of the “observed” model (96.2 μHz).
Fig. 12 Offset of frequencies of the 10 best fit models from those of the “observed” star. 

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The difference between the observed frequencies and the model frequencies , often referred to as the “frequency offset” is plotted in Fig. 12 for the all 10 models. These offsets can be extremely large even though the interior structure of the models agrees well with that of the “observed” star. This reflects the fact that the large separations Δ of the models are larger than that of the “observed” star by 1.4 − 7.4 μHz; to lowest order the frequencies ν_{nℓ} are approximately (n + ℓ/2)Δ and for modes of radial order n = 20 this gives an approximate offset in the range 30 − 150 μHz, in agreement with that shown in Fig. 12.
6. Separation ratios and internal phase shifts
The frequencies of a spherical star satisfy an Eigenfrequency equation of the form (Roxburgh & Vorontsov 2000) (6)is the acoustic radius of the star and c the sound speed. The inner phase shift δ_{ℓ}(ν) is determined by the structure interior to some fitting point r_{f}, and outer phase shifts α_{ℓ}(ν) by the structure above r_{f}. This is exactly true for modes of degree ℓ = 0,1 where the oscillation equations reduce to 2nd order (Takata 2005; Roxburgh 2006, 2008b), and holds to high accuracy for ℓ = 2 provided r_{f} is in the low density outer layers of the star. Eq. (6) is identical in form to that of Eq. (3) with ϵ_{ℓ} = α_{ℓ} − δ_{ℓ} and . The important difference is that since the density, sound speed, and curvature effects are small in the outer layers, the α_{ℓ}(ν) are almost independent of degree ℓ.
To see this we define α_{ℓ}(ν,t),δ_{ℓ}(ν,t) at any radius and frequency in terms of the functions ψ_{ℓ}(ν,t) = rp′/(ρc)^{1/2} by (7)where p′ is the Eulerian pressure perturbation, t is the acoustic radius, τ = T − t the acoustic depth, and ν any frequency (not restricted to an eigenfrequency). In the outer subsurface layers the functions ψ_{ℓ}(ν,t) are almost pure sine waves with constant phase shift, and the phase shifts are almost independent of the choice of fitting point r_{f} (or t_{f}) (Roxburgh & Vorontsov 1996). In Fig. 13 we show the differences α_{2}(ν) − α_{0}(ν) and δ_{ℓ}ν) − δ_{0}(ν) for the “observed” 1.2 M_{⊙} star, evaluated at radii x_{f} = r_{f}/R = 0.93,0.95, 0.97; the differences α_{1}(ν) − α_{0}(ν) ≈ [ α_{2}(ν) − α_{0}(ν) ] /3.
As can be seen in Fig. 13 the α_{ℓ}(ν) − α_{0}(ν) are less than 1% of the differences δ_{ℓ}(ν) − δ_{0}(ν), and the values of δ_{ℓ}(ν) − δ_{0}(ν) are almost independent of the fitting point x_{f}. So, to a good approximation, the phase differences (8)and are therefore determined by the interior structure of the star.
Fig. 13 Phase shift differences α_{2}(ν) − α_{0}(ν) and δ_{ℓ}(ν) − δ_{0}(ν) at fractional radii x_{f} = 0.93,0.95,0.97 for the “observed” “star”. 

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7. Conclusions and comments on model fitting
The separation ratios are an approximation to the phase shift differences ϵ_{0}(ν) − ϵ_{ℓ}(ν) which are determined by the structure of the stellar interior and are almost independent of the structure of the outer layers. One should not compare the ratios of an observed frequency set with the of a model at the same n values since one is not comparing values of ϵ_{0}(ν) − ϵ_{ℓ}(ν) at the same frequencies. This can be corrected by interpolating in the model values to determine the model ratios at the observed frequencies and then comparing observed and model values. Alternatively one could compute the phase shift differences from the model and compare the observed ratios with the values of the model phase shift differences at the observed frequencies.
The error in comparing ratios at the same n values depends both on the scale of variation of ϵ_{0} − ϵ_{ℓ} and the difference in frequencies at the same n values (Eq. (5)). For the model fitting examples given here, where the models and the “observed” star have almost the same governing physics, we find many models that have both and less than one (with an error on the frequencies of 0.2 μHz), but even so we find examples where but and the models do not have the same interior as the “observed” star, the difference being large enough to lead to erroneous conclusions on the structure, age, and physical processes that govern stellar evolution.
The fact that we find so many models (91) with indicates the limits of model fitting using ratios. We have only 20 data to match, each being an integral over the interior structure, different structures can therefore give approximately the same ratios, differences in structure in one region being compensated by differences in other regions. To overcome this one needs considerably higher precision on the data – in the examples given here a σ_{ν} ~ 0.06 μHz – or considerable more data points.
In the above analysis we have taken the error estimate on the frequencies to be independent of frequency whereas, in general, both the error estimates on observed frequencies (σ_{ν}), and the error in comparing ratios at the same n values, are larger at high frequencies leading to a somewhat smaller value of than obtained using a constant average value of σ_{ν}. But the general point remains valid; a model that has the same values of the separation ratios as an observed set when compared at the same n values, and which does not have the same frequencies, does not have the same internal structure as the observed star.
A further comment is in order. Since the objective of using separation ratios is to subtract off the unknown effect of the outer layers of a star it can only yield a best fit to the interior structure. It is inconsistent to impose strict constraints on the radius, effective temperature, surface gravity, metallicity, and large separations derived by observations, since these all depend on the structure of the outer layers and nonadiabatic effects, which we seek to eliminate by using ratios. The luminosity on the other hand is determined by the interior structure alone, as is the mass which, in principle, can be estimated from surface gravity and surface radius; these can provide additional surface layer independent constraints on the model fitting.
Acknowledgments
We thank Andrea Miglio who generously provided the large data base of models used in the model fitting. I. W. Roxburgh gratefully acknowledges support from the Leverhulme Foundation under grant EM2012035/4.
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All Tables
All Figures
Fig. 1 Internal structure of the “observed” and model stars. 

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In the text 
Fig. 2 Top panel: the separation ratios r_{01}(n) for model99 (triangles) and scaled model (squares) superimposed on those for the “observed” star; bottom panel: the same but for the r_{02} ratios. 

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In the text 
Fig. 3 Top panel: the separation ratios r_{01}(ν) for model99 (triangles) and scaled model (squares) superimposed on those for the “observed” star; bottom panel: the same but for the r_{02} ratios. 

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In the text 
Fig. 4 Top panel: phase shifts ϵ_{ℓ}(ν_{nℓ}), and interpolated curves ϵ_{ℓ}(ν); middle panel: phase shift differences ϵ_{0}(ν) − ϵ_{1}(ν) and the 9 separation ratios r_{01}: bottom panel as above but for the 11 r_{02}. 

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In the text 
Fig. 5 Comparison of “observed” with model99 ratios interpolated to the “observed” frequencies. 

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In the text 
Fig. 6 “Observed” and model ratios r_{01},r_{02} as functions of radial order n for 4 models within the L,T_{eff} error box in the HR diagram and their separate values. 

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In the text 
Fig. 7 “Observed” and model ratios r_{01},r_{02} for the 4 models within the L,T_{eff} error box in the HR diagram interpolated to the “observed” frequencies, and their separate values. 

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In the text 
Fig. 8 Internal structure of the “observed” star and that of the 4 models obtained by fitting at the same n values. 

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In the text 
Fig. 9 Internal structure of the observed star and that of the best 10 models obtained by fitting at the same n values. 

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In the text 
Fig. 10 Comparison of “observed” ratios with those of the best 10 models interpolated to the observed frequencies. 

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In the text 
Fig. 11 Comparison of the internal structure of the observed star with those of the best 10 models. 

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In the text 
Fig. 12 Offset of frequencies of the 10 best fit models from those of the “observed” star. 

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In the text 
Fig. 13 Phase shift differences α_{2}(ν) − α_{0}(ν) and δ_{ℓ}(ν) − δ_{0}(ν) at fractional radii x_{f} = 0.93,0.95,0.97 for the “observed” “star”. 

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In the text 
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