A hydrodynamically pulsating 0.6 solar mass model of a typical RR Lyrae variable was studied with a radiation transport-hydrodynamic computer program to predict theoretical T sub 3 and colors at many phases and to find the proper methods for getting mean colors and the consequent mean effective temperatures. The variable Eddington radiation approximation method was used with gray and with multifrequency absorption coefficients to represent the radiation flow in the outer optically thin layers. Comparison between observed and computed B-V colors indicate that these low Z population 2 models are reasonably accurate using King 1A composition opacities. The well known Oke, Giver, and Searle relation between B-V and T sub e reproduced. Mean colors were found by four different averaging methods. The method that gives a mean color and the mean T sub e closest to the nonpulsating model was the separate intensity means of B and V

Cox, A. N.

Davis, C. G.

English

NASA Technical Reports Server

THEORETICALMEAN COLORSFOK RE LYRAEVARIABLES
C. G. DavisandA. N. Cox
Los AlamosScientificLaboratory
Universityof California
ABSTRACT
A hydrodynamicallypulsating0.6 M@ modelof a typicalRR Lyrae
variablehas been studiedwith a radiationtransport-hydrodynamic
computerprogramto predicttheoreticalTe and colorsat many phases
and to find thepropermethodsforgettingmean colorsand the con-
sequentmean effectivetemperatures.The variableEddinEtonradiation
approximationmethodwas usedwith gray andwith multifrequency
absorptioncoefficientsto representthe radiationflowin the outer
opticallythinlayers. Comparisonbetweenobservedand computedB-V
colorsindicatethat theselow Z PopulationII modelsare reasonably
accurateusingKing IA compositionopacities.The well knownOke,
Giver,and SearlerelationbetweenB-V and Te is reproduced.Mean
colorsare foundby fourdifferentaveragingmethods. Themethod
that givesa mean colorand themean Te closestto thenonpulsating
modelwas the separateintensitymeansof B and V, Just as the case
forpreviousstudiesof classicalCepheids. Thebest mean for Te,
whichis knownforall pulsationphasesfrom fourcolorobservations
of realRE Lyraevariablesor fromthecalculatedmodel,is a time
averageofT e withoutanyweightingfunction.
293
https://ntrs.nasa.gov/search.jsp?R=19800016752 2020-03-21T18:35:42+00:00Z
I. L_TRDDUCTION
The problem of obtaining the mean color of PallLyrae scars, in
order to get the correct non-pulsating temperature, is similar to that
for Cepheids (Cox and Davis 1975). In our nonlinear dynamic models we
know the original nonpulsating temperature, and by using a multi-
frequency snapshot approach w%th the appropriate filter responses,
in this case for the UBV filters, we can study various ways to obtain
mean colors. "'Inthis paper we study a model using nonlinear gray and
multifrequency hydrodynamic transport calculations and Population II
(Y- .299, Z = .001) King IA mixture opacities. The discussion concerns
the taking of color averages as well as the question, if we know the
effective temperatureat many phases, how best to obtain an average
for the non-pulsating temperature of the model.
II. METHOD
.Theradiationflow for our RR Lyrae model istreatedby a non-
equilibrium diffusion approximation where the radiation field is not
directlycoupledto the materialenergyfield as in the equilibrium
diffusionapproximation.The methodlimitsto equilibriumgray dif-
fusionin opticallythickzones and to streamingin opticallythin
zones. The forwardpeakingof the radiationfieldis correctlydes-
cribedusing variableEddlngtonfactors. In our zodel we use a plane
geometrycharacteristicray calculationfor the Eddlngtonfactorsat
each timestep and for each frequencygroup. The multlfrequencycal-
culationis carriedout for 13 frequencygroupsselectedso that
ionizationedges and the Planck functionemissionare well resolved
(Davis1971). Some attempthas been made to includeeffectsdue to
llne radiationas perceivedto be importantby }_halas(1969). This
effectis includedusing a formulationproposedby Cassinellland
describedin Davis (1978). 294
Effectson colorsdue to shockwavestransitingthe atmosphere
are approximatedusinga numberof opticallythinzonesoutsidethe
photosphereand the Richtmeyer-VonNeumsnmethodof pseudovisoosity.
The phaseof shocktransitingoccursbetween0.4 and 0.5,where
phase 0.5 is approximatelythephaseof peak luminosity.Thereis
some evidencethata UV excessoccursduringthisphaseto affect
the continuumcolors(Davis1975). Conditionsforhydrogenllne
emissionand Ca linedoublingdo existduringthisphase (Hill1972).
111. MODEL
The selectedmassis 0.6 M@, and luminosity(logL/LG - 1.6) is
equivalentto an _oi " 0.72where_oi% " 4.72. This is in reasonable
agreementwith Oke,Giver,end Searle(1962)estimatefor SU Dra. A
Telf - 6840K (logTe - 3.835)thengivesa fundamentalperiodof 0.44
days. The innerradiusof the modelis less than10% of thephotospheric
radius. No convectionis allowedas appropriatefor thisTe value
(Deupree1977). A zero pressurehydrodynamicboundaryconditionis
applied.
In our structurecalculationswe have foundthat72 zoneswith
5-10 zonesin the opticallythin atmosphereare sufficientto resolve
the luminositycurve. Somenoisestillremainsnear lightminimum
when the ionizationfronthas approachedto 1 or 2 zonesfromthe
star'ssurface. The shockescapingduringthe phasenear lightminimum
throughlightmaximumresultsin the observationof H7 lineemission.
In our modelswe have not resolvedthe detailedatmosphericstructure
duringthisphase (Hill1972). The effectsthereforeof llneemissions
295
during the phase of light minimum to rising light are not treated
exactly but the effects on the colors are expected to be small.
Spectra using the calculated structures at many phases and 30
different frequency groups are convolved with the B and V filters.
Raw colors b, v are corrected for their relative transparency by the
formula
B - V= b - v+ 0.65
as for the classical Cephelds consideredpreviouslyby Cox and Davis
(1975).
Figure 1 shows the calculated_oi for the model where the vari-
ation is > 1.8m. This is larger than the usual observed range of V
for RR Lyrae stars (~ lm). Figure 2 shows the variations of Teff
versus phase which go from approximately5600 to 8700 K. In Figs. 3
and 4 we show the radius and velocity variations as calculated. The
variation in radius is llke 10-15% around a value of 4.5 R@. The
observed range in velocity is 62 km/s. Thelower line is the velocity
calculatedat I = 4404 _ at an optical depth of 0.10 (the location
of the metal lines).
IV. RESULTS
The transformationformula for the conversion from B-V colors
to Teff, as derived by Oke, Giver and Searle (1962) for SU Dra with
the assumptionthat the star is unreddenedis @ = 0.62 + 0.51 (B-V)
where Telf = 5040/@. We use this formula for our comparisons. In
Figs. 5 and 6 we plot the OGS relationshipagainst the calculation
for the gray and multifrequencystructures,respectively. There is
296
PERIOD= 0,d44 R.1GREY¢0|.16
-.4 I I I I I
10 _
m
.4-- m
m • m
.8
o
1.2
i
1.6 m
I I I !o Io.o .5 1,o 1.s 2 2._ 3,0
PHASE
Flg. i. Calculated bolometrlc magnitude for three periods (P = 0d44).
Note wiggles near light minimum (see text).
297
PERIOD= 0d44 Rf,1(;R_'Yco,.T6
4.0 ....
I I I ! I I I i I I i i i i
.3,8 n
m m
I111111,11!1111
3'70 1 2 3
PHASE
Fig. 2. Log Teff versus phase with variations in Teff calculated as
5600 - 8700 K. ,
298
PERIOD= 0,d44 _iO_EYCOlTG
m i
I I I i I
1= PHOTOSPHERICADIUS
,38-- 2 = LINERADIUSAT4404A --
3= SPECTERPHOTOSPHERICADIUS
-- ATOPTICALDEPTH= 0.10--
.22_ --! !, r ! (
0,0 .5 1.0 1,5 2.0 2,5 3.0
PHASE
Fig., 3. Calculated phbtospheric radius (Req Z 4.5 Re).
299
PERIOD= 0d.44 RR,M_(ZI_coz.-T6
6o I I I I I
40_ ----
^ 20--
U
W
:K
0, "
0
L_
> -20
1= PHOTOSPHERICVELOCITY
-40'-- 2 = LINEVELOCITYAT4404A
ATOPTICALDEPT(I= 0,10
( I I (
-6_,0 ,5 1:0 1,5 2.0 2,5 3,0
PHASE
o
Fig. 4. Calculated velocity at T = 2/3. 2 = line velocity at 4404 A,
T = 0.i0.
300
RRI GREY COt. T6
,6 I I I I I I I 1 I I I I I I
! I I I !,1 ! I I _ !'I I I
oo.7 3.8 3:9 : 4.0
LOG(T_F,F)
Fig. 5. B-V versus log Teff calculated using gray transport structure.
Solid llne shows Oke, Giver, and Searle relation. Calibration
values (0) for static models at L = 38 L and Teff = 6800, 6500
and 6350 K are plotted. Squares (D) areecalibration values with
convection. Dashed line is fitted to Kurutcz's latest results
(unpublished).
301
RRIN_(Zl)COI. T6
116 nIIIII'.IIIIIILII
12 .I •
LOG(TuF)
Fig. 6. B-V versus log Tef f calculated uslngMF/O transport structure. Solid
llne shows the Oke, Giver, and Searle relation. Calibration values
(0) for statlc models at L = 38 L® and Tef f = 6800, 6500, and 6350 K
are plotted. 302
an improvementin slope forthemultifrequencystructureas anti-
cipatedby CoxandDavis(1975).
The various averages of B-V that we obtained are shown in Table I.
Averagesoverthethreeperiods,fortheintensitymeans<B>int - <V>int,
forthegraystructurewhichis essentiallythesameforthemulti-
frequencystructureis 0.264.TheaverageTeff calculatedfromtheOGS
relationis 6678K within162K of the_odelTeff (6840K). Theother
meansareat least600K low. Thestaticfinezonedradiativemodel
givesa(B-V)of .244andan OGSTef of 6770. In Fig.6 we alsoplot
calibrationpointsforstaticmodelswithL = 38 L@,and6800,6500
and6350K effectivetemperatures.
Directaveragesof Teff wereobtainedusingI00phasesover
threeperiodsfora morereasonableamplitudevariationof RE Lyrae
•(N1.1_ andforourKingIAlargeamplitudemodel(1.8m). Teff is
determinedfromtherelationship:
L = 4_R2 Teff4
whereL is themultifrequencytransportcalculatedluminosityandR,
thephotosphericradiusdeterminedat T - 2/3. Theweightingsused
werenone,directweishtingon L anda weightingsimilarto thatused
by Lub (1977),i.e.,
eq =
8eff [(L/<L>)I/2^ _1/2. _effj •
The resultsforthe1.1m model(Fig.7),averagedoverthreeperiods
are: Teff " 6800,7115,and6670K, respectively.Onlythedirect
luminosityweightingiswelloutofline,beinghighby 300K, but
303
PERIOD=0,d44
-1.0 I I 1 1 1
w
f_
_0,0
e_
z
o
u
mm_m
=1.0
2,= I I 1 I
0 1,0 2.0 3,0
PHASE
4,0 I 1 1 1 I 1 I 1 l I 1 I I I
3,9
%
oD
5,8
m
-llltllIiI011111 -
3'0 i 2 3-
P_$E
Fig. 7. Bolometrlc _itude and log Tef f for an "ob-
served amplitude" _ Lyrae used to obtain
averagesof Teffover phase. (Thismodelused
opacities that we believe are too high).
304
TABLE1
B-V Teff(K) Teff(K)
<B-V> .3907 6151 -689
mag
<B>int - <V>int .2640 6678 -162
<B-V>int .4018 6108 -732
i
-<V-B>int .4532 5925 -915
using the Oke, Giver, and Searle relation
8 = 0.62 + 0.51(B-V) where Teff = 5040/8e e
305
theunweightedmeanis thebest. ForourlargeramplitudeKingIA
modelthedirectTef averageis 500K lowandtheLubaveragewas
not attempted.
V. CONCLUSIONS
The slopeof our (B-V),log Te relationis closeto that given
by Oke, Giver,and Searleif one usesa full transportsolutionfor
the atmosphericstructure.It appearsthatan intensitymean on B
and V is the most appropriatemean to use for RR Lyraestarsas for
Cepheids. From <B>int - <V>int we obtaina calculatedTelf within
about160K of the knownnonpulsatingTeff. A modelcalculatedthat
agreesin amplitudevariationswith RR Lyraeimpliesthata directtime
averageof.Te is preferableto any otherweighting.
306
REFERENCES
Cox, A. N. and Davis,C. G. 1975,MamoiresSociete,Royaledes Sciences
de Liege,6e SerieTomeVIII,p.209.
Davis,C. G. 1971,J.S.Q.R.T.ii, 647.
Davis,C. G. 1975,Proceedingsof the IAU Colloquium#29,Budapest.
Davis,C. G. 1978,Ap. J. 221,929.
Deupree,R. G. 1977,Ap. J. 211, 509.
Hill, S. 1972,Ap. J. 178,793.
Lub, J. 1977,Thesis,Universityof Leiden.
Mihalas,D. 1969,Ap. J. 157, 1363.
Oke, J. B., Giver,L. P., and Searle,L. 1962,Ap. J. 136,393.
307
I
Discussion
Baker: Can you say anything in a general way about how the Cox-Davis opacities
compare to the Cox-Stewart opacities?
Davis: They seem to about double the opacities in the region below 5000°K,
and we're not ready to say what the reasons are. But it appears to be a
difference in the treatment of the line wings from that used in the original
Cox-Stewart opacities. The Cox-Davis opacities are a great improvement be-
cause molecules are included, so that the line blanketing treatment for Cepheids
is improved. We found good agreement, where we didn't without the Cox-Davis
opacity, in Cepheids. Here for RR Lyrae stars we didn't expect it to make
much difference, and it made a big difference, so we were surprised. We're
studying the question of what really went into the Cox-Davis opacities.
Spangenberg: That opacity effect might be subject to the effect of zoning
when.you're doingthe low photoshere. Your temperature structure could be
quite a bit different if you had significantly more zones in one case than
in the other. But if you kept the number of zones the same, then you would
lose resolution in the one case, but the opacity would be adding a lot of
temperature-dependent features which would get lost in the zones. If you
changed the atmospheric zoning, you would understand these opacity effects
better.
Davis: The zoning was done in the same manner as for the Goddard Cepheid
model -- we used 72 zones with a 10% inner radius. Art has looked at the
opacities in this region and there is a difference of a factor of two. Your
point is well taken concerning the position of these effects.
308
Spangenberg: When you're trying to estimate the optical depth in order to get
Teff, it could be quite zone-dependent.
A. Cox: You're right, Bill, but unfortunately we seem to have gotten into a
glitch and we hope it will be straightened out soon.
Davis: Thenew opacities did improve the amplitude . . . I wish we could keep
those opacities.
A. Cox: Do you think your amplitudes will decrease ifyou use a non-zero
boundary pressure?
Davis: No. It Just disturbs my light curve. I did not try_the Castor
boundary condition.
A. Cox: I'd like to elicit something from Pel about criticisms we have on
how you take your temperature means.
Pel: What we did is very simple. As soon as you have the temperature and
radius variation, it is clear how you have to average. If you assume the
luminosity mean over the cycle is exactly the time average over the luminosity
curve, and the radius mean is approximately the time average over the radius
curve, then the Stefan-Boltzmann relation tells you what the temperature mean
is. Where we may come out with different results is that our definition of the mean
radius is not exactly where the equilibrium radius was. I think that's all
the play there is in the definitions, and I'm a bit surprised that there is
a difference of about 190@K for the RRLyrae stars. Did I hear that correctly?
Davis: 170@K cooler than Teff.
309
A. Cox: He took three means. One was the time average, one weighted with
the luminosity (which weights the higher temperatures more), and then your
technique.
Pel: You would prefer the intensity time average of the individual bands and
wewould prefer the average, not in the colors, but inthe temperature and
gravity curves, working back to see what that meant in the colors. That is a
little bit closer to the straight time average of the color itself.
A. Cox: I should tell the audience that he [Pel] is talking about Cepheids,
whereas Davis was talking about RRLyrae stars.
Pe___l:Yes, but this recipe works for both.
310


Theoretical mean colors for RR Lyrae variables

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