The dynamically zoned pulsation code developed by Castor, Davis, and Davison was used to recalculate the Goddard model and to calculate three other Cepheid models with the same period (9.8 days). This family of models shows how the bumps and other features of the light and velocity curves change as the mass is varied at constant period. The use of a code that is capable of producing reliable light curves demonstrates that the light and velocity curves for 9.8 day Cepheid models with standard homogeneous compositions do not show bumps like those that are observed unless the mass is significantly lower than the 'evolutionary mass.' The light and velocity curves for the Goddard model presented here are similar to those computed independently by Fischel, Sparks, and Karp. They should be useful as standards for future investigators

Adams, T. F.

Castor, J. I.

Davis, C. G.

English

NASA Technical Reports Server

LIGHT CURVES FOR "BUMP CEPHEIDS" COMPUTED WITH
A DYNAMICALLY ZONED PULSATION CODE
T. F. Adams
GroupJ-15,MS 420
Universityof California,Los AlamosScientificLaboratory
LosAlamos,NM 87545
J. I. Castor
JILA, University of Colorado
Boulder,CO 80309
C. G. Davis
Group J-15, MS 420
University of California, Los Alamos Scientific Laboratory
Los Alamos, NM 87545
ABSTRACT
The dynamically zoned pulsation code developed by Castor, Davis,
and Davison has been used to recalculate the Goddard model and to cal-
culate three other Cepheid models with the sameperlod (9.8 days).
This family of models shows how the bumps and other features of the
light and velocity curves change as the mass is varied at constant
period. This study, with a code that is capable of producing reliable
light curves, shows again that the light and velocity curves
for 9.8-day Cepheid models with standard homogeneous compositions do
not show bumps llke those that are observed unless the mass is signif-
icantly lower than the "evolutionary mass." The light and velocity
curves for the Goddard model presented here are similar to those com-
puted independently by Fischel, Sparks, and Karp. They should be
useful as standards for future investigators.
Also Los Alamos Scientific Laboratory Consultant.
175
https://ntrs.nasa.gov/search.jsp?R=19800016746 2020-03-21T18:35:28+00:00Z
I. INTRODUCTION
Cephelds with periods around ten days are known to show secon-
dary bumps in their light curves around the time of maximum light
(Cox 1974). Until recently, however, theoretical light curves for
Cephelds have been unsatisfactory because of contamination by spurious
"zoning bumps" (Keller and Mutschlecner 1971). The new dynamically
zoned stellar pulsation code developed by Castor, Davis, and Davison
(1977) is capable of producing reliable light curves without these
artifacts. We have used a somewhat improved version of this code to
recalculate the Goddard model and to calculate three other 9.8-day
Cepheid models. We will discuss here the systematic trends that
appear in the light and velocity curves for this family of models.
II. THE MODELS
The family of models represents a one-parameter sequence with
mass as the independent variable, subject to the constraints that:
l) the period is fixed at 9.8 days, and 2) the models lie along a
llne in the HR Diagram near the center of and roughly parallel to the
instability strip. The characteristics of the models are given in
Table I. The actual masses of the models range from 62 to 99 percent
of their respective "evolutionary masses" (corresponding to their
luminosities with a standard mass-luminosity relation; see Table i).
The light and velocity curves for the four models are shown in Figures
1 and 2.
176
TABLE i
MODEL PARAMETERS
Mass(Mo) 4.O 5.0 6.O 7.4
Luminosity (Le) 3187. 3849. 4421. 5210.
Effective Temperature (K) 5700 5690 5687 5682
Evolutionary Mass (Me) 6.46 6.82 7.10 7.44
Zones in Model (Opt. Thin) 65(12) 72(15) 73(12) 74(10)
Artificial Viscosity Coeff._ 1.0 2.0 2.0 2.0
Fractional Core Radius 0.12 0.12 0.13 O.11
Period (days) 9.79 9.78 9.79 9.78
Peak KE (1042 ergs, Expanding) 1.78 3.30 5.30 9.70
Periods Calculated_ 46 52 37 51
Time Steps per Period 1670 i170 1160 1230
Periods per Minute (CDC 7600) 0.54 0.67 0.64 0.67
Comments Goddard Evolutionary
Model Mass Model
Log [M(evolutionary)/M]__ = 0.287 log (L/Le) - 0.195.
tStellingwerf (1975) cutoff at 0.i.
_Includes periods of amplification.
177
-3.2 ' ' J
o•0 0.5 1.0 I 5 2.0. 2.5 3.0
PHASE (TIME/9 .8 DAYS )
Fig. i. Bolometric light curves for four 9.8-day Cephelds. Zero phase
is at maximum light, except for the 4.0 Me (Goddard) model,
where zero phase was shifted 0.2 units earlier for easier com-
parison. The model parameters are given in Table i.
178
0" I I I I i
20. 7.4 Mo
O,
_ 20 . 6 .0 M8
Z O.
v
20. S.O M8
O.
0
w 20. 4.0 M_
> _ U
O,
-20.
-40. ' " ' ' '
0.0 O.S 1.0 l.S 2.0 2.S 3.0
PHASE(TIME/9 8 DAYS)
Fig. 2. Velocity curves for four 9.8-day Cepheids. The velocities
(positive for radial expansion) are interpolated values that
show the motion of the matter at optical depth two-thlrds. The
phase convention is as in Fig. i. The model parameters are
given in Table i.
179
The original version of the code, used by Castor, Davis, and
Davison (in Deupree 1976; see also Davis and Davison 1978) to calcu-
late their Goddard model, incorporated an exponential artificial
viscosity cutoff. The principal change in the code since then has
been to go to the standard Stellingwerf (1975) cutoff. In addition,
the value of the cutoff has been increased from one to ten per cent
of the isothermal sound speed. These changes account for the higher
limiting amplitude and the slightly different appearance of the light
and velocity curves for the present Goddard model compared with the
earlier one (their model II).
The light and velocity curves for the Goddard model in Figures i
and 2 should be useful as reference light and velocity curves for the
Goddard model. They are quite similar to those presented by Fischel,
Sparks, and Karp (in Deupree 1976). It is comforting that there is
now reasonable agreement between the light and velocity curves calcu-
lated by (at least two) independent investigators for the same model..
Ill. DISCUSSION OF THE LIGHT AND VELOCITY CURVES
Let us first discuss the features in the light andvelocity curves
for the Goddard model. Starting at minimum llght, we see the phase of
rapidly rising light, interrupted by the so-called "artificial viscos-
ity dip." The dip is associated with the compression wavethat stops
the infall of the envelope. The artificial viscosity plays a role in
that it spreads out the compression wave in the mesh. However, the dip
itself may not be entirely artificial, since the compression wave un-
doubtedly raises the effective gravity briefly and can thus cause the
light output to drop.
/
180
In this model, peak light is preceded by a shoulder in the light
curve. There is a broad maximum in the velocity curve that begins at
the phase of the shoulder and lasts through the light maximum. The
light curve near maximum seems to be governed by changes in the effec-
tive gravity at optical depth two-thirds, as there is an increase in
the gravity between the shoulder and the peak and a gravity minimum at
the peak itself. However, the physical origin of the shoulder is not
yet clear.
After maximum light, there is a dip and then the "Christy bump,"
which is associated with the reflection of a compression wave off the
core. The bump in the velocity curve actually agrees in phase with
the dip preceding the bump in the light curve. Therefore, in agreement
with previous investigators, we identify the feature in the light curve
as a "Christy dip," due to the compression that occurs when the re-
flected wave reaches the surface. Although the light and velocity
curves are similar, they differ in detail. This raises some questions
about the practice of comparing observed light curves with calculated
velocity curves.
We now"discusshow thesefeatureschangeas themass is increased.
In general,the Christybump or dipmovesdownthe descendingbranch.
At the sametime,it becomeslesspro_nent,especiallyin the veloc-
ity curve. In fact, there is no bump in the evolutionary mass (7.4 MO)
model. This confirms the conclusion that, with a standard homogeneous
composition, the mass must be less than the evolutionary mass to show
bumps llke those that are observed.
181
Another interesting change that occurs as the mass is increased
involves the light curve near maximum. In the 5.0 M model, what was
o
the shoulder in the Goddard model has become the peak, while the peak
in the Goddard model has evolved into a bump following maximum light.
This bump then moves down the descending branch and becomes less
prominent as the mass is further increased.
The ascending branch also changes with increasing mass. It be-
comes steeper and the "artificial viscosity dip" moves up the ascend-
ing branch until it nearly doincides with the peak in the evolutionary
mass model. The compression wave associated with the dip increases
greatly in strength as this happens. It should be emphasized, though,
that this phenomenon is probably amplitude-dependent, and that the limit-
ing amplitude, which also increases with mass (cf. Table i), may depend
on subtle physical or numerical effects in the calculation.
Another interesting phenomenon occurred in the 4.0 and 5.0 M models,®
although it is not apparent in the light or velocity curves. In these
models_ at the phase of rapid expansion, when the hydrogen ionization
front is racing inward, a shock forms on the neutral side following the
front. This can be understood as a transition in the nature of the
ionization front from D to R type (Kahn 1954, Castor 1966). Conditions
change too rapidly for a steady-state R-type front to develop, and the
front soon returns to its usual D-type behavior. This transition occurs
about the time of the Christy dip in the Goddard model, and about the
time of the dip just after maximum light in the 5.0 M model. The front®
is near optical depth unity at this time, so there should not be any
major changes in the emergent spectrum or energy distribution. However,
182
the following shock may be connected with the weak redshifted Ca II
emission features that appear Just after maximum light in Beta Doradus
(Gratton 1953). The ionization front transition will be discussed in
detail elsewhere (Adams and Castor, in preparation).
IV. CONCLUSIONS AND FUTURE PLANS
We have calculated a family of 9.8-day Cepheid models with a code
that can produce reliable theoretical light curves. We find that only
if the mass is significantly less than the "evolutionary mass" do the
bumps in the light and velocity curves occur near maximum light, where
they are observed to occur in real ten-dayCepheids (Cox 1974). How-
ever, only models with homogeneous population I compositions were cal-
culated. It thus appears that we require either reduced masses, inhomo-
geneous compositions, or some change in the physics in order to match
the observed light curves of ten-day Cepheids.
We plan to continue this program by calculating families of models
at other periods to see how the light and velocity curves vary with
period. Our goals will be to learn more about the mechanisms that pro-
duce bumps in the light and velocity curves and, in general, to see if
the Hertzsprung sequence (variation of bump location with period) can
be reproduced theoretically. Wewould also like to calculate some in-
homogeneous models to study Art Cox's proposed solution to the "mass
anomaly."
Finally, we conclude that there is nowa consensus standard Goddard
model that can be used as a benchmark for any new pulsation code. It
is entirely fitting that this consensus should be reported here at the
183
Goddard Space Flight Center only two conferences after the idea of a
standard Goddard model was originated.
ACKNOWLEDGEMENTS
This work was supported under the Los Alamos Scientific Laboratory
New Research Incentives Program. The authors would like to thank
S. S. Bunker, A. N. Cox, J. P. Cox, D. K. Davison, R. G. Deupree,
C. F. Keller, and D. S. King for their helpful comments and suggestions.
REFERENCES
Castor, J. I. 1966, "Atmospheric Dynamics in a Model RR Lyrae Star,"
Ph.D. Thesis, California Institute of Technology, Pasadena.
Castor, J. I., Davis, C. G., and Davison, D. K. 1977, "Dynamical Zoning
Within a Lagrangian Mesh by Use of DYN, a Stellar Pulsation Code,"
Los Alamos Scientific Laboratory Report LA-6664.
Cox, J. P. 1974, Rept. Progr. Phys., 37, 563.
Davis, C. G., and Davison, D. K. 1978, Ap. J., 221, 929.
Deupree, R. G. 1976, in "Proceedings of the Solar and Stellar Pulsation
Conference," A. N. Cox and R. G. Deupree, eds., Los Alamos
Scientific Laboratory Conference Proceeding LA-6544-C, pp.243-270.
Gratton, L. 1953, Ap. J., 118, 570.
Kahn, F. D. 1954, Bull. Astr. Soc. Netherlands, 22, 187.
Keller, C. F., and Mutschlecner, J. P. 1971, Ap. J., 167, 127.
Stellingwerf,R. F. 1975, Ap. J., 195, 441.
Discussion
J. Wood: Do you have a temperature range for this 8 Dor Goddard model? From
the G-band, I found about 1500 ° over the cycle, but Rogers and Bell found
about 450 ° from Ha profiles. Sidney Parsons found about 900 ° from UBV work.
Adams: First of all, I didn't make any attempt to match a particular star.
This was simply a theoretical sequence, so the parameters may not be correct
for a particular star. The temperature data are available, and could be used
to plot loops in the color-color diagram. However, there's a larger question
that has to do with limiting amplitudes. The light amplitude shown here was
rather large for a Cepheid; therefore, any behavior that depends on the
limiting amplitude is questionable. But I don't know whether the temperature
range depends on the limiting amplitude.
Wesselink: Have you made any comparison between your light curves and observed
light curVes?
Adams: l've just taken a cursory look at the observed light curves. I can't
claim that I can match any observed light curve. The most important conclusion
is that wecan calculate good light curves, and they still show the traditional
mass anomaly with the bumps. Let me add thateven the features seen in the
superb light curves produced by Pel and Lub still leave a little freedom for
the theoretician to imagine various things happening. It's an extremely diffi-
cult problem to get sufficient data.
Connolly: What is causing that sharp dip at rising light and is it a dip,
not actually a bump?
185
Adams: It is a dip. In the theoretical calculations it is due to the fact
that when the envelope is falling in, it is stopped by a compression wave or
shock wave. The intense compression for a short period of time steals some
luminosity and puts it into ionization or compression. Therefore there is a
sharp drop in the luminosity as the shock passes through the photosphere at
T = 2/3.
Wesselink: I know one observed light curve -- VZ Her -- which resembles one
of yours. It has a dip on the rising light.
Adams: That is interesting. However, I would not go so far as to say that
the dip has to exist theoretically, even though it shows up in the models. I
think that it is difficult to find any observer who has a well-documented dip
in the rising light because that's a veryhard thing to see. It looks a lot
llke a cloud going by, lasting only a very short time.
Pel: But they exist for longer period stars, with 15-30 day periods. There
is a "standstill."
Adams: That feature may be a bump, related to the bumps that define the
Hertzsprung progression.
186


Light curves for bump Cepheids computed with a dynamically zoned pulsation code

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server