Fundamental properties of solar-like oscillating stars from frequencies of minimum $\Delta \nu$: I. Model computations for solar composition

Low amplitude is the defining characteristic of solar-like oscillations. The space projects $Kepler$ and $CoRoT$ give us a great opportunity to successfully detect such oscillations in numerous targets. Achievements of asteroseismology depend on new discoveries of connections between the oscillation frequencies and stellar properties. In the previous studies, the frequency of the maximum amplitude and the large separation between frequencies were used for this purpose. In the present study, we confirm that the large separation between the frequencies has two minima at two different frequency values. These are the signatures of the He {\small II} ionization zone, and as such have very strong diagnostic potential. We relate these minima to fundamental stellar properties such as mass, radius, luminosity, age and mass of convective zone. For mass, the relation is simply based on the ratio of the frequency of minimum $\Delta \nu$ to the frequency of maximum amplitude. These frequency comparisons can be very precisely computed, and thus the mass and radius of a solar-like oscillating star can be determined to high precision. We also develop a new asteroseismic diagram which predicts structural and evolutionary properties of stars with such data. We derive expressions for mass, radius, effective temperature, luminosity and age in terms of purely asteroseismic quantities. For solar-like oscillating stars, we now will have five very important asteroseismic tools (two frequencies of minimum $\Delta \nu$, the frequency of maximum amplitude, and the large and small separations between the oscillation frequencies) to decipher properties of stellar interior astrophysics.


INTRODUCTION
Every object in the universe oscillates in its own way. For stars, the increasing sensitivity in detecting oscillations in solar-like objects by the space missions Kepler and CoRoT is ushering in a new era in stellar astrophysics. Determination of fundamental properties of single stars from oscillation frequencies (ν) is among the promises of asteroseismology. The relation between the mean density and the large separation between the oscillation frequencies (∆ν) is well known (Ulrich 1986). Kjeldsen & Bedding (1995) proposed a semi-empirical relation between fundamental properties of stars and the frequency of maximum amplitude (νmax). In the present study, we suggest two new frequencies (νmin1 and νmin2) which, together with νmax and the mean of ∆ν ( ∆ν ), can be used to derive expressions for these fundamental properties. νmin1 and νmin2 are the frequencies at which ∆ν is minimum. We will show that they have excellent predictive power for stellar mass (M ), radius (R) and effective temperature (T eff ) of single stars, especially if the observations yield accurate values of νmax. ⋆ E-mail: mutlu.yildiz@ege.edu.tr Kjeldsen & Bedding (1995) estimate amplitudes of solar-like oscillations and then interrelate νmax and ∆ν to the stellar mass and radius. Relatively simple expressions for M and R as can be written as functions of νmax, ∆ν and T eff (e.g. see Chaplin et al. 2011): These expressions for M and R are applied in many studies.
Kepler and CoRoT data provide us ∆ν and νmax for enough stars to confirm that there is a significant difference between the mass found from modelling of these stars and their mass given by equation (1) Corsaro et al. 2013), it is still uncertain if these relations are sufficient as written, or are sensitive to other stel-lar parameters not included in them (see, e.g., White et al. 2011).
The scaling relation may depend on, for example, metallicity (see section 5.8).
For some of the hottest F-type stars, it is reported that power envelopes have a flatter maximum (see, e.g., Arentoft et al. 2008;Chaplin & Miglio 2013). For Procyon, for example, photometric and spectroscopic methods give different νmax values: ν max,phot = 1014 µHz, νmax,RV = 923 µHz (Huber et al. 2011a). For such extreme stars, it is difficult to use the scaling relations given in equations (1) and (2). For the stars later than F-type, however, νmax is much more precisely determined from the observations than for the hot F-type stars.
Sound speed throughout a star changes due to a variety of factors. Abrupt variations can occur due to structural reasons, e.g., transformation between the energy transportation mechanisms and ionic states of certain elements. It is thought that such acoustic glitches induce an oscillatory component in the spacing of oscillation frequencies (Houdek & Gough 2011;Mazumdar et al. 2012). In particular, changes in physical conditions in the He II ionization zone are very efficient in creating detectable glitches. Variations of ∆ν around the minima are almost entirely shaped by variations of the first adiabatic exponent throughout the zone (see Section 4).
In this paper we suggest two new frequencies and show their diagnostic potential by relating them with the fundamental stellar parameters. The paper is organized as follows. In Section 2 the basic properties of stellar interior models and Ankara-İzmir (ANKİ) stellar evolution code used in construction of these models are presented. Section 3 is devoted to analysis of oscillation frequencies, the method for determination of νmin1 and νmin2, and diagnostic potentials of the reference frequencies and their mode order differences. In Section 4, we consider how the He II ionization zone influences oscillation frequencies and hence their spacing. Section 5 deals with relating the asteroseismic quantities to the fundamental properties of stars. Finally, in Section 6, we draw our conclusions.

Properties of the ANKİ code
The models used in the present asteroseismic analysis are constructed by using the ANKİ code (Ezer & Cameron 1965). Convection is treated with standard mixing-length theory (Böhm-Vitense 1958) without overshooting. ANKİ solves the Saha equation for hydrogen and helium, and computes the equation of state by using the Mihalas et al. (1990) approach for survival probabilities of energy levels (Yıldız & Kızıloglu 1997). The radiative opacity is derived from recent OPAL tables (Iglesias & Rogers 1996), supplemented by the low-temperature tables of Ferguson et al. (2005). Nuclear reaction rates are taken from Angulo et al. (1999) and Caughlan & Fowler (1988). Although rotating models (Yıldız 2003(Yıldız , 2005 and models with microscopic diffusion (Yıldız 2011;Metcalfe et al. 2012) can be constructed by using ANKİ, these effects are not included in the model computations for this study. For only the solar model, diffusion is taken into account in order to use its known values for the convective parameter (α), hydrogen (X) and heavy element (Z) abundances.

Properties of Models
Interior models are constructed by using the ANKİ code. The mass range of models is 0.  chemical composition is taken as the solar composition: X = 0.7024 and Z = 0.0172. The heavy element mixture is assumed to be the solar mixture given by Asplund et al. (2009). The solar value of the convective parameter α for ANKİ is used: α = 1.98.
We have computed adiabatic oscillation frequencies by using ADIPLS oscillation package (Christensen-Dalsgaard, 2008) for each mass when the central hydrogen is reduced to Xc = 0.7, 0.53, 0.35 and 0.17. We can compare models with different masses having the same relative age (t rel ). Define tMS as the main-sequence (MS) lifetime of a star. Then, for a star having age t, relative age becomes t rel = t/tMS. The first value of Xc marks essentially the zero-age main sequence (ZAMS) age of each stellar mass and therefore t rel is very small. By definition, t rel = 1 for terminal-age main sequence (TAMS) models. The other Xc values (0.53, 0.35, 0.17), however, nearly correspond to t rel ≈ 0.3, 0.5 and 0.75, respectively.
In the construction of solar models, diffusion is taken into account. The maximum sound speed difference between the solar model and the Sun is 1.7 per cent. The base radius of convective zone (CZ) and surface helium abundance are 0.732 R ⊙ and 0.25, respectively. These values are moderately in agreement with the inferred values from solar oscillations: 0.713 R ⊙ and Ys = 0.25 (Basu & Antia, 1995;Basu & Antia, 1997). Improved solar models (and also models for α Cen A and B) by using ANKİ are obtained by opacity enhancement (Yıldız 2011).

FREQUENCIES OF MINIMUM ∆ν AND THEIR DIAGNOSTIC POTENTIAL
The asymptotic relation describes the relation between frequency of a mode (ν nl ) and its order (n) and degree (l). According to this relation, the large separation between the frequencies (∆ν = ν nl − ν n−1,l ) is to a great extent constant. This is true for the Sun and other solar-like oscillating stars. We compute ∆ν in the mode order range n = 10−25. ∆ν is plotted with respect to n and a constant function is fitted. For the BiSON solar data (Chaplin et al. 1999    µHz for l = 1. These results show that ∆ν is independent of l. In this study we compute ∆ν from the modes with l = 0. The range of ∆ν for degree l = 0 is 133-138 µHz. Although this is a very small interval, there are very significant changes through it. Variation of ∆ν with the frequency is plotted in Fig. 1 for l = 0, 1 and 2. The aim of this paper is to make links between such changes and stellar parameters. The common feature of the three curves is that there are two minima. We call the minimum having high frequency as the first minimum and the other one as the second minimum. The frequency of the first minimum (νmin1) is around 2600 µHz, and the second (νmin2) is around 1900 µHz. Do these minima also exist in the eigenfrequencies of a solar model? In Fig. 2  Sun also exist for the oscillation frequencies of 1 M ⊙ model. We now consider whether this kind of variation also appears in ∆ν − ν graphs for other solar-like oscillating stars of different mass. In Fig. 3, ∆ν is plotted with respect to ν for 1.0, 1.1, 1.2 and 1.3 M ⊙ interior models with Xc = 0.35. The eigenfrequencies are for the l = 0 modes. This is the case throughout this paper, if not otherwise stated. The values of νmin1 and νmin2 for 1.0 and 1.3 M ⊙ models are marked in the figure. As mass increases the minima regularly shift towards lower frequencies. While νmin1 is 2600 µHz for the Sun, νmin1 for a 1.3 M ⊙ model is about 2000 µHz. νmin2 for the Sun is about 1900 µHz and it is about 1500 µHz for the 1.3M ⊙ model. The ZAMS model of a 1 M ⊙ model has νmin1= 3400 µHz and νmin2= 2500 µHz. For interior models of stellar mass up to 1.4 M ⊙ , the minima shift again towards lower frequencies as model evolves within the MS.

Determination of νmin1 and νmin2
For oscillating stars, we have a discrete set of eigenfrequencies. In such a case, say, νmin1 does not have to correspond with any of the eigenfrequencies. Then we must determine where the minimum occurs in the ∆ν − ν graph. Suppose νmin1 is in between the frequencies ν1 and ν2. We use the slopes of the frequency intervals adjacent to ν1 and ν2 to determine νmin1. The intersection point of these two lines gives us value of νmin1. In Fig. 4 two examples for the determination of νmin1 are sketched. These are 1.0 M ⊙ models with Xc = 0.17 and 0.35.
In Figs 1 and 2, it is shown that there are three very similar curves for ∆ν for different values of oscillation degree l (l = 0, 1 and 2). However, the values of νmin1 for different l are slightly different. Such a difference may be considered as negligibly small but it may be important if one wants to determine fundamental stellar parameters. Therefore, we should try to find a single value for νmin1. In Fig. 5, the frequency difference parameter ǫ3 is plotted  with respect to ν. Here, ǫ3 is defined as where ∆ν l is the large separation for degree l. For each n, (∆ν0 − ∆ν )(∆ν1 − ∆ν )(∆ν2 − ∆ν ) is computed. ǫ3 has a much clearer minimum than ∆ν. This minimum is the first minimum. The second minimum for the Sun is missing in Fig. 5 because it is very shallow. The frequency of the first minimum for the Sun from ǫ3 is obtained as 2493.2 µHz for the BiSON data. However, in some evolved stars, the mixed modes that are observed render this method inapplicable.

The diagnostic potential of the mode order difference
The relation between the frequencies of two minima is approximately given as Although the order of oscillation modes is not determined from observations, the existence of νmin may solve this problem, entirely or in part. Both νmin1 and νmin2 shift regularly as stellar mass and age change. The difference between νmin1 and νmin2 of the models we consider is Its mean value is 5.6. The value of ∆n12 is a function of both M and t. However, the depth of the CZ, dBCZ = (R⋆ − RBCZ)/R⋆, is also a function of M and t. Here, R⋆ and RBCZ are the stellar radius and base radius of the convective zone (BCZ). Indeed, there is a linear relation between ∆n12 and 1/dBCZ. ∆n12 is about 5 when dBCZ ≈ 0.3 and ∆n12 is about 7 when dBCZ ≈ 0.1. For the mass of the CZ, however, a stringent relation is found with the mode order difference (∆nx1) between νmax and νmin1. We define ∆nx1 as  Mass of CZ (MCZ) is plotted with respect to ∆nx1 in Fig. 6. For the models of mass M < 1.2M ⊙ , there is a linear relation between MCZ and ∆nx1, at least for the MS stars. This relation arises from the fact that both MCZ and ∆nx1 are related to T eff . It is a very strict constraint for interior models of solar-like oscillating stars. While MCZ of the 1.0 M ⊙ model with Xc = 0.35 is 0.025 M ⊙ , the fitting curve gives it as 0.024 M ⊙ . For the MS models of mass M > 1.2M ⊙ (∆nx1 < 0 ), MCZ is negligibly small and therefore the method is not applicable.
A similar method can also be obtained for ∆nx2 = (νmax − νmin2)/ ∆ν , the difference between νmax and νmin2. The fitting curve for MCZ is MCZ = 0.0091∆nx2 − 0.0540. For the model given above MCZ is found from ∆nx2 as 0.025. This result is in good agreement with MCZ obtained from ∆nx1. One can take the mean value of MCZ from ∆nx1 and ∆nx2 as a constraint to interior models of solar-like oscillating stars.
In Fig.6, T eff is also plotted with respect to ∆nx1. There is an inverse relation between T eff and ∆nx1: T eff,∆n = (1.142−9.63 10 −3 (∆nx1 + 4) 1.35 )T eff⊙ . This relation is very definite and may be used to infer T eff from asteroseismic quantities alone. The difference between T eff,∆n and model T eff is less than 100 K.

SIGNATURE OF THE HeII IONIZATION ZONE ON THE ASYMPTOTIC RELATION
The sound speed within a stellar interior is given as c = Γ1 P ρ . The first adiabatic exponent Γ1 is to a great extent constant and very close to 5/3 in the deep solar interior. Near the surface, however, an abrupt change in Γ1 occurs at about 0.98 R ⊙ , as a signature of the He II ionization zone. Such a change significantly influences the sound speed profile near the stellar surface and behaves as an acoustical glitch for the oscillation frequencies.
The effect of the acoustical glitch induced by the second helium ionization zone on the oscillation frequencies is extensively discussed in the literature (see e.g., Perez Hernandez & Christensen-Dalsgaard 1994, 1998. In particular, Dziembowski, Pamyatnykh & Sienkiewicz (1991), Vorontsov, Baturin & Pamyatnykh (1991), Perez Hernandez & Christensen-Dalsgaard (1994 successfully obtained the helium abundance in the solar envelope from the phase function for solar acoustic oscillations (see also Monteiro & Thompson 2005). Houdek & Gough (2007) consider the second difference as a diagnostic of the properties of the nearsurface region. In this section we consider how the glitch shapes the variation of ∆ν with respect to ν.
The large frequency separation of a star depends on the sound speed profile in its interior. It can be written down in terms of acoustic radius as where acoustic radius dr c is the required time for sound waves to travel from the centre to the surface.
As stated above the acoustic glitches induce an oscillatory component in the spacing of oscillation frequencies. Therefore, we are facing a deviation from the asymptotic relation. The reason of the oscillatory component is essentially due to coincidence of the He II ionization with the peaks between the radial nodes.
Let ξr be the radial component of the displacement vector. It gives us the positions of the radial nodes. The solution of the second order differential equation yields (Christensen-Dalsgaard 2003) where r2 is the outer turning point and In this equation, ω is the eigenfrequency obtained from solution of the wave equation. c, N and S l are sound speed, Brunt-Väisälä and the characteristic acoustic frequencies, respectively. The influence of the He II ionization zone on ∆ν can be understood from Fig. 7. Square of ξr, given in equation (8), is plotted with respect to relative radius around the zone, for eigenfrequencies of the 1.0 M ⊙ model with Xc = 0.35 near νmin1. Also seen is the first adiabatic exponent Γ1. The horizontal axis is chosen so that the effect of the zone on Γ1 is clearly shown. In the zone, Γ1 has a local minimum where number of He II is the same as number of He III. The largest deviation from the asymptotic relation occurs for the mode that has one of its peaks closest to the minimum. This is the mode with n = 17. νmin1 is between ν17,0 and ν18,0 (see also Fig. 4). We note that the minimum of Γ1 takes place between the points where ξ 2 r of modes with n = 17 and 18 is maximum. Therefore, variation of Γ1 in the He II ionization zone shapes the variation of ∆ν. In order to relate quantitatively the expected local frequency decrease for specific modes to the minima in ∆ν, further analysis is required.  (8) is arbitrarily chosen to obtain ξ 2 r about unity. Also plotted is the first adiabatic exponent Γ 1 (thick solid line). Γ 1 has a local minimum about r = 0.98, due to the He II ionization zone. The location of the local peak in ξ 2 r relative to the dip in Γ 1 determines the departure of the frequencies from the asymptotic relation, with a decrease in the frequency that is larger, the closer the peak is to the minimum in Γ 1 . While the circles show peaks of the oscillations, the filled circles represent their projections on Γ 1 .

The relations between the reference frequencies
In our analysis, νmax of models is computed from equation (1), using model values of ∆ν , T eff and M . In Fig. 8, νmin1/ν min1⊙ is plotted with respect to νmax/ν max⊙ . The solar values of ν min1⊙ and ∆ν ⊙ are found from the BiSON data as 2555.18 and 135.11 µHz, respectively. ν max⊙ is taken as 3050 µHz. The dotted line is for νmin1/ν min1⊙ = νmax/ν max⊙ . They are correlated but there is no one-to-one relation. However, we confirm that the difference between νmin1/ν min1⊙ and νmax/ν max⊙ increases as stellar mass is different from 1.0 M ⊙ . The closest models to the dotted line are 1.0 M ⊙ models. If we plot νmin1M ⊙ /ν min1⊙ M with respect to νmax/ν max⊙ , a linear relation is obtained. In Fig.  9, νmin1M ⊙ /ν min1⊙ M (filled circle) is plotted with respect to νmax/ν max⊙ . It is shown that νmin2 obeys the same relation with νmax as νmin1. The solar value of ν min2⊙ is taken as 1879.52 µHz, again from the BiSON data. This implies that νmin1/M and νmin2/M are equivalent to each other. Furthermore, they can be used with and without νmax in new scaling relations.

Stellar mass from just ratio of the frequency of minimum ∆ν to the frequency of maximum amplitude
A very important result one can deduce from Fig. 9 is that the ratios of νmin1 and νmin2 to νmax are constant. The ratio is independent . ν min1 M ⊙ /ν min1⊙ M (filled circle) and ν min2 M ⊙ /ν min2⊙ M (circle) are plotted with respect to νmax/ν max⊙ . This shows that ν min1 M ⊙ /ν min1⊙ M = νmax/ν max⊙ . This equality is a very important tool for computation of stellar mass using asteroseismic methods.
of evolutionary phase and it simply gives stellar mass M : This implies that we can obtain stellar mass in two new ways: one is with νmin1 and the other is with νmin2. The masses computed from equation (10) in terms of νmin1 (M1) and νmin2 (M2) are listed in Table 1. Equation (10)   stellar age, at least for the MS evolution. That is to say the fractional changes of νmin and νmax in time are the same. However, the effect of chemical composition (X and Z) and the convective parameter on equation (10) should be tested. Such a test is the subject of another study. As seen in equation (10), νmin1/ν min1⊙ and νmin2/ν min2⊙ are equivalent to each other. Therefore, we hereafter prefer to use νmin1 only but νmin2 can also be used provided that it is divided by the solar value.

Scaling relations in terms of νmin 1 , ∆ν and T eff
In previous studies, the main asteroseismic parameters used to infer the fundamental stellar properties have been ∆ν and νmax. If we have high quality data, then one can also extract the average small frequency separation. In addition to these, νmin1 and νmin2 increase asteroseismic ability to predict stellar properties.
νmin1 and νmin2 can easily be determined from oscillation frequencies. We show above that νmin1/M is equivalent to νmax in new scaling relations. Then, equation (1) can be written in terms of νmin1 as The masses computed from equation (11) are plotted against model masses in Fig. 10. The agreement is good between the two mass estimates. There is a very slight deviation from a linear relationship. For better agreement, the power in the right-hand side of equation (11) should be modified to The computed masses (Me12) from equation (12) are also listed in Table 1. The maximum difference between equation (12) and model mass is about 2.5 per cent (see Fig. 14 in Section 5.9). In Table 1. Masses and radii computed by using asteroseismic methods. They are in the solar units. M mod and Xc in the first and second columns are the model mass and central hydrogen abundance, respectively. M 1 and M 2 , given in the third and fourth columns, are masses computed by using ν min1 and ν min2 (equation 10), respectively. M 2 of some low-mass models is absent because the second minimum is not seen in the eigenfrequencies of these models. M e12 is computed from equation (12), and M sis is obtained from equation (15) or (16). M given in the seventh column is the mean of M 1 and M 2 . We give the percentage difference between M mod and M in the eighth column. In the last two columns, model radius and radius derived (R e17 ) from ν min1 and ∆ν (equation 17) are listed.   1.30 0.17 1.31 1.32 1.27 1.27 1.31 -0.9 1.59 1.60 the mass interval for solar-like oscillating stars near the MS, two structural transitions occur. While the CZ becomes shallow in the outer regions as stellar mass increases, a convective core develops in the central region. Therefore two separate fits may in turn be required (see below). If we insert the expression we derived for νmax in equation (2), is obtained for radius. We insert equation (11) in equation (13) and then find The uncertainty in the above expression is 3.5 per cent. In order to raise this uncertainty we plot a figure similar to Fig. 10 but for radius. We obtain more precise results than given by equation (14) if we reduce the right-hand side of equation (14) to the power of 0.95.

Mass and radius in terms of νmin 1 and ∆ν
The T eff values of many Kepler target stars are not determined very precisely. If we assume a typical uncertainty ∆T eff ≈ 200 K, the uncertainty is about 3 per cent for T eff = 6000 K. This causes an uncertainty in M about 4 to 5 per cent. To reduce this uncertainty in M , here we try to derive expressions for M and other fundamental properties in terms of purely asteroseismic quantities ∆ν and νmin1. These simple relations are obtained to illustrate the diagnostic potentials of new asteroseismic parameters (νmin1). They are not the final forms that one can derive. For a lower uncertainty, two separate formula may be derived for two mass intervals 1−1.2 and 1.2−1.3 M ⊙ . If M < 1.2 M ⊙ , then If M > 1.2 M ⊙ , then The uncertainties in equations (15) and (16) are less than 2 per cent; see Table 1. Again using only the asteroseismic quantities ∆ν and νmin1 we try to obtain an expression for stellar radius. Indeed, many relations can be found by similar fitting procedures; the most precise one we obtain is The maximum difference between equation (17) and model radius is 1.5 per cent. Similarly, we also derive an expression for gravitational acceleration at the stellar surface (g): Equation (18) is also a very precise relation. It is in very good agreement with the model g values. The maximum difference between them is less than 2 per cent.

Effective temperature and luminosity in terms of νmin 1 , νmax and ∆ν
T eff is one of the very important stellar parameters and it is not precisely determined in many cases. Luminosity, however, is one of the essential parameters if one compares stellar models with stars. It is the most rapidly changing parameter throughout MS evolution and therefore is considered as an age indicator. In order to show the diagnostic potential of asteroseismic properties, we also derive fitting formula for T eff and luminosity L. For T eff of models with M > 1.0M ⊙ , The maximum difference between equation (19) and T eff of the models is 150 K, but the mean difference is about 50 K. The method for determination of T eff from ∆nx1 gives much more precise results (see Fig. 6). The fitting formula obtained for luminosity as a function of the asteroseismic parameters is In Fig. 11, the luminosity derived from the asteroseismic parameters is plotted with respect to the model luminosity. The agreement seems excellent at least for the MS models with solar composition. This result is very impressive because luminosity is one of the most uncertain stellar parameter derived from observations. 5.6 Age in terms of νmin 1 , ∆ν and δν02 The age of a star is one of the most difficult parameters to compute. It is very sensitive function of stellar properties, such as mass and chemical composition. The number of stars for which we know these properties is unfortunately very small. Therefore the promise of asteroseismology to better constrain stellar age is very important. The mean value of small frequency separation, δν02 , is a very good age indicator. We obtain a fitting formula for the stellar age, which is a function of νmin 1 , νmax, ∆ν and δν02 . Defining we obtain the fitting formula for the stellar age as where M/M ⊙ is computed from equation (10) and therefore a function of νmin 1 and νmax. The ages computed from equation (22) are plotted with respect to model age in Fig. 12. For some ZAMS models, equation (22) based on the asteroseismic parameters gives negative values for the age. Age in such a case is considered to be very small and can be taken as the ZAMS age. For the other models, the difference between the age derived from equation (22) and model age is less than 0.5 Gyr. For the Sun, equation (22) gives its age as 4.8 Gyr. This result is in very good agreement with the solar age found by Bahcall, Pinsonneault & Wasserburg (1995), 4.57 Gyr. One should notice that the models are constructed with solar composition. The effect of metallicity on these relations (and chemical composition in general, for example effect of X) should be studied further. a bit different. δν02 depends also on which interval of n is used, and it is not very certain in many cases. We suggest a new asteroseismic diagram (AD) in terms of ∆ν and νmin 1 in Fig. 13. The vertical axis is ∆ν in the new AD and the horizontal axis is chosen as ∆ν /νmin 1 in solar units. This form of the AD is very compatible with the classical HR diagram. The ZAMS line is in left-hand part and TAMS line is in the right-hand part of the AD. Furthermore, the low mass models appear in the lower part and high-mass models are in the upper part of the AD. Thus, evolutionary tracks of stars in the HR diagram and the AD are very compatible with each other.

The effect of metallicity on the relation between stellar mass and oscillation frequencies
Eigenfrequencies of a model depend on many stellar parameters. This can lead us to expect that metallicity may influence the relations we derive in the present study. Equation (10), for example, can be rewritten as where βz is the parameter to be determined from model computations. Our preliminary results show that βz is about 0.1.

On the uncertainty in νmin1 and relations between asteroseismic quantities and fundamental stellar parameters
The main uncertainty in our results comes from uncertainty in νmin1. As an example, we plot the difference between the model mass and mass derived from M1 =  Figure 14. The mass difference between M 1 = 1.188ν min1 /νmax and model mass (filled circles) is plotted with respect to M mod . The uncertainty in M 1 is about 0.025 M ⊙ . ∆ν /(2ν min1 ) is also about 0.025. This implies that uncertainty in M 1 depends on how accurate ν min1 is. The uncertainty in ν min1 is about ∆ν /2. than 0.025 M ⊙ . This must be due to determination of νmin1. For comparison, ∆ν /(2νmin 1 ) is also plotted. It is also about 0.025. This implies that νmin1 is uncertain by about ∆ν /2. This amount of uncertainty seems reasonable considering our method for determination of νmin1.

CONCLUSION
In the present study, we analyse two frequencies (νmin 1 and νmin 2 ) at which ∆ν is minimized. These frequencies correspond to the modes whose one of radial displacement peaks coincide with the minimum of Γ1 in the He II ionization zone (see Fig. 7). They have very strong diagnostic potential. If we divide any of them by the frequency of maximum amplitude (νmax) we find stellar mass very precisely. In the previous expressions in the literature, M is found in terms of νmax, ∆ν and T eff . The precision of stellar mass found from asteroseismic methods depends the precisions of the inferred frequencies (νmin 1 , νmin 2 and νmax).
Both νmin 1 and νmin 2 are functions of stellar mass and age in particular, and in general depend on all the parameters influencing stellar structure. Such dependences in some respects complicate the situation, but they become very strong tools if the relations between parameters and frequencies are well-constructed. Variations of both νmin 1 and νmax with evolution are the same and therefore their ratio remains constant and yields stellar mass.
The method we find is in principle very precise. Fundamental properties, such as mass, radius, gravity and T eff are determined within the precision of 2 to 3 per cent. This is the level of accuracy for the well-known eclipsing binaries. We also derive a fitting formula for luminosity (equation 20) and age (equation 22) as functions of asteroseismic quantities.
Frequencies νmin 1 and νmin 2 are equivalent to each other. They obey the same relations, at least for the MS stars. The mode order difference between them ((νmin 1 − νmin 2 )/∆ν) is related to the depth of the CZ. However, the mass of the CZ is best given by any of the mode order differences ∆nx1 = (νmax − νmin 1 )/∆ν or ∆nx2 = (νmax − νmin 2 )/∆ν. For example, MCZ = 0.066∆nx1M ⊙ (see Fig. 6). ∆nx1 and ∆nx2 are also very important tools for precise determination of T eff .
We obtain scaling relations using asteroseismic quantities, νmin 1 , ∆ν, νmax and T eff . We also derive expressions for fundamental stellar parameters by eliminating T eff .
We also suggest a new AD. The y-axis is the large separation ∆ν and ∆ν/νmin 1 is the x-axis. In this form of the x-axis, the evolutionary tracks and ZAMS and TAMS lines in AD are compatible to those in the traditional HR diagram.
The present study is essentially based on the oscillation frequencies of models with solar composition. Our preliminary results on the models with higher metallicities than the solar metallicity show that relations between asteroseismic quantities and fundamental stellar parameters are changing with the metallicity. A similar test should be carried out for variations in the hydrogen abundance. In the next paper of this series of papers, we will do this test and apply the methods developed in the present study to the Kepler and CoRoT data and also test the effects of chemical composition.