Detre Centennial Conference
Commun. Konkoly Obs. N°. 104
© Konkoly Obs., Budapest, 2006

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Fundamental parameters of pulsating stars from atmospheric models

Szabolcs Barcza

Konkoly Observatory of the Hungarian Academy of Sciences
P.O. Box 67, H-1525 Budapest, Hungary

A purely photometric method is reviewed to determine distance, mass, equilibrium temperature, and luminosity of pulsating stars by using model atmospheres and hydrodynamics. T Sex is given as an example: on the basis of Kurucz atmospheric models and UBVRI (in both Johnson and Kron-Cousins systems) data, variation of angular diameter, effective temperature, and surface gravity is derived as a function of phase, mass M = 0.76±0.09 Msun, distance d = 530±67 pc, Rmax = 2.99 Rsun, Rmin = 2.87 Rsun, magnitude averaged visual absolute brightness <Mv> = 1.17±0.26 mag are found. During a pulsation cycle four standstills of the atmosphere are pointed out indicating the occurrence of two shocks in the atmosphere. The derived equilibrium temperature Teq = 7781 K and luminosity 28.3±8.8 Lsun locate T Sex on the blue edge of the instability strip in a theoretical Hertzsprung-Russell diagram. The differences of the physical parameters from this study and Liu & Janes (1990) are discussed.

Introduction

Except for supergiants and cool stars with effective temperature Te<4000K a large grid of model atmospheres is available in the literature (Kurucz, 1997 for the latest versions). The conversion of theoretical fluxes to observational magnitude and colour indices has been solved (e.g. for Johnson and Kron-Cousins photometries, Castelli, 1999) opening the way to a determination of the atmospheric parameters effective temperature, Te, and surface gravity if broad band photometry is available for the target star.

The stellar angular diameter can be derived in the following manner from atmospheric models. Monochromatic flux F{lambda}(0) of the models at optical depth {tau}=0 is given in tabular form for a large grid of Te, g, metallicity [M], etc., furthermore, by integrating F{lambda}(0) for a number of photometric systems the fluxes FX(0) were computed in physical units erg s-1 cm-2 as well, and the corrections from interstellar absorption, reddening were included (Kurucz, 1997). X represents the photometric band, e.g. V in the Johnson system. On the other hand, observed physical fluxes, IX, of stars in band X can be derived by using the absolute flux calibration of Vega (Tüg et al., 1977). If the interstellar extinction is AX magnitude, the half angular diameter of a spherical (i.e. non-distorted) star can be derived from the simple formula where d is the distance to the star, R is the radius of {tau}=0 (Barcza, 2003).

For giant, main sequence, or white dwarf stars the stellar atmosphere is a thin layer in comparison with R. The gravitational acceleration is practically constant in the atmosphere, i.e. at r~R, where {rho}(r), p(r), M, G are density, pressure, mass, and the Newtonian gravitational constant, respectively. The surface gravity, g(R), is an important parameter of the static atmospheric models given conventionally as log|g| and it can be used to weigh a star if the stellar radius, R, is known. Relation (eq2) was applied for a number of non-variable stars enhancing our knowledge on stellar masses: e.g. for white dwarfs purely from log|g(R)| of model atmospheres and R Mwd=0.480±0.014 average mass was derived (McMahan, 1989) for a sample of 53 stars while other methods gave somewhat larger mass, Mwd=0.58±0.10 for 64 stars (Koester et al., 1979).

Problems, dependence of g, {theta}, M on E(B-V),[M] were discussed in numerous papers, e.g. Kinman & Castelli (2002), Decin et al. (2003). Accuracy of the data from broad band photometry is inferior to that from fine analysis of spectra or interferometry, respectively, and M can be determined much better from binary orbits. However, if the more accurate methods are not available, the photometric determination of g, {theta}, and M must be appreciated.

Implementation for UBV(RI)C photometry

The main parameters of atmospheric models are Te, g, [M], and in addition to them the interstellar reddening, extinction (e.g. E(B-V), AX) must be known. From observational point of view colour indices, Ci, (i=1,2,...) are the input parameters and the inverse problem of ordering a model to them must be solved. In optimal cases the typical errors of interpolation are 0.1 in log |g| while a few hundredth in FX, respectively, however, the successful solution depends on the shape of the functions Te(C1,...) and log|g(C1,...)|. The final step is that theoretical FX(C1,...) belonging to the selected model must be compared with IX from observed magnitudes according to (eq1).

From inspecting the Kurucz tables some general rules can be inferred, we mention a few of them. For main sequence or giant stars UBV photometry must be preferred if 7000 < Te < 104 K while for cooler stars BVIC gives reliable results as well. Below Te=9000 K the dependence of log|g| on U-B becomes more pronounced, U-B itself is sensitive to [M] because of the appearance of metallic lines in ultraviolet. In favourable cases Te and g can be determined from a two colour diagram if [M], E(B-V) were already known while the procedure is more uncertain if [M] and E(B-V) must be estimated from colour indices as well, in this case more than two colour indices are necessary.

To demonstrate reliability of the method, some date are given in Table 1: {theta} was derived from observed V, B-V, U-B (Hoffleit, 1982) by using FV(U-B,B-V) of the Kurucz models while for comparison the value {theta}HB is given from intensity interferometry corrected for limb darkening (Hanbury Brown et al., 1974). Using the Hipparcos parallaxes (ESA, 1997) R=3.086×1013{theta}/{pi} km was calculated and the mass was obtained by (eq2) from log|g(U-B,B-V)|. It can be noted that the weakest sampling of the spectrum is provided by (U-B,B-V) since the wavelength interval ~360-540 nm is covered in this case. Of course an inclusion of other colour indices could improve the values of {theta},log|g|.

Half angular diameter of some stars from intensity interferometry ({theta}HB) and V, B-V, U-B of Kurucz models ({theta}), the units are 10-9. The mass was calculated from the Hipparcos parallax {pi} (ESA, 1997) and using R=3.086×1013{theta}/{pi} in (eq2).

HR V B-V U-B {theta}HB {theta} {pi} M{sun}
2421 1.93 0.00 0.04 3.37±.21 3.55 0.031 1.51
2491*[1] -1.46 0.00 -0.05 14.28±.39 14.31 0.379 3.47
2943*[2] 0.38 0.42 0.02 13.47±.42 11.80 0.286 1.7
3685 1.68 0.00 0.03 3.85±.17 3.89 0.029 4.07
4534 2.14 0.09 0.07 3.22±.24 3.33 0.090 1.70
6556 2.08 0.15 0.10 3.95±.31 3.79 0.070 2.24
7001 0.03 0.00 -0.01 7.85±.11 7.97 0.129 2.47
7557*[3] 0.77 0.22 0.09 7.22±.34 7.65 0.194 1.18
8728 1.16 0.09 0.08 5.09±.34 5.29 0.130 1.55

For pulsating stars with periods exceeding 0.1 day the characteristic time of changes of the main atmospheric parameters Te, ge, FX is long in comparison with the formation of radiative (and convective) equilibrium, therefore, the static atmospheric models account satisfactorily for the changes of ge and {theta} as a function of pulsation phase {phi} (Buonaura et al., 1985). The subscript in ge(=-{rho}(r){nabla}p(r)>0) indicates that this is an effective outward acceleration produced by the pressure gradient in the atmosphere. On appropriately selected two colour diagrams the loop of a variable star gives logge({phi}), Te({phi}), FX({phi}) etc. of the models while photometric observations can be reduced to IX({phi}) giving finally {theta}({phi}) by (eq1).

- Mass and distance from {theta}({phi}), ge({phi}) by using the radial momentum balance in the pulsation

If {theta}({phi}j), j=1,2,...,N points are available in a sufficient number to numerical differentiation of {theta}({phi}) the angular acceleration can be converted to R(t)={theta}(t)d, t={phi}P, where P is the period in seconds. Numerical experience showed that N~500 or more points gave reliable R({phi}) if they were distributed uniformly in 0 <={phi}<=1.

Now the main innovation is that R({phi}j), ge({phi}j), j=1,2,... are introduced in the radial component of the Euler equation of hydrodynamics and we exploit that the atmosphere must be in standstill at least twice in a pulsation cycle. At these phases {phi}i (i=1,2) the radial component of the non-linear term vanishes: the non-radial motions are expected to contribute negligibly to the momentum balance in radial direction, therefore, (v,{nabla})v{proportional to}{dot}R~0 and we have the equations: The fundamental stellar parameters: distance and mass (Barcza, 2003) follow from (eq3) by elementary operations. The apparent accelerations Ra can be neglected if the characteristic spatial extension of the atmosphere - i.e. the size of the interval 0<={tau}Rosseland<=1 varies slightly during the whole pulsation cycle. If there are more than two standstills, then i>=3 in (eq3). Eventual different values of M,d from i=1,3, i=2,3 etc. can indicate e.g. non-radial pulsation because radial pulsation was assumed throughout the procedure outlined here.

The fundamental stellar parameters equilibrium luminosity and temperature (Carney et al., 1992) are obtained from {theta}({phi}),Te({phi}),d by where {sigma} is the Stefan-Boltzmann constant, Teq differs slightly from the average effective temperature <Te({phi})>. Leq,Teq allow to locate a variable star in theoretical Hertzsprung-Russell diagram: of course on the basis of [colours-Te, FX] calibration of the used atmospheric models.

- Comparison with the Baade-Wesselink method

The determination of {theta}({phi}) is common in both methods. The difference lies in its further use.

In the BW method the radius change is derived from integrating the radial velocity curve and it is equated with {theta}d. The kinematic equation must be solved for d where pp is the projection factor of converting radial to pulsation velocity and v{gamma} is the barycentric velocity of the star. Physical input comes from the time-dependent projection factor. A much more serious uncertainty of kinematic nature is imported in this procedure by the error {Delta}v{gamma}. If {Delta}v{gamma} << {overline}|v{gamma}-vradial(t)| a negligible error is propagated into {delta}R, however, it is problematic to achieve this desired accuracy because the observed radial velocities are an integral of the radial component of true, non-uniform motions in the atmosphere contaminated by apparent velocity changes from varying opacity during phases of different compression. The difficulties from the uncertain value of v{gamma} could be circumvented by differentiating pp(t)vradial(t) and substituting it for {theta}d in (eq3). However, because of low number, large scatter of the observed radial velocities, and eventual time dependence of pp(t) this principally correct use of (eq3) cannot result in d,M of acceptably small error.

By using {theta},{theta},ge in (eq3) radial velocity observations and their problematic conversion to radius changes are not necessary at all at the price of differentiating {theta}({phi}) twice plus more physical input: ge({phi}) and the assumption of radial momentum balance must be used. Finally d and M are obtained by solving algebraic equations. The method was described in detail and applied for the RRab star SU Dra by Barcza (2003). We mention two remarkable details from this study. (i) At minimum radius, i.e. at maximum compression in the atmosphere, ge=50.1 ms-2 >> GM/R2~8 ms-2 reduces (eq3) to d < ge/{theta} giving d < 718pc. This upper value is very close to d=647pc from the solution of (eq3) for three observed standstills. From them the unequivocal d,M show that a non-radial mode, if there is any, is negligible in comparison with the radial oscillation of SU Dra. (ii) During a time interval t the small uncertainty {Delta}v{gamma}~+5.9km s-1~0.04v{gamma} (Liu & Janes, 1990 vs. Oke et al., 1962) leads to a phase dependent error {Delta}R=pp{Delta}v{gamma}t<=3.43×105km (~(Rmax-Rmin)/2 !) which has a considerable effect on d. If the method by Liu & Janes (1990) is followed to solve the kinematic equation, the increment of d is 1.26 from {Delta}v{gamma}=5.9 km s-1. This results in correction <MVmag>=0.78->0.28 for the magnitude average absolute magnitude. (Roughly this is the difference between the short and long extragalactic distance scales - Gratton, 1998. The distance 815 pc corresponding to the long distance scale is ruled out by d < 718 pc.)

Application to T Sex

Now we apply the method for the RRc variable T Sex. Good quality photoelectric observational material was collected from the literature, the sources are given in Table 2.

V,UBV,BVRI,UBV(RI)C data of T Sex used for constructing average light and colour curves, k is the serial number of the segment, nk is the number of V points in the segment, {delta}k,Ek are the phase shift and epoch in (eq9)

k HJD-2400000 nk
{delta}k Ek Source
1 34311.8675-.9315 31 V -0.0160 6 TS58
2 34350.8051-.9402 66 V -0.0247 126 TS58
3 34363.7245-.8531 64 V -0.0300 166 TS58
4 34508.5948-.6598 10 V -0.0766 612 TS58
5 35190.7392-.8208 2 UBV 0.0593 2712 TS58
6 35191.6423-.8871 72 UBV 0.0586 2715 TS58
7 35195.6440-.9000 106 UBV 0.0470 2727 TS58
8 35513.7227-.9620 40 UBV -0.0286 3707 TS58
9 35514.6815-.9477 60 UBV -0.0275 3710 TS58
10 35516.6802-.9387 73 UBV -0.0272 3716 TS58
11 38017.9584-8.0870 43 UBV 0.0322 11417 PP64
12 38035.8141-.9985 55 UBV 0.0280 11472 PP64
13 38038.8007-.8928 28 UBV 0.0227 11481 PP64
14 40678.6990-.7931 10 V 0.0778 19610 EE73
15 40680.6390-.7382 8 V 0.0757 19616 EE73
16 41013.7358-.7565 4 V -0.0127 20641 EE73
17 43525.792-56.694 11 V 0.0588 28418 E94
18 45387.7850-.9212 76 BVRI 0.0010 34109 BM88
19 45388.6578-.7634 69 BVRI 0.0026 34112 BM88
20 45389.6382-.8760 113 BVRI 0.0029 34115 BM88
21 45393.7134-.8596 98 BVRI 0.0000 34127 BM88
22 45400.6689-.7235 45 BVRI -0.0021 34149 BM88
23 46845.6886-.7489 6 UBV(RI)C 0.0540 38598 LJ89
24 46846.8005-.8859 7 UBV(RI)C 0.0429 38601 LJ89
25 46847.6521 1 UBV(RI)C 0.0480 38604 LJ89
26 46848.6547-.9614 25 UBV(RI)C 0.0432 38607 LJ89
27 47197.7625-8.0035 16 UBV(RI)C -0.0301 39682 LJ89
28 47226.6368-.9153 3 VRC -0.0258 39771 LJ89
29 47488.8955-.9059 7 BVRI -0.0881 40578 BM92

- Period, O-C, average light and colour curves

The periods given by TS58, PP64, EE73, BM88, LJ89 indicated period changes of O(10-6). To clarify it string length minimization (SLM) was applied, since good conversion of the observations to average light curve and colour-colour loops is crucial to determine {theta}({phi}),ge({phi}). (The adaptation of SLM to pulsating stars was described by Barcza (2002), the notation of this paper will be used here. The essence of SLM is that first the segments k=1,2,... of the observed V magnitudes are projected onto the phase axis {phi}, 0 <={phi}<=1 by the saw tooth function which is perturbed by a term {delta}k=(O-C)k << P0 accounting for the phase shift of segment k and next the neighbouring points are connected by straight strings. Finally the normalized sum {nu}l of the string lengths is minimized numerically as a function of period P0 and phase shifts {delta}1,....)

The V file contains 1149 observations in 29 segments, SLM gave for the maxima with P0=0.3247796±0.0000032 and the values {delta}k in Table 2. The summed string length was {nu}l0(P0,{delta}=0)=0.330 with standard deviation SD=0.069 magnitude, the folded light curve belonging to it is plotted in Fig. 1a. Applying the values {delta}k of Table 2 reduced {nu}l(P0,{delta}) to 0.062 with SD(P0,{delta})=0.012 mag, dots of Fig. 1b are a plot of the folded light curve, its SD=0.012 does not exceed the expected random error of a V point, it is lower than the claimed amplitude 0.028, 0.015 mag for the second and third periods of T Sex (Hobart et al., 1991). Thus, the conclusion must be drawn that between HJD 2434311-2447488 the light curve of T Sex can perfectly be reproduced if the light curve segments in Fig. 1a are shifted to the HJDmax given by (eq9). The colour indices were shifted by {delta}k of the segment, Figs. 1c-f are their plots. The line in Figs. 1b-f was obtained by fitting high order (<=9) polynomials to the points. These drawn light and colour curves were used to construct colour-colour loops and to interpolate the physical parameters of the Kurucz models.

Two remarks on the homogenization of the colour curves. (1) In U-B of segments 5,6,7 a zero point correction +0.08 was applied while for 8,9,10 it was +0.05 in order to bring the TS58 observations in coincidence with those of PP64 and LJ89. (2) The Johnson V-R,V-I colours of BM88 and BM92 were converted to V-RC,V-IC by Taylor's (1986) empirical formulae.

 


Panel (a): 1149 V observations folded by P0=0.3247796 days, {delta}=0. Panels (b)-(f): dots: V,U-B,B-V,V-RC,V-IC observations folded with the values {delta}k given in Table 1. Lines: High order polynomial fitting curves.

 

- The physical parameters

The colour-colour loops (U-B,B-V),(U-B,V-RC),(U-B,V-IC),(B-V,V-IC) could be used to interpolate Te({phi}), logge({phi}), FV({phi}), FRC({phi}), BC({phi}). Using a midpoint formula the half angular velocity {dot}{theta} and acceleration {theta} were determined from {theta}({phi}). Fig. 2 is a plot of the results for E(B-V)=0.095,[M]=-1.2. From the different loops the scatter of logge was the largest: ±0.11,±0.03 at {phi}=0.3,0.9, respectively, the scatter in FV,FRC,Te was <=0.02. Therefore, logge, {theta}, Te were averaged at each {phi} from the four loops. The scatters result in an error of ~0.15 for the derived mass and distance if one pair of standstills is used.

Average {theta}, {theta}, logge at {phi}= 0.31, 0.56, 0.65, 0.90 for E(B-V)=0.095,[M]=-1.2,-1.0. The units are: radian, radian s-2, cm s-2.


{phi} {theta}×1010 {theta}×1018 logge
[M]=-1.20
A 0.31 1.273 -0.334±.013 3.38±.11
B 0.56 1.242 0.189±.012 3.37±.10
C 0.65 1.247 -0.257±.020 3.36±.07
D 0.90 1.222 1.56 ±.02 3.71±.03
[M]=-1.00
A' 0.31 1.264 -0.331±.013 3.45±.10
D' 0.90 1.217 1.58 ±.02 3.74±.03

Average physical quantities from the pairs of standstills AD,BD,CD assuming M=-1.20 and different reddenings. The units are: K, K, pc, solar mass, radius, and luminosity. Our final choice is E(B-V)=0.095.

E(B-V) <Te> Teq d M <R> Leq
0.08 7607 7625 452±73 0.37±.06 2.54 19.6±6.9
0.09 7712 7728 514±84 0.55±.08 2.86 26.5±9.4
0.095 7765 7781 530±67 0.76±.09 2.93 28.3±8.8
0.10 7816 7834 558±65 1.06±.16 3.07 31.9±6.9
0.105 7871 7886 892±130 3.04±.44 4.89 84 ±26

 


The average variable parameters of T Sex for E(B-V)=0.095,[M]=-1.2 from the loops (U-B,B-V),(U-B,V-RC),(U-B,V-IC),(B-V,V-IC). The outlier points of panels (a,c,d) at {phi}~0.36,0.72-0.76 are artifacts of interpolation, they were not smoothed out to show the effect of eventual uncertainties in the interpolation.

 

- Distance, mass, equilibrium luminosity and effective temperature

The standstills of the atmosphere were found at {phi}~0.31,0.56,0.65,0.90 by searching for zero average angular velocity {overline}{dot}{theta}({phi})~0. {theta},logge were averaged here in an interval {Delta}{phi}~0.02 (i.e. ~10 min). Table 3 reports the results for some values of E(B-V)=0.095,[M]=-1.2,-1.0.

The characteristic size of the atmosphere is ~104 km, Ra << ge, therefore, Ra can be neglected.

<Te({phi})>,Teq and from pairs of standstills AD,BD,CD the average d, M, R, Leq are given in Table 4 to some values of E(B-V). The estimated errors from the averaging are in accordance with that of the interpolation of logge, the other quantities propagate negligible errors in (eq4),(eq5).

Discussion

- Dependence of the physical quantities on reddening

The strong dependence of Te({phi}),logge({phi}) and the derived d,M on E(B-V) is a surprising result of this study while the value of [M] is of secondary importance. E(B-V)=0.05±0.02 by Liu & Janes (1990) gives very small d,M which cannot be reconciled with our present day theoretical knowledge on RR Lyrae or other type of pulsating stars. Reddening E(B-V)=0.07 and 0.09 were suggested by Hobart et al. (1991) and Hemenway (1975), respectively, our finding is that E(B-V)=0.09-0.1 is the only possible choice.

- Effective temperature, surface gravity

Te({phi}),logge({phi}) of the present study are significantly higher by some 650 K and ~0.4 than those of Liu & Janes (1990). Since essentially the same Kurucz tables were used we attribute the difference to the different philosophy of the interpolation procedure:

  • [] Liu & Janes (1990) determined ge({phi})=0.6M{sun}G/R2({phi}) from kinematics with assuming the canonical mass 0.6M{sun} of RR Lyrae stars. Next, one arbitrarily chosen colour index, V-K was taken as sole source of Te({phi}) belonging to logge({phi}) from the kinematics and Te({phi}) from the other colour indices were neglected leading to their very low <Te>=7137 K. To check this procedure their Te(B-V),Te(V-RC),Te(V-IC),Te(V-K) tables were all used for logge=3,3.5: according to our opinion no reason can be found to reject any Te since the difference was <=250 K with no systematically decreasing trend when going more and more to the infrared indices. Therefore, averaging could have been more appropriate. Theoretical colour indices differing systematically from the observed ones by 0.02-0.04 mag correspond to their lower logge({phi}) values.
  • [] In the present study two colour diagrams were used to determine a pair of Te({phi}), logge({phi}) simultaneously taking into account the dependence of Te({phi}) on logge({phi}) automatically and the small differences (<=250 K in Te, <=0.11 in logge) from the four colour-colour loops justified averaging. This is a self-consistent procedure trusting on the Kurucz models exclusively. By using four colour-colour loops, different parts of the whole spectrum F{lambda}(0) were sampled in the {lambda} interval 360-1000 nm.

- Position in a theoretical HRD, absolute brightness

The values of the fundamental parameters logTeq=3.891, logLeq=1.452, M=0.76{sun} for E(B-V)=0.095 place T Sex just to the blue edge of the instability strip given by Tuggle & Iben (1972) and Lee et al. (1990). This position is expected for an RRc variable and it was our main argument to accept the large reddening. With distance d=530 pc we get R({phi}=0.31)=2.99R{sun}, R({phi}=0.91)=2.87R{sun}, and magnitude averaged visual absolute brightness <MmagV>=1.17±0.26 mag. These values come purely from atmospheric models which are trustable to a few percent level and verify figures from stellar structure calculations. The values d=578.5 pc and <MVmag>=1.06±0.38 of Hobart et al. (1991) agree with the present results within the quoted errors. The physical parameters of Liu & Janes (1990), (0.47M{sun}, d=667 pc, <Te>=7137 K, <R>=4.05 R{sun}, <MV>=0.76±0.27 mag <log ge>=2.98) are unreliable.

Figs. 3a-c are plots of the variable acceleration, velocity, radius in absolute units for a pulsation cycle if d=530 pc. The sharp undulation of the curves in 0.72 < {phi}< 0.78 is not real, it originates from the interpolation artifact indicated in the caption to Fig. 2.

 


Panel (a): line: ge({phi}), dashed: -0.76M{sun}G/R2({phi}). Panel (b): {dot}R({phi}), filled circles: 25-vradial({phi}) from Liu & Janes (1989). Panel (c): R({phi}). Panel (d): circles: 28-vradial({phi}) from Liu & Janes (1989), triangles: 38-vradial({phi}) from Barnes et al. (1988), crosses: 38-vradial({phi}) from Tifft & Smith (1958), dashed line: mean of 38-vradial({phi}) from Barnes et al. (1988). The outlier points of panels (a)-(c) at 0.72 < {phi}<0.78 are artifacts of interpolation, they were not smoothed out to show the effect of eventual uncertainties in the interpolation.

 

- Open problems

There is a systematic difference in the vradial({phi}) curves of Barnes et al. (1988), Tifft & Smith (1958), and Liu & Janes (1989): v{gamma}=38, and 28.0±1.4 km s-1, respectively, which exceed the expected observational errors. On the basis of Fig. 3b v{gamma}=25 km s-1 seems even more probable. The amplitudes 28,24,24 km s-1 of TS58, BM88, LJ89 do not differ significantly. Fig. 3d shows that subtracting vradial({phi}) from the different barycentric velocities does not bring the different observations to full coincidence. The stability of the light and colour curves is, however, obvious from Fig. 1, therefore, secular change of the shape of vradial({phi}) seems to be improbable. A variable v{gamma} i.e. duplicity cannot be excluded in spite of the large scatter of vradial({phi}) of TS58, BM88, furthermore, the scatter of {delta}k is some 3 hours. This is significantly higher than the imaginable error of a {delta}k value and for its explanation barycentric motion i.e. light-time effect can even be considered.

On speculation level the extreme sensitivity of the BW method on {Delta}v{gamma} may be guessed as the main source of the discrepant d of Liu & Janes (1989) which was propagated into their M, <R>, <MV>. Since the radial amplitude of T Sex is only some {delta}R=80000 km for a BW analysis v{gamma} ought to be known by an accuracy O(100)m s-2 which was obviously not reached, another factor is their low Te({phi}). (Their quoted error {Delta}v{gamma}=±1.4 km s-1 leads to an error ~±18000 km in {delta}R resulting in an error {Delta}d/d~0.21.)

The product pp[v{gamma}-vradial({phi})] is the comparable quantity with {dot}R plotted in Fig. 3b. The spectral lines originate from 0 < {tau}Rosseland < 0.4, therefore, we expect pp|v{gamma}-vradial({phi})|< |{dot}R({phi})|. It is satisfied if pp(t)~1, however, the large scatter of the radial velocities gives weak basis for this very small value. Remarkable is that the most accurate radial velocities i.e. those of Liu & Janes (1989) show small humps at the extreme values of {dot}R at {phi}~0.2,0.4,0.6 and {dot}Rmax-{dot}Rmin agrees better with the extreme values of v{gamma}-vradial if v{gamma}=25 km s-1.

The rather loose correlation of {dot}R and 25-vradial({phi}) is similar to that found in SU Dra (Barcza, 2003) and it raises a serious question concerning the basic equation (eq8) of the BW method. A qualitative explanation can be guessed from gas dynamics and the technique of measuring radial velocity.

  • {dot}{theta}({phi}),{dot}R({phi}) reflect the motion of {tau}=0 while the spectral lines originate from the surroundings of {tau}{lambda}line~0.3. Non-negligible velocity gradient is definitely present in an RR Lyrae atmosphere (e.g. Oke et al., 1962) and the limb darkening integrates the non-uniform motion of the layers 0 < {tau}< 0.5 into a single value vradial({phi}). (Dynamical atmospheric models are not available to treat quantitatively the conversion of the pulsation velocity to vradial({phi}).)
  • CORAVEL technique is itself accurate for stars of non-variable spectra while applying it for a variable spectrum may result in systematic errors which are not easy to survey. The coarse agreement of vradial({phi}) from spectroscopy and CORAVEL is obvious in Fig. 3d, however, the large scatter indicates that some caution is appropriate, especially, since fine details of {theta}({phi}), vradial({phi}) play some role in a BW analysis.

Conclusions

The purely photometrically derived fundamental parameters of T Sex (see Table 4) have been found from ATLAS atmospheric models of Kurucz (1997) and their calibration to stellar photometric systems (Castelli, 1999). They have been found to be in consensus with our knowledge on stellar models and pulsation theory of asymptotic giant branch stars. In addition to bridging over these remote branches of astrophysics some details have been revealed on RR Lyrae type pulsation: at the RRc variable T Sex fine structure, definite footprint of two shocks have been found in the variable stellar radius R({phi}) (i.e. in the distance of zero optical depth from the stellar centre). In a previous study of SU Dra similar details were found concerning the fine structure of the atmospheric pulsation: there is at least one pair of temporal, intermediate, minor standstills of the pulsating atmosphere between maximum and minimum extension. This seems to be a common feature of RRab and RRc stars at phase ~0.55, it was not considered (or it was smoothed out) in the previous BW studies, presumably because it is a sub-oscillation in the upper stellar atmosphere which is scarcely reflected in the radial velocities. (The radial velocities give information on the motion of the deeper layers.) To derive the fundamental parameters the less accurate radial velocity observations and their problematic conversion to radius changes had not to be used at all, however, an indication of eventual variable barycentric velocity of T Sex has arisen.

Acknowledgements This research has made use of SIMBAD database operated at CDS, Strasbourg, France and ADS of NASA. The author is grateful to J. M. Benkö for comments on the text.

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