4 Models

Figure 3: H-R diagram showing the models calculated. Different symbols show chemical mixtures used in Table 3. Boundaries of the instability region for this range of masses and chemical composition are also shown.

Homologous stellar pulsations obey the period-mean density relation

$$\Pi= {Q} \sqrt{\frac{R^3}{M}}\cdot$$ (3)

$\Pi$ is the pulsation period, R and M are the stellar radius and mass and Q is a factor depending on the structure of the stellar envelope.

Since luminosity and effective temperature are related by the identity,

 

$$4~ \pi \sigma {R}^2\mbox{~\em$T_{\rm eff}$ }^4$$                    L = (4)

$$\mbox{~\em$T_{\rm eff}$ }^4 4~ \pi (\Pi/Q)^{4/3} {M}^{2/3}$$ = (5)

then,

$${\rm log} L = 4~{\rm log}\mbox{~\em$T_{\rm eff}$ }+ {a_1},$$ (6)

where, ${a_1}= {\rm log} ~~(4\pi\sigma (\Pi/{Q})^{4/3}{M}^{2/3})$.

We note that models which have the measured radii and masses of V652 Her and BX Cir do not give a period corresponding to that observed. However, period is the best observed constraint. For this reason, we have chosen to investigate families of models obeying Eq. (6), with $\Pi=0.107$d to within $\pm$0.002 days and masses in the range 0.38- $1.16~\mbox{$M_{\odot}$ }$. We have also investigated models with a range of chemical composition, including a standard mixture of 21 metals (Grevesse & Noels 1993) with Z=0.01 and Z=0.02 in both cases X=0.0 (z1 and z2 in Table 3) where X, Y, Z are mass fractions of hydrogen, helium and metals. Mixtures with enhanced carbon and/or nitrogen (fv, n1, n2, c1 and c2 in Table 3) were also considered.

The models are described in Table 4 (see also Fig. 3).

One model (11) was computed for direct comparison with that of Fadeyev & Lynas-Gray (1996). This model has the same stellar parameters as the model they consider to be in best agreement with observations at the time, i.e. ${\rm log}L/L_{\odot} = 3.03$, $M/M_{\odot} = 0.72$, ${\rm log}\mbox{~\em$T_{\rm eff}$ }= 4.37$ and mixture B13.002 (our mixture fv in Table 3). Our model has velocity and luminosity amplitudes similar to those obtained by them, showing smooth maxima in velocity and luminosity and a characteristic spike after the maximum in the luminosity curve. Although this model was already in good agreement with observations, new models were calculated specifically in an attempt to match the composition of V652 Her and either the lower mass or the radius suggested by recent observations (models 12-20).

Another five models were computed to match the composition of BX Cir and either the mass or the radius (models 21-25). Certain models were computed simply to explore further certain areas of the parameters space.

Table 4 also gives the corresponding value for the factor Q, the surface gravity, the pulsation period, $\Pi$, and the amplitudes of the radial velocity, $\Delta U_{\rm ph}$, and logL/$L_{\odot}$, $\Delta L_{\rm ph}$, at the stellar photosphere. If the model is stable against pulsations (12, 13, 21, 22), as happens for those mixtures with very low iron abundances, the latter values are omitted. If the model was not computed until it reached a limiting amplitude, its amplitudes are preceded by a < sign indicating an upper limit.

 

 

Table 3: Chemical composition adopted: relative abundances by number.

	fv	n1	n2	c1	c2	z1	z2
$\mbox{~$n_{\rm H}$ }$	0.0060	0.0115	0.0076	0.0000	0.0000	0.0000	0.0000
$\mbox{~$n_{\rm He}$ }$	0.9900	0.9862	0.9883	0.9961	0.9934	0.9976	0.9952
$\mbox{~$n_{\rm N}$ }$	0.0020	0.0015	0.0027	0.0003	0.0005	0.0002	0.0003
$\mbox{~$n_{\rm C}$ }$	0.0005	$7\times 10^{-5}$	0.0001	0.0029	0.0048	0.0006	0.0012
$\mbox{$n_{\rm Fe}$ }$(1)	4.72	2.37	4.17	2.49	4.18	5.19	10.45
$\mbox{$n_{\rm Z}$ }$	0.0040	0.0023	0.0041	0.0039	0.0066	0.0024	0.0048

$^{{\rm (1)}}$ / 10^-5.

 

 

Table 4:

Model	log  $L/L_{\odot}$	$M/M_{\odot}$	log $\mbox{~\em$T_{\rm eff}$ }$	$R/R_{\odot}$	Q	log g	$\Pi/d$	$\Delta U_{\rm ph}/~\mbox{$\mbox{km}~\mbox{s}^{-1}$ }$	$\Delta L_{\rm ph}$	Compos.(1)
1	2.879	0.42	4.380	1.592	0.0346	3.66	0.1073	127.17	0.16	z1
2	2.826	0.42	4.367	1.592	0.0345	3.66	0.1068	129.20	0.17	z1
3	2.719	0.42	4.340	1.591	0.0344	3.66	0.1065	98.02	0.17	z1
4	2.886	0.52	4.363	1.734	0.0336	3.68	0.1065	123.80	0.20	z1
5	3.142	0.70	4.400	1.965	0.0331	3.70	0.1090	118.33	0.20	z1
6	3.102	0.70	4.390	1.965	0.0331	3.70	0.1089	120.81	0.22	z1
7	3.210	0.90	4.400	2.125	0.0328	3.74	0.1072	107.73	0.24	z1
8	3.121	0.90	4.380	2.104	0.0328	3.75	0.1055	126.31	0.31	z1
9	3.089	0.90	4.370	2.123	0.0328	3.74	0.1070	130.85	0.32	z1
10	3.152	1.16	4.367	2.310	0.0327	3.78	0.1066	185.14	0.39	z1
11	3.026	0.72	4.371	1.965	0.0331	3.71	0.1074	104.75	0.22	fv
12	2.886	0.52	4.363	1.734	-	3.68	-	-	-	n1
13	3.089	0.97	4.363	2.190	-	3.74	-	-	-	n1
14	2.890	0.62	4.350	1.851	0.0334	3.70	0.1068	<82.20	0.17	n2
15	2.940	0.62	4.360	1.873	0.0333	3.69	0.1084	80.50	0.16	n2
16	2.920	0.60	4.360	1.830	0.0335	3.69	0.1070	<79.50	<0.16	n2
17	2.869	0.64	4.340	1.892	0.0333	3.69	0.1084	<69.28	<0.15	n2
18	2.950	0.64	4.360	1.894	0.0334	3.69	0.1088	<82.70	<0.17	n2
19	3.000	0.67	4.370	1.916	0.0333	3.70	0.1080	<83.67	<0.17	n2
20	3.010	0.70	4.370	1.938	0.0332	3.71	0.1071	84.69	0.17	n2
21	2.826	0.42	4.367	1.588	-	3.66	-	-	-	c1
22	3.152	1.16	4.367	2.310	-	3.77	-	-	-	c1
23	2.838	0.38	4.380	1.519	0.0351	3.65	0.1065	55.96	0.08	c2
24	2.740	0.40	4.350	1.558	0.0347	3.65	0.1065	<36.75	<0.07	c2
25	2.820	0.50	4.350	1.708	0.0338	3.67	0.1068	53.38	0.10	c2
26	3.136	0.90	4.380	2.140	0.0320	3.73	0.1056	170.68	0.27	z2

⁽¹⁾See Table 3.

4.1 Variations of velocity and luminosity curves

The object of this investigation was to study whether the amplitudes and the shape of the stellar radial velocity and luminosity variations could help to determine the stellar properties.

Because the period is the parameter most accurately determined by observation, fixing it and assuming the factor Q approximately constant allows us to fix the stellar luminosity and to compare models in a small range of temperatures and masses around the measured values.

The changes in the light and velocity curves for 0.42 and 0.90 $M_{\odot }$, as functions of temperature are shown in Fig. 4.

For 0.42 $M_{\odot }$, we have found that for the higher effective temperature, a characteristic bump appears after the luminosity maxima, which causes a flattening effect over the light curve at log $\mbox{~\em$T_{\rm eff}$ }= 4.38$. This shape is similar to the one observed for BX Cir. Since this feature is almost negligible for higher masses (see luminosity curve for 0.52 $M_{\odot }$  in Fig. 5) the hypothesis of BX Cir having mass less than (because luminosity amplitudes increase with the mass) or equal to 0.40 $M_{\odot }$ and effective temperatures around log $\mbox{~\em$T_{\rm eff}$ }= 4.38$ seems suitable. However, this feature disappears when varying the chemical composition, as is shown in Fig. 6 models 25 and 4.

For 0.90 $M_{\odot }$, both velocity and luminosity amplitudes increase when temperature decreases. In the velocity curves, maxima and minima are rounder than for lower masses, whereas luminosity curves are, in general, sharper. These effects can be observed in Fig. 5 for different stellar masses and approximately constant effective temperature, around log $\mbox{~\em$T_{\rm eff}$ }\sim~~$4.37.

Our experiment to investigate the effect of increased carbon and/or nitrogen content within the metal component was not entirely successful since in the cases with mixtures n1 and c1, closest to the measured ones, models were stable.

The reason is that the increase in carbon (or nitrogen) means a reduction in the iron-group elements (for constant total metallicity $n_{\rm Z}$). Therefore in these mixtures n1 and c1, $n_{\rm Fe}$  is approximately half the solar value. When $\mbox{$n_{\rm Z}$ }$ is increased to 0.0041 (for the nitrogen-rich mixture, n2) and to 0.0066 (for the carbon-rich mixture, c2), the iron abundance increase to $\mbox{$n_{\rm Fe}$ }\sim 4.17\times10^{-5}$ and the iron-group bump mechanism becomes effective. In these cases, models were unstable and velocity and luminosity curves were consistent with the observed ones.

The behaviour of velocity and luminosity with increasing iron abundance is illustrated in Fig. 6. The velocity curves show a faster growth and sharper maxima and minima when the iron abundance is higher (models 4, 6 and 26). These models also show a characteristic spike after the maximum in the luminosity curve.

Figure 4: Variations of the velocity and luminosity curves with effective temperature, for models with $M=0.42~\mbox{$M_{\odot}$ }$ (1, 2 and 3) and $M=0.90~\mbox{$M_{\odot}$ }$ (7, 8 and 9). Model number in Table 4 is indicated in brackets. In Figs. 4-7 the vertical axes have been shifted by an arbitrary amount.

Figure 5: Variations of the velocity and luminosity curves with stellar mass for effective temperature approximately constant (log $\mbox{~\em $T_{\rm eff}$ }\sim$ 4.37).