With the models presen ted , i t is not easy to deduce the q u a n tita tiv e e f f e c t o f using the Carson o p a c itie s on the p rop erties o f h orizon tal branch models. In ad d ition to p rop erties of models ca lcu la ted w ith d iffe r e n t input physics by o u rselv es, tab le 4.6 contains d e ta ils o f models ca lcu la ted by other authors who have used Cox-Stewart o p a c itie s . i
Model IR (taken from IR70) was ca lcu la ted using lin e a r in terp o la tio n in the Cox-Stewart (1969) o p a c itie s and energy generation rates from FCZI.
t
. . . ’IModel IR (from Rood, 1973) was ca lcu la ted w ith a m ultip oin t in terp o la tio n # procedure for the o p a c itie s , and energy generation rates a ltered by
- 6
1) a 1 2% in crease in the p-p rea ctio n r a te , 2) a fa cto r 1 0 decrease in
the W ( ) y ) F r a te , and 3) fa cto r exp (-0.138/T ^ ) decrease in the ^ r a te . Model SG (from SG76) was ca lcu la ted w ith sim ila r physics to IR'^ 5
except that lin e a r in terp o la tio n was used for the op a city . The su rface A
4
p rop erties o f SG at zero-age agree w ell with those o f IR (noting the mass d iffe r e n c e ), although the in te r io r p rop erties (e .g . L ) do not agree so w e ll.
The remaining models mentioned in tab le 4.6 were ca lcu la ted using lin e a r in terp o la tio n in the Carson op acity tables* Models Z, X and W were ca lcu la ted w ith energy generation rates from FCZII. These adopt a sm aller ' value for ^ for the O reaction than used by IR70 and SG76, ;
I
and the rate is reduced by a fa cto r 1.41 exp (-0.117/T ^ ) w .r .t . FCZI, & The reduction for the rate is 50% a t cen tral temperatures for HB s ta r s , w h ile the same reduction used in ca lcu la tin g models IR ^ and SG is 'V 70%, f
Model Z was ca lcu la ted w ith the rate restored to the FCZI valu e,
1
I
1
i
Table 4 .6 . A summary o f r e su lts for the h o rizon tal branch stru ctu re and evolu tion o f 1) a 0.62 N@ sta r witVi Carson o p a c itie s (Z, Z ', X, W), « 2) a 0.625 M© sta r w ith Cox-Stewart o p a c itie s (IR, I R '), 3) a 0.62 M#
sta r w ith Cox-Stewart o p a c itie s and sem i-convection (SG), A ll sta rs have Me = 0,475 M , Ye = 0 .3 , Ze = 0.001. 2) and 3) are adapted from Iben and Rood (1970b) and Sweigart and Gross (1976) r e s p e c tiv e ly . Physics fo r each sequence are described in the te x t.
Property
• Sequence
1 Z X W IR IR' SG
Mass (/M© ) 0.62 0.62 0.62 0.62 0.625 0.625 0.62
log Teff (ZAHB) 3.840 3.877 3.870 3.758 3.864 3,823 3.854 log L/L© (ZAHB) 1.741 1.713 1.720 1.777 1.634 1.719 1.713 log L (ZAHB) 1 .208 1.218 1.159 1.143 1.173 1.215 1.183 log g (ZAHB) 2.807 2.985 2.949 2.443 3.(92. 2.765 2.878 Mcc/Ho (ZAHB) 0.107 0.107 0 . 1 0 0 0 . 1 0 0 0 . 1 1 0 0 . 1 1 0 0.109 log Tc (ZAHB) 8.068 8.062 8.067 8.066 8.065 8.078 8,072 log ÇC (ZAHB) 4.308 4.280 4.315 4.322 4.291 4.323 4.327 tn8 yrs) 47.3 44.0 47.7 6 8 . 0 104,3 Hcc <Yc=O.S) 0.113 0.113 0 . 1 0 0 0.119 ^0.163 X,^c (Yc«0 .05) 0.231 0.163 0.235 0.5 0,504
ilo g Teff (blue loop) 0.143 0 . 1 0 1 0.230 0 . 1 2 0 0 . 2 2 1
4 l o g L/L@ (blue loop) -0.092 -0.077 -0.127 -0.04 ^4,07
excluding seHiconvective zone.
I
- -T-• -■ - — ...— ■- - •> ■■ ■ ■ -■ • . !• -,
- '
Î
The r e su lt o f in creasin g is to in crease the core lum inosity % and the cen tral d en sity and tem perature. However the to ta l lum inosity L | and, consequently, the s t e lla r radius R are decreased. The drop in the % hydrogen-shell lum inosity L,^ follow s from the in crease in the helium -core | radius w ith . Changing has no e ff e c t on the mass in the con vective core, . The in crease in L a lso leads to a decrease in the | h orizon tal branch life tim e .
Model X was constructed in the same way as Z, except that a d iffe r e n t % method o f in terp o la tin g between the op acity tab les fo r core com positions fï was used, which led to an in crea se in the core op acity (as illu s tr a t e d in |
fig u re 3 .2 ). Since the in crease is non-uniform, i s a lter ed (RS72
and ta b le 4 .2 ). In th is case the larger in crease occurs at a lower S temperature, leading to a reduction in , in agreement w ith the r e s u lts ï o f RS72. In ad d ition , the in creased opacity leads to a drop in L , | hence to an in crease in L ^ , L and R. This i s in con trad iction to the f
r e s u lts o f RS72 (see ta b le 4 .1 ). ;
Model W was constructed in the same way as X, except that the helium I p r o f ile in the hydrogen-shell was m odified by the ad d ition o f carbon in
proportion to th e ir r e la tiv e core abundances. This represents the s itu a tio n where helium and carbon are both fu lly mixed (in the p ost-h eliu m fla sh phase)-
throughout the helium core and in to the base of the hydrogen envelope, and w ill be d iscu ssed in the next se c tio n . For the p resen t,, we fin d that i t s immediate e f f e c t is to s ig n ific a n tly in crease the hydrogen-shell
lum inosity L y . L|.|^ is reduced. L and R are in creased . Combined w ith the drop in due to the in creased core o p a city , the reduced
core mass leads to v ir tu a lly no change in w .r .t . sequence Z,
With th ese r e s u lts from the v a ria tio n o f the input physics we may now atten p t to understand the e f f e c t o f using Carson o p a c itie s in stea d o f
Cox-Stewart o p a c itie s on the p rop erties o f zero-age m odels. Models Z ' i
94
the o p acity . The a lte r a tio n s to the input physics to produce IR ' and SG led to an in crease in L and R. However in th is case L was a lso in creased .
By in terp o la tin g the behaviour o f L and T^^ for models IR and IR^ to models w ith to ta l mass M - 0,62 M , we see th at a lin e in the
( ) plane connecting models constructed w ith Carson o p a c itie s , but w ith v a ria tio n s in other aspects of the input p h y sics, l i e s above or to the l e f t o f th at connecting sim ila r models constructed w ith the Cox-Stewart (1969) o p a c itie s (fig u re 4 .1 ). At log T ± 3.85 th is s h if t is approximately = 0.025 at constant lu m in osity, or L. ~ 0 . 0 2 a t constant tem perature. I t should be noted that th is r e su lt is d issim ila r to a sim ila r comparison fo r main-sequence sta rs (S to th ers, 1974a) where
^ "" 0 .0 1 . We also note that the small d ifferen ce in the lo c a tio n o f the ZAHB models is u n lik ely to create a s ig n ific a n tly new p ictu re o f the ZAHB. We noted in se c tio n 3.2 that in terp o la tin g in the op acity ta b les for
O <. 2. < 0.01 may lead to an op acity which i s too sm all. I f th is is the case, an in crease in the envelope op acity w ill lead to a s h if t to lower lu m in o sities and e f f e c tiv e temperatures (R efsdal and S ta b e ll, 1972) for a ll models constructed w ith the Carson o p a c itie s .
In terp retin g the behaviour o f the core-lu m in osity i s more com plicated. With the Cox-Stewart o p a c itie s the r e su lt o f reducing , and other
rea ction rates was to in crease , and a lso to in crea se . With the Carson o p a c itie s , reducing led to a reduction in L , but an in crease in L^ . The paradox appears to l i e w ith the Gox-Stewart m odels, as in a ll other cases o f in creasin g we fin d a decrease in L . This may be caused by the reduction in used in ca lcu la tin g IR^ . However the op p osite e f f e c t on L of sim ila r changes in the reactio n rates s t i l l requires an explanation which li e s beyond the scope o f th is present study. We may add that i t seems u n lik ely to be an op acity e f f e c t , although th is is