Christy, R. F.

English

Caltech Authors - Main

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A MODEL OF W VIRGINIS WITH RV TAURI CHARACTERISTICS 
Non-linear calculation methods (Christy 1964) have been applied to a model intended 
to represent W Virginis. The original objective was to find if a model with mass near 
1.0 M 0 could show the long period of W Virginis. The radius, effective temperature, and 
luminosity of the static model were chosen with reference to the study of Abt (1954), but 
T e and L0 were adjusted to arrive at instability with approximately the right period. 
The chosen model had M = 1.75 X 1033 gm (0.88 M 0), Lo = 7.0 X 1036 ergs/sec (M b = 
-3.4), Ro = 3.2 X 1012 em, and Te = 5500° K. The composition was given by Y = 0.45 
and Z = 0.002. This led to a period of 18.5 days and a surface gravity of 11.5. In addition 
to this model, a few nearby models with varied T e were examined for instability. 
Although the model was violently unstable at 5500° K, increasing T. to 6000° K with 
the same L 0 gave a model that was stable in the fundamental and in the first harmonic. 
This model, however, showed an increasing amplitude (instability) in the third harmonic 
(the only case I have found). The zone structure with thirty-eight zones is not reliable 
for such a short wavelength, and no attempt was made to study this instability with an 
adequate structure since it would involve at least twice as many zones. Models with 
other masses than 0.88 M 0 were not investigated with the high luminosity needed for a 
W Virginis model. Thus the results obtained here in no way indicate that the chosen mass 
is correct. It is merely a possible value. 
The model chosen to represent W Virginis with T. = 5500° K showed an exceedingly 
rapid amplitude growth, increasing from half to full amplitude in about three periods 
(see Fig. 1, a). This was the most rapid growth (and most violent instability) of any 
models so far studied; in general, however, rapid growth characterizes unstable models of 
large luminosity-to-mass ratio because of their abnormally low-density envelopes. An-
other peculiar feature of this model was the very strong relative driving of the hydrogen-
ionization zone. The work developed in the hydrogen zone per period was 4.4 per cent of 
the total radiated flux, whereas it was only 2.8 per cent in the helium+ ionization zone. 
When the motion was fully developed, the pulsation showed a very large amplitude with 
some irregularity which then settled down to a fairly regular modulation of the funda-
mental in which alternate periods had larger and smaller amplitude (Fig. 2). 
The period found (about 18.5 days) was remarkably long. The period quoted here is 
the fundamental period of the small-amplitude oscillation which is the meaningful quan-
tity in pulsation theory. It is not the repetition period of the motion which was twice as 
long. The pulsation constant Q = Py'(p/ p0) = 0.0560 compared to values near 0.038 
in most of my RR Lyrae models (Christy 1966). Calculations by Schwarzschild (1941) 
indicate that the ratio of first harmonic to fundamental period is reduced to near 0.6 for 
such a large value of Q, whereas the ratio is near 0.75 in the RR Lyrae models. This ratio 
is probably significant in the modulation that would be excited by a combination of first-
harmonic and fundamental excitation. A ratio of 0.66, for example, would lead to a 
modulation every second fundamental period, as found in the model, if a mode combining 
the first harmonic and fundamental were excited. 
It is known observationally (Arp 1955) that the longer-period population II Cepheids 
demonstrate irregularity in oscillation similar to that of RV Tauri variables. The fact 
that this irregularity, in the form of an alternation of maxima of successive cycles, ap-
peared in this model suggests that the model is a qualitatively correct representation of 
337 
© American Astronomical Society • Provided by the NASA Astrophysics Data System 
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E 
0 
C\1 
-0 
(f) 
:::> 
0 
<{ 
0:: 
37 
36 
34 
20·1=7~------~1~8--------~1~9~----~~2~0--------~2~1--------~22~------~23 
TIME (106 sec) 
FIG. 1.-(a) Initial growth of the motion in the calculated model, shown by the radius of mass zone 34; 
(b) ultimate behavior of the radii of the outer mass zones, showing the ejection of zone 38. 
l:~ 
~ 0~----~----~----~----~----~~--~~----~----~--~ 
'Q · 8 9 10 II 12 13 14 15 
~ TIME (106 sec) 
§ 16[ 0 12 
z 
~ 8 
::;:) 
..J . 
4 
QL-------------~----~----~178-----=----~2=0~--~----~22~ 
TIME (106 sec) 
FIG. 2.-Final behavior of the luminosity, showing the alternation of successive periods 
© American Astronomical Society • Provided by the NASA Astrophysics Data System 
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NOTES 339 
the real population II Cepheids. Since the principal peculiarity of the model was its very 
large luminosity-to-mass ratio, I conclude that it is probably this property (a very low 
value of g) that distinguishes the population II Cepheids and also the RV Tauri vari-
ables. The small metal content of the model is probably unimportant in its direct effect 
on the pulsation but is of prime importance in determining the evolutionary history of 
the model and thereby its large L/M. It would be necessary to study a series of models of 
varying Land Mas was done in the study of RR Lyrae pulsation (Christy 1966) in order 
to refine the conclusions about the luminosity and mass of these variables. 
(f) 
::::> 
0 
<t 
a:: 
w 
a:: 
w 
:r: 
a. 
(f) 
0 
1-
0 
:r: 
a. 
20 L18 ______ _L19------~20--------2L1-------2~2~----~2·3 
II 
f.. -40 
~ 
~ -20 
0 
Q) 
"' 
' 0 E 
>- 20 
1-
g 40 
..J 
w 
> 18 19 
TIME (106 sec) 
20 21 
TIME (106 sec) 
22 23 
FIG. 3.-Time behavior of the photosphere radius and of the velocity, showing the alternation of 
successive periods. 
The pulsation was particularly interesting to investigate in detail because of its re-
markable violence. At full amplitude, the photosphere varied in depth from 60 gm/cm2 
at maximum light to 1000 gm/cm2 at minimum light. The material that was in the photo-
sphere near minimum light was heated to 45000° Kat maximum compression. The shock 
in the photosphere was very strong, with a shock velocity of about 19 km/sec and a com-
pression ratio of about 16. However, these results are not reliable since the optically thin 
regions are allowed to radiate strongly in the diffusion treatment used. This resulted in an 
almost isothermal shock in the photosphere. 
Because of the very great variation in depth of the photosphere, the behavior in time 
of a fixed mass layer is significantly different from that of a fixed optical depth. The time 
© American Astronomical Society • Provided by the NASA Astrophysics Data System 


A Model Of W Virginis With Rv Tauri Characteristics : Notes

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A Model Of W Virginis With Rv Tauri Characteristics : Notes

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