USE OF THESES - Open Research: Home · 2019-10-02 · theses sis/library telephone: +61 2 6125 4631...
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THESES SIS/LIBRARY TELEPHONE: +61 2 6125 4631 R.G. MENZIES LIBRARY BUILDING NO:2 FACSIMILE: +61 2 6125 4063 THE AUSTRALIAN NATIONAL UNIVERSITY EMAIL: [email protected] CANBERRA ACT 0200 AUSTRALIA USE OF THESES This copy is supplied for purposes of private study and research only. Passages from the thesis may not be copied or closely paraphrased without the written consent of the author.
Transcript of USE OF THESES - Open Research: Home · 2019-10-02 · theses sis/library telephone: +61 2 6125 4631...
THESES SIS/LIBRARY TELEPHONE: +61 2 6125 4631 R.G. MENZIES LIBRARY BUILDING NO:2 FACSIMILE: +61 2 6125 4063 THE AUSTRALIAN NATIONAL UNIVERSITY EMAIL: [email protected] CANBERRA ACT 0200 AUSTRALIA
USE OF THESES
This copy is supplied for purposes of private study and research only.
Passages from the thesis may not be copied or closely paraphrased without the
written consent of the author.
TUB MAXINIJI E:tl''flle'!T Ol! OONVIOTIO:I
IM 8Tili.I,AR A'l'ltO!IlPlil!lR:IS
Oli 'l'lilil OBSERVll:D PROPIB'l'IJ!ll 0:11'
Ill 'l' B I. L A R S P l!l 0 'l' B A •
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
'!' h e 111 i a
subm:ltte4 by
A n t o n 1 P ~ 5 F b F 1 a k 1
ot Bogosno (Poland)
to the
A'IJS'l'BAI.IAB UNIVERSITY
'
• • • • • • Mean absol:'pt1on eoeff1etent used 2.1. 1111E~an il!lbsorption eoe:ffiotsnt 2.2. ftetul abe~ol')'ltion ooeff:l.eient
• • •
• • ··' .. • • • • • • • • • •
3. Oonatruotion of models • • • • • • • • • • • • 3.1 • R.adiati VIII !!lOd'ils
3.2. Adiablltio models ''• 1\fonoehrom!i!t:l.<:) f'lwtet!ll
• • •
• • • • • •
• • • • • • • • •
• • • • • • • • • • • • • • •
• • • • • • • • • • •
• 6.2. !on11iat1on • • • • • 6.3. Damptns • • • • • •
7. R411lil'll tm • • • • • • • • 7 .1. M!odel stelllllr atl'l'll')sphcu•es
1.2. ~onoehromat1o fluxes • •
• • • •
• • • •
• •
• •
• • • • • • • •
• • • • • • • • • •
• • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
s. Oompar:hlon or observation wtth eomptttl\ltions • • •
8.1. '!'heorll'lt1aa.l gm!u.ii.ent-equivalent width relatt.on
s. 2. Obeevational delta • • • • • • • • • s. 3. 0~1114111"\\'eel grad.ient-equiTalent width Pl!llation •
• • • •
•
• • • •
• •
• 8.h. Oompar1aon. of th~~t obse:Mred t~J.nd oomnuted relations
9, Ste:tli!lll' tl'l!mpl!il~atlll:"e eeale • • • • • • • • • • 1 o. Bllllnkettns ef'foat • • • • • • • • • • • •
• • • •
• •
•
• • • • • • • • • • •
• • •
• • • •
• • • • • • • • • • • • •
• •• •• •• •• •• • • • • • • • • •
• • • • • • • • • • • • •
• • • • • • • • •
2 3 h 6 7 7 9
10 11 12
12
1ft 15 Hi 11S 16
17 18 19 19 20
2h
25 27 29
Itt e~tte of m~ ~ttemptato solve the pPObl .. of eon•
Vli'$ctlon in l!lt(lllli!lr at1110apl"un•el!l our PNillllllllt lmowle<~ct~ of its
Eiff'«<IOtl!l Oil thllll l!ltROt~l"lll Of thllB cute!' l~t;rtU'Iil Of liltliU."Iill :hi still
theorlllltioal lnvestiga Mil lllhow cl•vl;r that th1n:•e 111 IU'l tmll!tlilble
:a~ono it~ ll'ltellar atmomphere111, in wh.lch tile tNufel' of anerg;r
'1111 Pllll'tl;r dl'le to NtU.ation and 't'UJ' to conTeotion antl in whhh
tbel'lllfc:n.•e the actual ttllllperatln•e s;ra41ent hall! a V&lue 1nte:med1atll!
bl!ltWIII(ll!'l th.e two limit:lftg Vii~liUili\1 Of' the rad1!ii't1TIII IIUld iHlililb!ii'tiO
gradhlnt111, but we h§!'lfl!l no eu:t"e.ball!1a on whlch to inveeti,;i'iltlll
th411o:t"l!lthally th!!i: 1ntloenee or eonvection • th<l'l temp!lllr>atiiM
howeTI!IP the enel"J;;r d1atr1but1on in the ~Jtellar spectrua ie
a :f'uotta ef' thl!l temperatli!M 41etr11)uUon 1n the outer
la;r~i~M of 111telbr atmcu~phl!lrlilll!• 1 t o~aa be hope4 that the
analJ'ide o.f stellar illpi!IOtft ma;r le!i.d to an approx:l:mte
eval11a tto~:a of th<~ influl!lnce of eo:nvaet1.a on the etrt:u:t'IAI.t•e
of' the convective 21o~·u1t it there aM •n:r prop•;n•t11!1111 or atellar
lliP!llotra, whioh are aasibl;r llllffllllcted b7 convection and which
o~n be 81!1117 11101!1111\t:t'Mo ~hit p:l'lilli!lmlt inVOBtiga.Uon ll!hliiiWI!I
that tb111 ehllal' colo~r-s;ndiwt lli'Ml the II!Qllhl'l.lent width111
of th111 Cl!llciua line111 lll!ii't11!1f7 both the above eonditions and
theHtoH 817 hill ~I!IM for that plal']()OiiUh
properties
lllltDolllpherll!i!!, thlll 1110delu ot every Pllllir hav:lDg the saue murteolll
tel!l'l>lerat~.are a~ad the lill\De Vlll.lue of uul'faH SHVit;v, but dU'tlllriq
b7 dtsres;ard:lng convection ia on• l!lllldel altogether, and b,
taktng it at 1tlil hilhlill!lt amount, i.e. the adiabatic value,
in the convective sone in tho othel'; the colour> srad.ionts
ot 1"1\lllll st&lllllr a tmosphlllrea 111t1.11t then :u111 betwe&n th111 two
atmosphlllr>llll!l were oomputl\l4 tCill'l years mao bJ' 1. 8t~l!lll'Gn an4
M.e oo:Ual:loNtors {l), but the;r soon b•oame obaobte, wbon the
that the old ?alu~ of absor~tion coeff1e1ent tor n&gattve
hydrogen 1on uaed by B. zH>romgrllln Wfililll cnl;r 111 rough approxi ..
models hl!Ull!!d on reo4!1nt valn4!1a of abaorptioo oo111t':ttctente
. l'I'IU\l computed.
21 1\i:ea:n ab~&Ol'!i!t:l on CQE~f'f'icient llfttfjl.
··· 'l'he n.a tul:'lill conata:nte e{ veloe1t;r of Ught) ~; a( shotro:ntc
ohtu•m;•, mo( el•u:rtrcn m~un1), h(ll'lanck9 e oonetant}.
k(Boltmma:nn•a eon.5tant), 11\i(m!UMil of th• H-atom) appearing
tn the formulae for abaorot1on were taken from the critical
work ot Je~~~~~ ~. Mend and E. iobard Cohen (3}, whose
valuea d.t:rfer !illligh.tl;r :f'l'O!!I thoall! ll!i!llld b;r Stril~~~gren. In
order to achieve ag~t btltwMn the adopted vdul!llll of
neturlll.l conmtllntlll and the tibeln•vel! :Uml till ot spl!lotroeco,to
eertes of h;rdrogen the wluo of 13. 595 Volt~:~ ·mu11 a~tll!.lllllll!ld
:tor the 1oo1111ation ootenttal of h;rdrogoo; thb aeem~1 to be
confi
As in the work of D.
~~bsorotioo coeffictentl'l
ion W\11':1'111 taltl'!!n into acecnnt, the ef'ffllct of' oth!lll' c~ou:t'ces of'
abaorottoo ttetns rei?Ht:Mr<!>d tor fu:rth~?~r :l.nv~?~mttsa tl <mill. The
metG.l 111. tom1.1 Will:!'!~ conllliderllld onl;r Ill. I! Ill lllUppl;v of clectt•ool!l,
thi!!Jr 1u"e Vf;f!J' impcl'tant :tn lcMl:' temp~?~n ttu•el!l 1n which
a s~:;W!lllld legt1r1 thm
hvdr<>~ll~ particll!ll!l to
-,, ..
01:1.ly to
el'Ultl'd ( 5) to
to the
-" = Av kT
.u.= .X•/Jc T
& = 5"0/1-t) T
_.4 (/1) = J. 7'13 ·ltF'~'&~ ~-uj"'iJ,e·"--du . •
f$i!Otorl1! e roe t<> t • the
to the Pl~nek m•en oen
t
eontt'iblltion o:r gebmi.eflt::ll,-.
~~l
- ? -
Wlill'l') •ummed 1:n D and the l'e!'l'llili in:lng with "">/ f
b;r the corl'lilll1lponding integMl,
eq11U:I.stant point•;
found alg~:~b1'1i teall;v • In
(6 •
WIU'I ova.lua ted li'IY!n'oxima tel;r b;r
oontlt'ib11tlons of th!!ll further' continua
Planok me!ln a(ll) W!UI OO!f~'PUt<!.>d from 8 ,. O.l~.t
to 8 "" 0.12 in inte:rTI'Illil of o.o2 !lind :from 8 " 0.7'4 to
(3 ., 1 0 08 in inte:rT!ii:llil of' 0.011., lill:ld th111n :l.ntorpoltxt!!td f'Ol'
inhl'lllt~diah values of e. 'fb.ie :tnte:rpol111t1<m ftl!ll T;~l'il:t:':tormil!ld
b;r tilt~ of thl!! function &jD,e-'j/u, which vl!l onl;y l!ilowly
in the whole :tnte:t:'\'15!1 ot' e. of
!liMn llli::Hi!tol'jUon oo~ff1ei•nt of nt~tutnl h;,drog~n pe:r a tom Ul. 2
01!1 •
log 111 >< !ega (H) log Ill
o.us 'D.33h 0.!.9 11.319 o.ao tr.'7l2 0.19 • o •. so • 0.81 .592 o.2o • 0.51 .096 o. .h71
.107 o. '13.9111 0.8:5 .351
.026 0.53 .871 o.Sh .230 ~.91•.:5 o. 51! .758 0.85 .109
• tt57 0.55 .61!5 0.86 "· 98111 o. .769 0.56 .~5:$1 o.a7 • 8E'i6
o. • 0.57 ·'•17 o.aa .7115 0.27 .589 o.sa .303 o.&9 • 62,3 0.23 • ~>96 0.59 .us 0.90 .501 o.29 .M::I2 o.6o .013 0.91 • .319 0.30 .307 o.61 'li.'. 9513 0.92 .257 0.31 .210 0.62 .&M! 0.93 .l3h '"'• .113 o.63 .726 0.911 .012 o •. 33 .O~h o.6h .6()9 0.95 '9.889 o •. :5h n.911. 0.65 .t.93 0.96 .776 o •. 3.:i .au o. • 316 0.97 .6111! 0.36 .7ll o.67 .259 0.98 • 5:?1 o.n .608 o.68 .1111 0.99 .397 o.:se • 501! 0•69 .Q23 1.00· .27ti 0.39 .MXl 0.70 1'!'.905 1.01 .151 o.M) .295 0.71 .181 1.02 .021 O.h1 .1.89 0.12 .661 1.03 1J.901~ O.h:? .08! o.n •. 550 l.Oh .780 o.h::s 11.975 0.71! • h31 1.05 ·''' O.hh .867 0.75 •. 312 1.06 .532 o.J.5 .758 0.76 .192 1.07 ·"08 o.~t6 .61t9 o.n .on 1.08 • 2811 0.111 • 51!0 0.18 "'· ,,,, 1.09 .160 o.Ml •''30 0.79 • 1.10 .035
(H)
to ono l!ll!lllt!:'ll.l a tom li!lld unit ot el£c!(~tron prlill!l!iiU.re -2 p 111 (in d;wn. om ) liU1d, otn'l"M1atd fen• the l!!timulated em1.!i!a1l\m
was tabul~;,ted by s. \lhll.ndr!Ul<!lkhar and ll'.t!. Breen (2).
Dlllfining 1tl'! mean at,:~;ain as - J .. .ll Af/1) = .x, .:z/·d:v , and lUling equl!l t1l\m ( 1) we f'ind
- Iff"!- -"~_,e""' ...-a r/7) = 7Ty J(. 1- ..e·· .du •
• For the sutkll' of convenionc<!l ( er !ntl!l~olaUM
f'or 1ntem<!Hii&tl!! valullls) th4!1 aom;:mtaUonl!l li!UIIre oarricd
on with VllllUI!ll!l of x. refl!lrred to on(~!> ne~:a t1 ve llyd:w-ogllm
ion and then T'lilf'li!'!"red to oru11 lll!llltnl hy<ll•ogen atom b;r
faetol" _,, s-_4 I. 726 &
¢f&)= f./§tf·ltl e7 ._.e .
l!;ivinJi the I'Ul!llber of neaativl! :ton~< pN!II!ent for on~ '1\llllltN!l o.
Tl'Ull numl!!riaal evelutiolu; l:uttwel\m ). "" 912A · lilrtd
the I!HII!ll1t points ot div1mton au\1 tor tHIIUtral llydros•; ftve
equitUatlilnt pointm Wlll:ll'<ill used tu~twe~n 11 " 0 ( l "' oo) IU\.d 0
1 tlil val\ll! eorre~t~pon<li.DB to l .. 22800 A • 'fhe;y wore mude
betwc~e.n & ,. o. 50 an<l .& "' o. 72 1n 1nt~~trvalm of o. 02 111nd
in inte'!'Valt~~ of O.Oh.
in
pe~ neutral h7drogen atom
10-26 om1' c1;vm - 1
Q ' -, liiiJ'l e
0.50 o.615 o. 7h 0.51 o.665 0.75 o. " 717 \]. 0.76 0.5.3 0.772 0.77 o.sh . o. ,ll.:;Q 0.18 o.s5 0.891 o.19 0.56 0.956 o.eo 0.57 1.0211 o.s1 0,.58 1.095
<
0.82
0.59 1,170 o.e1 0,60 1.250 o.elt o.61 1.33:2 o.85 o.62 1.'•19 0.86 o.63 1.509 o.S1 o.6h 1. 6.01. 0.86 o.65 1.703 o.S9 o.66 1.806 0.90
o.67 l.91h 0.91 o.6s :c.o:n 0.92
o.69 2.1111.1 0.93 0.10 ::>.267 0.911
o.n, 2 •. 395 0.95 0.12 2 •. 5:27 0.96 o. 73 :?.666 0.97
'
a(H-) a a(H•}
2.509 0.98 8.288 2.959 0.99 8.620 3.111! 1.00 8.961 3.275 1.01 9.313 hh~ 1.02 9.670
3.61h 1.03 10.0lt1 3.79h l.Oh lO.h 2
., '''19 •"' • .,f ' 1.05 10.8111 h.l70 l, 11.215
''· 369 1,07 11.628 Ito 1.08 12.051 "·789 1.09 12.1t85 s.ou 1.10 12.933 5.239 1.11 13.389 5.1171 1.12 13.859 5.720 1.13 lit •. :Sh2
'
5.911 l.l'l 1!!.8311 6.231 1.15 15. 3hll
6.1 •. 99 1.16 15.663 6.715 1.17 16.396 7.061 1.16 16.9111! 7. 35.? 1.19 17.506 7.6;6 1.20 16.079 7.967
-1-
IU'lei 19, .. o.6, o. 7, o.s, 0.9, 1.0, ll.nd to:!' \hree g1111nta
With log g "' 3. 5 lllnd 6'. m~ 0. 6 t Oo 8, l. 0 t &. 'lu.ling thllt
reo:t proo111l to!llp<lilr"' ttU'I!I of th<lll El tt~llililll' sln•f&oeo. Thl!l
r~111ulh of eomputdtonl!! !l'l'l."l:lngea isa the torr~~ of tmbl~~B.
g1 VI!! the Yllll ues of (3 t p and p0 for a oorta1n ntabllil' Of
euitably ol"'ol!len V!i!lues of t.he 1114!\IUl. optical depth.
'1'\VO modob tor e. ., o. 6 weH oumput!lld in order to
I!!Xte~d on 111 un1fol"!!1 tuuth th111 avaUabl e (:\lli ta • though
pr>obably tlu:~;r do not O!H'HII!pond to roal at~>llar atmoll!phe:t>es
on account or the raot t]lat in hig'll.lil'r tlllmpeNtu.re the
to tnol~de the ert~ct of ttl.i~ sryu:roe ot opaGity in further
tn:vee ti(!;a tt on!ll.
't'he computation of mo.ifl!l lfl'tflllllU? etrttOI!!PhliiNilil in
re.d!l!l UVI!I equilibrium aonsiete in the numerical 1nttllll"i!i Uon
of \he e.q U!ll t1 on { 2.)'
j{ := :.., { 1-.Xk) { /11 (H) -t- ..a (H/jt-e. J (3)
1.111 the l!lbsorptton ooe:ffic:tent ner unit l!ltellar !Wlllill!l.
The :lonil!lllltion degx•elil .xH o.f hlfdl."og<tn and the electt•on
pH&?eure pl:il m1H.1t 1:1(11 t ound in t~~lu:th step ot integret1on
f:rom the V!!.l.IU\1 of & • given b)" Milne 'l!l tcormullll -~
& "' 6! (!r l -r) "';
1nstetu.'l. of the more r1!1oent O'll.andt'lllill!lkher• e ":fourth
~ppro::d.a t1on" ( 7) in oreier to I'UilVj!! a common bub for
comparison or our mot\ela with tl:ulse o:f Stromgre. llii'u:~
alll!lo llliUIId !Uln~t~• !.'l fnrmt~la for t~J~m:lii!INture diatributton.
'fhlll valu~~lll ot XH IUid Pe VHU'il oomput•d with the ~
-'<] A=- t1. ~77- /3. s?s-& + { ~ r, Jz-e .:: b x,., --& .f-/!!;-A/ ~-r ~~ (!r !f,r} J
.;z...,. I -.X" == _:_:.:._
j'l-..- ?"A
tha 1 ~:~nt::r:a tl. en d$1rt~e or 1116tala .XH l:!e1ng tall:!lln trom
'!:'he tntegretion of equlllt:l.cn (2) and { 3) oan be
et~reA:'!tl'lt!l by choosing 1' !'$!iii the independ.ent V$ll:'.1ablE11
IUld OOfll!)lltillg r fl:'l'>ll\l
,., /mHl;;.XH) { CI.{H)+ .a(#/fl•) ,dp ( lf}J
where the integ~nd d>il['!l'!lldfl oo.l;r on p and e ( dnoe
~ :1{1!'1' a{i!), fl. ·IU.'!!! fun e t:l. oru!l of i> and e or
liiVII!n e anl;r).. All! & does not va much tor the flrd
, .. u \'!ilttee ot -z: 1t 11! p01!1fl1hllll to Ill !'!fllllMI &"&. ' teil
ttz; oorreot vall!le in each liltep of 1ntegr~~tion 1!!! fcund
from li:l.lne'~t~ equt!lt:l.on. 1n wh1oh the extrl'll:)Olllt.ed Vlllllle C~f'
c: 1111 :l..ntl'oduoed, 1'4:11d then th.l'k ool'r'l!lit'!nondtng wlue
tnt~l:'llt1on ia to be l'fif!Htl~t•d, by choodng hoPVI!il" tl'l'll
i.ntlll!l'Vt~l of 1ntegra.tion rlillil!'iiOnabl;r s!lllllll 1 t 1s po~t~1111ble
to extt'ilpolate -c oor-r1111otly IUld thus avoid thlll rapetl Uoo.
of' OO!!!tllltlfi Uont.
l!lt!!lfllil inellldins tho nliilw st~&tl l!l:th't aeve:Mil previollll'\ 111tep111.
The 1ntt~lll1't~ted veJ.ue of 1:: 1JUld0d to :Us vdiU:I at the
blllg1nning of the intervlill. i.:nt<~~gl'lii'M on gives the new
'll!ltW Yiill u~ ct "C" wa!'B fo~md by using two 41t:f'~rst
• fol"!llulae ll.ppl. to two :U:ft'llrlllnt 1ntllii'Vels of
:l.ntegrl\lt:ton; thlll ~'~I'K~'~~m~~nt 'I:H:rtweE~n both v&llles thus
obtll ined :I.e a ohlilellt on
tn~at th.s tlotfllilt tol'lml~ae ll!'%tt4 1111'01"!11 suffj.ei.wntly &e.::naNtii't.
'!'his uthod of :tntegFation lll!ll!l!!ll.ll to be fl!ore conv~mist
than the uae of' other qua4Nture fol"!l!Ullile, :l.n which
reduot:u~ tho effeet ct !lUtmmt1on of erro:n.•111, which at'feeta
the other quadrature fcl'!mltu!l mcr!ll me,~icul!lll,7.
'!!hill 1nt~~trnl ot intl!lgNtiO!l whieh 1111'iUl chtuu11n mull
for the ini tisl steps of integr~'· t1on W!itS :l.nc!•<!:lai'Hid fo'l!
moder~tt!\ ve.ltt•UI of -r at whi.ch 1 t n.a po&1!11ble to
izun•el!llill!l e:t t~erati:IN w1 th the 1ne1'lltlll~>e of O})tioal
dtrpth affeetlll ~JerioUI'!ll7 the i,nt®grl!lnd in equl!lt1on (h) •
l'lild\101!1 the 1nt-~rvml t~'f i.ntegra t:!. on ag~atn, ueullJ' l:IJ'
hal v1~ it wl''l!m 1t b~~Jcems int1anvenientl7 llllrg111.
~hl!l mlllthad of inte!!;l".~At:ton del!!ei•ibed above givcu!
th$ opti,\'llilll depth -r: I'Uil 111 function ot total proseuN 'P•
l!'os- u:te of convon:l.enoe t.ile final tablfllll of !!!Qdel
l!ltellllll' lilt!!!Qil!tlhl!ll'l!l!ll I!U:•z~ li\I'Nllll!l4 ~UI to s;iv4t
a.nd los; Pe 111111 fm\cttiea ot " •
'!'he l!!telbl' lilt!!!Qelll:)I'U!Il'Q' COIIIDUt>~.~d l:md.$1" the IUII:IUl'll!)tiO!l
thi'lt th~~t tlil!ll'!)&:t?atu:rlll la contrellsd by t>!idiattve equ1libl'lum1
IH~>oomee Mahble at a eel"taln dapth ~ , at which t~
l'edietivlll tempel"attn.•e s;rei!11ent
tot t~e n = J . ...!..f-x(t+i•r' l 4 "? J>f...ut. T f
(5)
- ..
(6).
B7 oomputing both values of tho gr~dient fo~ ~•v~~al
det~th.fl of thil mil ill t1 ve model one f'tada 1!11Ul!U7 the level
!m(f i till! real
'lfdllfiS (5) (6}.
;radhnt Hea betwee.n tho Umittns:
Stneo tho rl!ldietivo val~e (5) 1~
atellllll" atmol!pherl!io, tua at41 t1onal "a<Ual:lotte" WliUl
eom!)Utl!ld tor, I!IVer:r ro.alat:tve ooe, tl!!ll!lttming that the
tempflrotlu•e g'f'!l.dte.nt 1n the unoteble a~eno is: liH'l.iabfiUe,
adi~batio modele ooneiata in n~rieal integration of
Uon (6}, log p hllltni the independwt "'''riabllll.
'the pt>o<HUUI of intet;rfltion ns &imillilr 11o that for
Ndiativlll modeh'l, the int!!!l"ftl!ll f'ol:' loa p being cho!llen,
tUUi!Ol'IU.!lg h ctreUrMtanoiiU!It betWIIIW 0.01 ( f'o:r &., = I 0 )
whish ~ and ~'-"' lllignif;r the optical de0th and the total
Pl"l!lf!lll!\U'III at thill tel) of t.he eonvootivel:r lAZU!IU~le sana.
The IUUII of l!lqua•uon (7) inohilu'l of equliltiW {11) tot' the
adi&bliltte models ill! due to the tact, that the oomputl!rtionl!l
of tUUI!Ibatic modeb give tho valuell! of T (en" & )1
f'-e
and .X11 llUI functhnl'l of log p, lllmking thus th!i! foftula
( 7} mo'f'!l s111 tll!ble ir1 pNotiee.
.. 11 ...
• th'l monoehrol!'llt,1o dlllptll!ll 7;, \lnl!rlil f'ounll lil!!l
f'unet1on!!l total pral!lauve p for two wavaler~th&
l "' 1.215! and il. .. 62521, the choice of' theee plilrt.iculiU"
wanllllnt;ths b~ling due to the tact that thl'!';y &1"11!1 4ivh10l't
pointe. ot the num<~\n•ied 1ntejil'ati0l't tor &(JI) liln4 a(lf")
~ in the Vii!!1ble rt of the ep!IIC.tl'WII iUld tbere:tore the11" mono
ohroatio eeeff1c1ats or ab111o:rpt1cn X, (JI) and x)I(Ji .. )
werl!l ime41atel;y ~t..rdl~lble without Ei\Xt~tnlii1Ve eomputii!Ucna.
The moooolu•oll'lllltic d~tpth!\11 tel' the RtUI!I.tive mdt~lllll wcn•e
:toad b;r intem;rati~ the equation (f,) rewri tton ira the
tom At,= 1 ~ ... (1-~J{J[X.(H) .,.X,IH1-j>-e.} .4}'
lind thoae tor adiabl!l tic models wUh the dd ot' f'omtlla
(7) r~wrH1tcm lUI 1
~/!' .
?;,(-!";!/"):: 7,;;M + /""'Hlflj-ef(I-X,){J<,(H}+X,.(/I)Je)ft d~J*" .d''ff>.<
1n whiCh ~r>') and .f,. 1dgni:f'l' the moooclu•omatic depth
anll the total preaeara lilt the top ot ooaveet1TG17 unl!ltlilbll!ll
'l'hlil moaoehNI'IIIi!Uo flue• F(v) weril fo.und b;y numrioal
evalllll tion of the 1ntlll&Nl J.tr
· flJ)J-== 2j13JIE2.fr,).dr)l
with the aid of Cot41!1ll* eight point fomtlla applhd. to
the internls :trom z;." P to z;"' 2. 8 t~~nd t'Nm T, .. 2. 8 to
7; .. 8.11. 'l'tto contribution of de~tplltl" h;r•r• to the
meraochromfttio tluxee wee omitted ainoe it 1e ftllll11gible.
Special t&blos of' B 11
wevelcmsthm in gnoetion.
Tho oolotu• IR<Uent h d.lllfineli b;, the equat10l't
¢ =-,d1{~ ).$~]
~·.rom that. ~~tqu®~Uon we :ti:rul lliUI an a.ppt•oximlltift ovo~:> a
.. 12 ...
p=-.>.
k ).._Fa.) -.l-;.z.)/F!j,)
:1 .. - /A,
1ntl"odno1n.g .lf' = ;fi F;
¢ =- .J~ ~-in t1~ - lt,
Intl'odnctng further mmtel":toal ftlUell! >.1
.. ''235fand
A.2 .. 62521 u.ari in computiiltion of monoohromliltle r'laxee
6. Computation of the equival~nt width of oal,~:tum l:lne.s:
6.1. Intro<iucto'!'7 l"l!lmlilrkl!l.
~~J1aut'lll, by oo.mp,n•ison with me:tuA;Ul'JIIIi!i o:r the oalour gt>ad1entll!•
if the cotUUlxion betwe~~m &., and speot!:'al typo were well
esta'l:lllehed.
The boet propert7 of a l!ltcllar epeotl"Um tor thin purpose
h thlil equ:l. 'l~llllmt width of so!OO sui t11ble speotNl l.ii:Ulll!l,
... 1.3 ...
on of dher•t• l!tfH~ounte:v<l/4, ean be appl111i4 onlY' tl) Une&J
resul ti~ from t rmnsi tions to the lowemt one:rgy leVI!IlS~,
l'linee other :U fil!llll 111.1'1!1 111ff111eted co:nst~lil!li"bl7 by the :l!t!u•k
eff'eet.
~he ellllm<!!l'lt for whieh line profiles llll"lll to be eom
putllld m1111t lu~ fa!.Fly $lilun&:mt in !!ltellar atmoll@ph!n•es, it
the th~!liVIillent wi.dthe of it1; line!!! IU"!!l to be dE~tt%l'!llined
r~~tlu!onably wel.l within a wide :v<lil.nge of sp111et~l typee.
Furth®r, tb111 lil'lllllll in que1:1t1on mu~:~t lh in th111 V1lll1bl~t
'l)!lll't of the sp~tetrU!!l and finally thll.l quantu.m met~h!!niet~~l
~tve.luatton ot the :run'rlllllli!!illld x·~<.dial wave function r~Nlilt
number ot elem®nta tor which the e!illoulattonlil~Y be unde~
t111km suoo~tll!!d'ull;r. to on111 1 !Ult!!ll!l<ly O<llcium, fin' which
' tbe profibe of the g line ( 1·227l1) aml lt Uno ( 3'B3fl) en
pet®nt.id (6.:U.l 1'olts) eve the I'tl!llilonaneG Une of xutut~l
oaleium is ve-ey fllli:r.rt. in l!lfU'lY t;rpe star111.
Sine<~ the li'ltl'®n~tb of 111 Une dlllp~tru'lli~ on the eblUldiUlce
or the a.bi!Un''bing element, wbieh ia ~:~nl!:nown and ll!.lll.y vaey
conddembly from I'! tal" to 111t" r, it lllould be advantlligeoua
to compute the t'~>tio or ttl'll equinlont wtdthe of two
lines o.t thE~ t!l&m'll eloment in two d1ff'llr~~tnt tontzs.Uon
l!ltagl!la. TM!i! ~tto does not var::~ gre!!!tl'Jf with the
l!l:mndli!nelll', ob entel"l!l intt':} ttHt tomull\l'll in l!l:l':ll!liltl'Jf
Thtill atd.culationa ot line 'I'J!:'Ofilea :followed olo~i!el7
the m111thod of ':;t'!"(}llllU'I!In ( 9} ae d~~tl'!orib<'iH! an;:i tllQstratod
''""'''"'"'"'• howev~n.•, w~tre 1ntrodtUlle4 beoaiU!Ill the
of' our e are tl'!Nn the t11:11:!!per<> to~~:r•
of the model eonah'i!il'rll:ld b;r ;~oolley. Th• .:th!tlnges coneem
tl1e lntmdttetton or :!"ti. U.en ftlnet1oi'U! :tn tb1!l! :fo'!"mnllll
for- ionil!lati<m l!ll04 !ilf Uu11 fliiW'tQP in th'!l t'o:~~!mnla
tel!> 'll!!lllisi oo l}l!l'l!fliRlh
i111 0.02 or all metvls.
Since th111 turel\1 ~~:~ome of our modele are
Pebt:tvel;v hts;h, it h nfi!H'll!ltUlA!I:r:f to eond.:'h!lr two l!ltllgl!UII
of: ionimat:ta b!i!sidea tbl!lt of the neutrllil ~1tom. The
ettleul~;t:tons are, howev!llr, s1mpltf1ed. b;v tl"HI taot thllt
the peroentag<~t of nl!lutr&l !litO!IIs ill! aMll oo aocount of
the low :l.onil!llllti<m potenti .. al of oaleium0 111:nd only in
one OIU!llii 11tli".:Mn149 o.llii;. It WliUl thoM:tcre lll1!11i!Ume4 tl'IN t
tll.<t neutr11l [tt0111!1! eoul!J b<~ disregl'lrdllld in thEI ocmptat.attoa
of thl!l dht:dbut:toa of' calcium in the dngl;r and doubly
cannot be computed exaotl7t e1nee the partition ftl.l'llotion
of do1fbl7 iontmed calcium 11!1 not mown (in the abll!ence
of Ru$11ell-Saun4erl!l coupling). It w&.i\'1 IU!Ienmed that tba
ratio of' the p~;rti t:ton funottonl!l in the ttrlllt and &eoond
ia1zat1on r~tagel!l h equal to the :ntto of the l!lt.atiatietiil
weights in the g:!'ound etate .. , Which in the hott<•l1!t medels
ma;r deviate donlid.derablJ' from u.~ true valu111. However,
eve U' the actulill number of Cali partteles were 1.5 til!llllf!l
tht~~ number found under our as~:~um!"'tionB, the error in the
equival~~mt width would not exee•4 ~,_J in eoole:r modela
the poei.'Jlble 11nl~n· h! even &!Nillel'. Froa the dbitrlbu:Ua
of &llllel~m~ in the first and seoond stl?lges of ioniutton,
tho :t:netia ot OI!I::U: ions in the gmund ll!ltate l~tiUJ :found
- 1!) -
ot thtt grounaj!"JttitiiJ and B:t the 'part! t1on f'unotion or the
fiNt stage et 1ont:;mt.,tol1.
Ill sim:\li)II' way by lllflplyiltg the samlll o&loulattons to the
distribution of' olllloiulll atoms between tho neutral s,nd
dngl7 i (Ill bed stlilges, 1. e. 1t Rill llltlil:!lllll!!:d that a sall
fraot:lon of' illhm!i! :tmmd to bill etnsl:r ionised ,in the
prev:I.Wil!l!l 1t~ ot ce1eulat1ons 1lll aotuall7 neutral. In
thCII!!llt eompatlllti<:llUI the tme vda&lll of tho part! tion
tunot:l.on!ll were Ul':!ed, llliruut tho;r au•e, known botb t'ol' neut1'1!!1l
atoma :ln the ground etate a!! agtra found b;r multil)lJ':lq
the total t'r1aet1oft of fteutral a.tom$ by the faotor -fd,':d:J0.
Apart fr";u radililtiM d!lmp:l.ag oftl:y eollb1M d!llllpiftl
'lty neutMl h;rdrogeft atmu wall taii:Eift into aconunt. 'l'b.e
collhiionf; with h~Uo:111 a toms 'Here dhrlllga:t-1!1G4, s:l.nee
the abund!lnee of hl$lium h not known lii!X&etly s.nd l!l:tnee
its polar1zab1ly ts onl:r 1/3 of the value of hydrogen.
The total effect or the eollia:l.on l:ra:•oadening by helium
cannot exceed hji of the broaden1ns due t.o h;rdl'OII!m•
broadaning by eleotron111 IIUld ions h oe!'ta:l.nl;r sm<~'<ll and
p!'obabl,- even I!IIIIIJillol' than the l:u•oadmtn.~ b;v heliwn.
111inoe in oool etarl!! the electron pl'eseure 1:'1 low and
in hott&l' wtlllrlll the &beorptton line 111 &l"lll tormqd in the
highest lll7'1'Jl'a, whioh haTe low l'h!l!'lfi!ity and :l.n which the
total ef'tent ot oolli don bt'ol!l:d!1lfting 1 s l!ll!!llll.l :ln com,...
~grisora with radiation 4amp:lftg.
'l!w ~1. at..U.• ·~~NIIII an ~:tvea ta 'fa~l•
'- u; ••• llf~l" flill"'t •t .... kt>le l'hft Ute Mett..-1
t!l'l~\\\lr1ltaQ tl • ~ t.lllltal. P~\\\111N p ma\1 Ute abet.Na
P!l'tllllilf!laN Pe aa haott~ of th<!! o,tlrral 6111ptla -z: •lMJV• tile - ef thf!l ._...-.ttTj!jlJ' UU;khle ._., tile lamt
Mtl"tlllls ta thl• pa:rt -·• tthe erttt•l vsla~~~~~J ~~ t2, lq fo -- lq f?e,~ " .,. 1 ... 1' pilll't atv• ..... ,. et.S. ...
atn..et.ttN ef tile naauve ~~ta\t •• •.Uantto _..1... 'l!w
aot.aal ~ktloa~~~ ef aeola w<!IN mtte w1tla oa111 •• two
f'lgai'IIIIIJ men tUB iiM gba ta Ule tahl•••
fte l!!ldOaz~lliiiii!II:Uo flu• fol' t."" UJJA ad
l • 6tJ2 A !IIIN Cbllla 1& thf!l t'fltllewl~ !'«iilhlflt }.
8, ~-o.i '·' o.o "'·' .. , k.5 o.a '·' 0.1 .... , o., '·' 1.0 '·' t.o '*·'
-
!!b\• L
IIM.,..Htan•l" la alta oft 1o-5 ... ~ M0-1 ..
- ·- -""" -'"'"' -- -· nMaUn~ atianUo~
···~-,-· ·-""" -). • 11.2)54 ,.. 1252A ). • t'I2JS& , .. 125M
-
''·'2 iO.M 50.09 '*7.&2
"·"· -.n ., .. 6, k7.JO ,,_,., hO.Oi •. ,. )2.26
31.02 25.91 16.&. 11 .. 6.3
26.hl ... ,., 16.9' 11.~
1J.n 16.00 10.-u 1h.7J
7.15 -10.10 '·"~ '·" 7.hh 10.6) '·'' 10.16
fte •lcJU> P'!lllltdte f4 tile ~tea ... la aN
Uatea 1a tk«t tel1P'1811 'fa1ltl«t 11.
~-- -• .. '"'' ' ··- --·---~---·-~-~---"---·-·•·""-' ~·
~fl'a (lGifl g .. tt.5} Olmtte (leg ... 3.5)
4 la41at. AUilHt. BlUe-~
18!411!1 ... Adlt<Ht,. DU't"e-mo«el '11191181 f'ML!lle ·. '11191hl '11191i8l ~
1.0 :r.oo t .... ..... 1.0 "·" J.tt 0"25 .
lo.t 1.71 J.OJ I o.t~
I I•·• "·"" .... , •. , ••• ... ,. .. ... 0"" o.7 ...... 1.67 0.57 o .. 6 o.tt 1.kl 0.56 . .. , O.fl 1.tt6 •••
J'o7! afta~taUo ~la tile lao~4!1e or SR41oate witll
tho Nliliip,....l t_,.ftten l«t qd to atf'ona; tkf!J' ahfm
alllo oa1sr aUpt art~ 1ft oalau t;Rdldto beweta
~7!1"8 aDI ataate.
'l'Jleae n~;ulart tlee 110 Mt ext•t t.'oP ~maattve mo4ellf.
'l'ke c~qe ef' gn4teata: w!tk tile netpl'o•l 'lapentuH
ta leaa nplal' tkM lR U.. atiaHUa aedes or •4ela• tke 1ate,.,.ldtca b:etRWer of: gra4.icate :tfill! 1ateme4t•te
valaoe ef: &a~ whick wUl H ll!lfOI!!;IlMI'J' f'ol' h.l'tihel" !moati
pti.Mfl_. !a quite pue1bla. 'ftte ll!idte b:eva 1owe1" P'!ll41Gtlll
the tn ~l"f:• wttk tb:e t'l:lt'OI!!;ptilllR of' eo •llale ntk
8a "' 0.6. wb~~ta tka ~l't" aaa. tba ll!illat b:eve thlll .._ en•
tieat etJI'MI!PlllR4iq to u at'lll9t~~pMn 1a whicJa aeatHl
b;r ... oa te pnot1oall.7 tn •11' a011ne of: op~~o1t.J'•
·'ftNI ti:tf'onuoo :ta oolllllll' t;H41Mttl# of' tb:l!l lltl#~diq
:ntia\bo an4 ntia!Ml\lo .aela a.n ••11.7 ao1141!aaU.le to
Obeena\1ea. ea'!)tilclsli.J' ta ll111tt~tll' I!!Vn'IJ• ta l'lrbic1a Co
~~Mtlllale uao IX~<&;i.lllmJ ~t ln llli\'ltlUl 4eptn; f:el" tn ••
-11-
.-ea te• ;~bats ~ liU.'fu_.n eN ~· tlHia te
.,...,. wl.tll tile --.~~- ., .... ,., ··- 1& WM.!Itll
a.,. a.- ta .-dl o • ._'tatte ft1tle (e'ttat o .. ") • ..,... nepndi• u o ·~ sa WM.!Itll tile eat:J!fl.nUea ef
aepU• ~tv~•t• te •• e,aeltv te usuat•te..
fte e,al'Ml&t wl4tll ef tile X 11M ftllll .... le4 t'ft
. all llletela 84 Umt et De a Uu tel' dx ~ta. fte eta!•
ftlat w1.4tll ttit tile a l.IM to:P tile tft !let.t.l ~let
•• e.,. 0.6. ta ee .-11 Umt ..... ,.._te ... - laae•
l'elli!flil1e 84 t!leRteM u ~1'1.._ ••we• tue17 aa«
•ul'\l'l\Ua b peui"'le,.
fte Nalt• et De •..-~au-. an ll•H4 ta h~lee
5 u« 6.
hbl.$ '·
"''D!Sl I&«D tf Df I Ulft ta A. - ,~,.
··~ -· ~-~~--·--'---~0·---- -- ~-- - ,,,_,,_ - '''''" --~ ··-._.,. (lq a ... h.!) lliuu (lq a .. .1.5) I! /RU&\. a<uUat..
i!
&., atiant. .. e. etil!lint.. 11 ' l•a"~e ~b .als .ata I! h II
0.7 o.O?J o.OIO 1:
II II li
II n (t o.a ... ... , o.a t:MJ 0.1, IIi. !i ,, ,,
II
~~ o., o.,, o.67 I
I :i
ii 1.0 '·" , ... 1.0 e. at o.to
'!'ole 6..
IU'l!:le!: 1&19 tf Df I 1!11 Ia ·fa
Dal'h (tq •• 1\ ... 5) .~. ... (lq ... ,.,,,
e. I Rtia\. &uat;;i; 11-&-· l'l!ltiat. al'!iaiM\~1
o :~l.a ~llf 1, • ~la l·uale I . o.6 I e.!, 0.61 .. , e., Ia O.!il ,:
i[ 0.7 .... , ••• I o.a
I 6..n '·'1 ... •. ,. '·'' I
e.t u.6 n.t 1.0 , •.. , .... 1.0 H~ .. t tt.l
.. 19 ..
eeyutv·al~t wi.dth tilt' th~ g ltruil 1111 not ~fteete4
ii!er1oull!l;:r by eoo,rec.tton even in the hott!':cat !!lOdelii! f'or
ble lumt
width of' tbiel
Th!':l lC. l:lne h moN s!!ln1!11t1v$ to th.e poe0H>le effect
of convl!lctiont but only in hot modl:lll'!lo In ooolep modde
the
f•r 1nstanee tl" o. 6
of about :?Oll( :tn the me1u~nrement ot' the oqn1valent width
is eyllite pos1!11ble, it is difficult to cheek thb v~,; tion
of the rliltio w•/1! in l!tellar apeot:ra.
The t'$etnroeal eurtaee te'£!Ptll"!;!ture ~ 111 a quaat1ty,
wh:l.eh ean:not be observed lilnd therefore the relation betwe•
-
I I I I \ I
\
\
•
•
J .. ; ~
I I \ \ \ \ \
\
' \ \
--
\ \ \
, I
I
\ \
• J I
\ <
\ \ \ I \
! '
' i '
),'
" ' '
I
:'' ,,
' ,,
(,
"'
l'i• I
':1 .,,
' ' ,I
' 'I) ,, i,
'<11 ,. ;l i'
I i
I .
'lj
j ' ' 1:1
: 111 ''I 1','
,, "
,,: ' 'I I''
' I
, I ,,, l:i,
,,,j
'
I ' '
.,
; 1 ' '
.,; ,,
i,
"· 'II>
I
, I
' ' i
I I
' :I ,I
'
, I
' I,
- .. thEt eq:d:t••l•t w:tflths ot the I: llnd s line~~~ tor the twt~
~llllillilllili of i11i!lill'O and ma:~~::lmmll oonvl!llotleJ~., lUi all theee qua•
ti tielll !il'l:'lill t~.OOlfll!lll!il'lle to obll!llll"V!IItion. Thlilll'l!lil Mag'l.'l!ll:lli!l U"l!ll
l!lhown ln x>1g. 1 •
It should 1:!1!11 noted1 that, for both the K and s U.nl!lla,
th~tu·ll! :ls a fai rl;r l.~U?Sili IU~para tion between the curves
drawn fop the ratiattve and adiabatic mcdl!llls, 1!118PI!IIoiall;r
f'ol' the h1shel" tompE~ntuHa. Thim 1111 V<lfl:'7 :r~:~rtunat•, e!no•
it ~~Wll:&ll! 1 t tlll!ill!!ier to dht1ngu1sh, whathep aotual !ll'tellav
a tmGI!Ipheree follow thf.t •• o:r> the otht~r 111E1t or oiU'VEil'l.
on tnt~ otbe~ hant, ror tbe K line thorlil 1111 no grtlat diffe
rtlnee between stante and dwaX"fa tor et uun• the radtliiUve
or the adlabatto models, and thia allowa ua to dieregar4
an-, di.tfe.renoea tn the lu1nod tiea ot the ~Bt&N in qttllll!il
tton. 'l'h:l111 1 a~~ again foPttnMte 1 !ilil'u:ut thf.t 1: lin.e 11!1 the
most 111111 ~M.• line f'ol' tlrle apeot:nl olaaii:IU'ieatioa of
thEUIO at&ra, IliAd ee whoa• egldvalent wtdtb oon be toad.
:tabl.,- usn.,-.
ll'OP the g line the I\IIII~NtiOO betWliiOD Si&ntlll !'Alld
d:Wiilrfe~ h ltu•ger than tor tlu~ I{ ltne; tor Nd:l.at1 ve ~~~o
del•• l'u'lwever, tt h et11l tolerabl-, li!!Wll 111nll the ol•ar
sei)~arat1oa. bdw!!ten tlH~t adilllbatic modela :ta a.ot lmpor~nt.
dnce f'llJ:>thllli' 1nvs&t1g~~ tion show that the :rE~tal st•llar
ntmosph<~:rlllla tollow the radiative models closel;r. The va
riation of' the l!lqutnlent width of thfl g line with coloul"
gradient ill} almost Une~u.·.
'rho comp,u•ison of the theore't1C1:11 !:'elation !:HIItweliiD
the eq,JS:vdent widths or· caloium Unes a,nd the gndil!lftt$
with the semll!l relation. sstabltshe4 from observatiafts eoulll
stve !lin ind1m:~t1oo of tho etfect of' eonveoUon oo the li'ltruo-
ture I'Jtelb:r atmol!rpherEuw. Unt'o:rtenmwl;r, lllltho~h the
lent widths of thlil .lt. and s lbu~a have be~~~n ~~t£11H't~~d o.nl-,
fo'l' flilw etarl!l. lven foP t'!'UtliUI the p:re.o1s1on of tile detel'minlll-
bll!i1\iht4
the wtqa
to the , l!l~qdvl'!llent 1rit'lt~ oa1mot blil :round wUb fi:Milll t aecct•Eut;v.
f'lll"the!', tt tho Unee al"~t bl.ad111il 1t h veey IH !'fle!tlt to
41:'tllw the ocmtinu;:;u4!1 bi!IOt.gHIATtd. eol"Hctl7• vM.cb aa!B!n l.•de
' to l-1 !'~1!1 ft:li't'ON.
Aettu~~<1l7 fcl" all tll"l"G wUh the l"'!!C:lj!U"'O!!il •ur-f'IIIOI!t to_..
f.I!IIN t11n w1 thia tho 1111'11 ta o. 6 ~an« 1 • 0, the K Uno 1 lll •=· l!lfldtn'lllbl;v blealte4 d th tlMII' fi!\1, K~; and H :u.nee, •n4 tl'lu1~
fore ''o:a-ll'E!ot 'mllll!llil or the equlftllil411t w14the of' lllll t!'.il!llllill
line• ciln t~o obt.atnd onl;v it' th* ovoll'l&!)l:}1q !}Nf11•a
ll411oo 11heoreUoal .adtet.cu•\®4 pl"!!lfUea of the nelilfb>o
botilrlq Une~a &n nl'.llt aT&U<lble, an ~tXIll41111 detemtnatla
or \he flqat:lTI!Il~t width ot tho Une till h&:~PdlJ' po~JI!IU.'Illll.
It 0<1411 bit iii:Ji:'!)li!Otedt ~1Gii!Wlill't that th<t Gl'!'~lf,'!lll Will not
~.tllcure4 25''~;, wM<lh ol'uU'*i:'IU11 tht~~ logarithm of' the eq~.t!V<~leat
wt4tl\l by o.ttL '!l'.:n• ttweFtlll wl e,, .. 1.0,. lt the •eatri•
b!lU~ of tho w:l&elll of the 1: Une frG~~ ov4h• :?Ol tn• the
e1111N of the :Ua• 111 M1 U:$4, tb«!! ''~~''1:'!111' 11!1 li'O:': H 1411 iiddl tion
the O<mt1411~.tOU baOlqfNOd b dma 7:1' toe low, t.he totd
erN!' woa14 n•• to about 35!1. :ror 'llho ri' d'th e0 .. o.a 0
tho oontr1blilU<m ot the wl:lllill fra 0\'!111" 10A fl'<m thtt ooH
la 1~, l!lft4 th4ll a!idUitm<~l t~~f'l"!!ll't U' the «h'ln111uuum b
dmea too low. 1111 II. An llH't'lU' ln t}'® IUI!!IUIHd ba~tqli'ea4
wo11ld, howeve~, 11111 e.t llllllllt r>tiell;r rUillltmUilld '~>7 thlll
raet tt1.111t the lllllflilflll or lis t~n.d 'K~ Uftelll 4<!!!111Ptm the~ eoH
et thl!l lit ltn111. 'l.'!u\ cnpa!"iiiiM ot the eqdvilllet 'lfldtlua
at U:llllllil filllt!'!Nii\U'II!I b;v two Oblll<i'U:''II'•N DhOW that tht~:!l!ll a.te~NioaUeail are not V!Yil"J' enot liUld \bllt e~n ~!U"t'4!11' ot
ts poe"'1blllll. l"hullh oompl\lrison no •·4• i:l;v ~ar>blll!'a l'!eU. ( H ) ,
Uon U.!UI'Illl tn the 1110b.r e~HtOt:r't•ll'l IU'I! 1117l!lt-.Uolllll,r b;v
151 hlgheP tho t.be dete~N1nett.we c:f a.w.AUon (tl).
An <IH'l!'Ol" of ~~tbout 111. hil!'llll!nrlll!r" ~ 11 tcl~~tmble. 111:lnoE~
the l!llol)G the ;m<'it~t .. eq1dvd.a\ wUlth 1&
The Stlflrlill, for Wh1!1!h the llll!lld.Vali!Jitlt 1ddtho of thl!l It
s Unl!lll! 1111"1!11 liVI!IU~tble IU'Ill lhtod in '1'1li.ble s:h.'he equ1 ...
nlent w14th of' the I: l:tne is hlilll'll 4lnl1pated b;r w• and that
Of tho S :U.ne b7 1J. 'fhilll l.ii~Hl!Ct:Nl tl"J)CII!I Of thi!JI lilt~U:'I!I ll!N thOiiUit
ot' s'!. Mor~~tlln, }'.c. KE!Ie!!tn and m. i(ellmi!i.n (1 ), 11!.) :l.f ~:r~tlll1
le.blilll. 'ol" ••• lilltavs th111y w~~t1"8 take from tb.e m:tprtbl11i!htt4
could be fGund t'!"OIII the !)ill'rall!!X• the l1.'1!111ncs1 t;r olaas wss
t~!ld8d, and in thtlt ca1:H!~ the ;Jpeotrel tne h wr1tte in pa•
fr0111 the l1!'1tl!l publhh.~~td. b;r the Oreewich Ob!!lttnat!)l'J' (15).
b;r 0.1.1. Gucoipllt (16) or i:l;r R.v.4.R. Woolley, ~~:.GoUUo'b
l.ilnd. A. Pra;rbyl~i~!d. (17). S:!.nee dl these l1111ts fom l.il un1fotS
gre<liet h givllln 1n two differlllnt Ulilt, 1tlil m~tM~n Vl.illul'l 11!1
Rho\lm. If no grllHl.ient has bee publh1hed. thlll valtH!I oo<l'l"lll&!lOI~.
d.tq to th.4i'l spectml t;rpo of th.~~t etar ~.Ill shoa in pEn'en-
which hlll.il a variable iJNdhnt; thilll h the mat t:u•obabl!!t
value f'o:r th!ll epooh. at Wl1icb tht~~ speotrollti'SlMI wf!>t'!!l tal!:llln.
Plotting the logarithm or the eqld.valent width, w• or
w, asa1nlllt th\11 &!:l"l!H'iil!lnt for Iii numbE~l' of l!ltlll'lil we o!:rta1n th~~t
Obliii\11I'VGd l''i\llllti®ll! 1:11111/l!lllllill'l. thi!lll!\8 (!ll!!iDti til!ltJ.
'l'h'i\1 Ell!U'll' detem1nat1onl!i ot' the equtval<~~nt width, !lllilde
b7 'f. Dunhtlm ,(US}, lil!in a. n. Pt~t~111 (19)* m. o. WUUau (21) 'J
(20) 111nd »U.e~~J I.T.n. Wil:UauJw:ttb tbe epEllctregl'li'lllfi of low
disperdoD, 4ev::hlte !lllillne:l.bl;r from reoet detemiMtions,
~tlll~'H!Ciflllll1 when the equtvl!ll~'nt width illl le»s thtla 1A. l!'cr
the lit Uno thea• «~~arl:r det~:~tw~inlll ttone &ll?'ttl!l t'ai rl;v Wililll
11'11 th tile reoet oruu1 lllnd the:ll'efQre they can 0till be uee4; "*) 0'7'1 4-C.ccnc*'l of ..Jt~ffle clcj/c ".cu//tl>J' l·.., o.6 lal171·,_!f J%e ,P'y.'t>-r
w,·tt(a nn; (C I) her Clh.s:r~vf<i-1/fh-r s WPr(' 4¥P7 ,:.,., cl..,. cfe4' ~ ,i--4te
wti! bt t"-n~l .... ,eeta' <~ fht ~ub!,·ca. T:.~ 64ft>~ coo~ :~'he :/h~.s(:s-.
c/ ~::rs C.7:R. ~hes<-S. 77.~y
•
\
•
•
•
\ I. \ \ I \ I I \ \ \ I I I
N·
Cl>
.. J:t.
I I I I : ,,
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.,. 6 In
6 .., 6
- 2) -
however, le~e weight was given to them when d~wing the
mem curve. J'ol" the 1 Une the11" devh uon f'l"'ttl the rutw
. dete1"tti1Bat1ona 1e eo exeeas1ve th~t they must be e:oluded
from furthel" oonelderation, even though the total n~er
ot stare f'ol" whtoh tlut eqdv~r~lent ·ddth of' the g line is
known h at~~~r~U.
Where many determination!! of the equivalent w1dthlll
were available for the same star, the mean value waa ueed&
The weight 1¥- wate given to the ea.rly determinationl!l ot the
X line when forming thh mean. The variable star RR Ly~e.
whose speotrum varies between Ao and P2 was treated as an
A6 at11.1', tollowtng th<'ll papel" by E.G. Williams (20}. ~ reJ"@c/Pof.
bifte117 eta!' H.D. 11tbii!08-9 wae olua1t1e4 u an 1'5 l'lw!u•t.
Eleven sta~ with abnormal spectra 1nvest1gsted by L.H.
Aller (22) wer11 inclu.ded 1n the diagram, since their K lin.:u1
were stated to be normal, although other metallic lines !!lhow
many pecularities.
The observed g~d1ent - equivalent width relatione are
shown in tig. a. l"or the liC. line thil!! relation 114 well ost!lblish.ed with
the exception of the region corre5pond1n,; to the A0 stllll"lll,
where thEII I!!O!!ltter1ng ie l11.rge. This scattering may be partl;v
due to the tact thlilt .for t~~~r~:ny A0 sta!'e the g~dients have
not been obearvt~d and the!:'efore they !v&rtl cllilel!li:f'ied lUll
stl.lrs with the gradient o.oo. on the other hand t~~~r~ny of
these e!!ct'ly type liltat>e are peculiar and the !!bu:nd!UlOI!l of
Clllc1um in their atmoapharell! may !:u11 abnormal. For eti>I'I!I
with gradten.ta exceeding 0.10 the unwe1ghtl!ld mem devta'tion
of logif+ fr>om the mean curve t'or a e.t.ngle atar 1111 o.zl.
The :tour largest deviations IU'e ehown by two at!!l"l!! for which
by two etl!lre with uncertatn I!'!;HUe:ntlll.
The curve for the g line cannot be drlllwn w1 th gr>eat
accuracy. Reliable determinattone ol' the equiv~•lent width
are ti\Vailable in a l!mf.:!'1c1ent number onl7
!l;!'!l <U ents exceeding o. ao. !l'or emrly type
thl'ete l'lCilttered !)<lints an'c llVH1.1&1ble thePe :i13 a M~
for tl! bet, ween o. ·1 0 o. 80 • .A ;•)tr;;tght line
m.~ ti on to the !'lilltl t 1 on b•· twel!ln the .F:l'1!CUent and the e<ml
V&lent width ot tlH! g line.
Since thlil obl!!!ll'VllH1 •11 en ts are retlll t1 ve ,~:~ correctl on
must b<!l lllppl1ed in o NlCI:' to compare them wl th oompu.ted
ebsolutl!l grf.uUents. This corHotion 1a of the ora.er of.'
1 .1 ;t 0.1 • A furthe:r correction mttet be applied to the com
puted C!ltl1 va.l®nt width:'l, if' the all!sumed abunli&noe of cal
a:!. um dii'fel:'e from tlH} !!.otual mellln alnmdHnce ot' this element
in I' teller '' tmoepher'ee.
It 1!!! eadl;r li!Gen that th.e o'l:leervGd enrve fol' the
K line fits the theoretical l'elat1on the tive
models closely, if a eor•rection +0.95 is applied to the
observed g!:'I:H1ltent!il and a ct~r'Hot!on -0.05 to the 1ogar•1thm
of the oomputod equivalent 1ddth. The l!grsement or the re-
lations the g ltne 1s then all!lo good; the observed l'lllrve
lies alit:htly b·:,low the thaoretictll one. whioh 11'1 !)!i.r>hllps
due to the fact thnt the ionization or calcium in etelllill"
eq~aa tioo.
on the other hand, the bEH't po:Jsihle flgreement between
ttHI observed t"elattone '>:>r the K and. g U.nes and the theore-
tical ourvll:le tor the a tuatio models need.s a correction
of 1. It fo'l:' the oh!HU•ved g:l"'ldien t,, • •JIM ch t a otn"tH 'nly too
high. r•;,.,en then the r1greememt lJetween the obeerve<l and the
oomput<!Hl curves is not R&<tisftlotoey, sl.noe there 1.1re a;rete
mattc ilH'f'erenaes at both of the \'iia~rr•;·.me. '!'he observed
ooloux• dhmt - oou1v~;~lent width rel£;t1•>ns both K !ilnd
g lines would then li.e partly below the t!Hl!orc;tic,;tl Ol.lil"VSI!
~he curve tleri ved .fx•om obsel'VPl t1 ~'nll!l r.~ls t l te between
the theoretical OnT'vee I'!x<:l.l!ltive 1md eHUubotio models.
.. 25 ..
Using the observed i~!'~ldhnt - equivalent d.d.th rE~lllltion:·,.of'
the K ltne dN~wn on trllln~;~p!u•ent paper and t:eying to l!!hift
:U eo as to ~Satht;v th.e ~>bove condition tot" higher valllell!
C>f' th~t gl'®dhnt (rj> 2.00) C>ne read11;v f1nde thll!t th1a 1a
polilai.hle onl;v, 1 f the obll!e!'Ved ourve follerwe the theoreti-
olil.l curve f'or radbtive model!!! closel;v o•HIU' a wide range.
For VS~lu.em of¢ over 1.1. 1 t is posl!lible to achieve this
lll!f!'llle~t~ent even i.f thG ll!ero peint correction for the obsE~!'Ved
gradients 111 allowed to t11ke values over !i\1 large l'l'Hili)IE!.t for
hotter stare, however, tlde ~<greement 11111 onl;r pol!llll1blet1f
the lltl"lldient correction dollls not dU'tnr gr&liltl;r :from 0.95.
'l'h" o'!Jet"Tei! curve for G <.0.10 :!.Ill, 'llowevel", unoeJ:>ta:tn rmd
tnerefol"e thl!l :~~ero polnt oorrl!Jction of.' the grtlu11enti!IJ cannot
he determined ex!llctl;r. A change 1n thet correction would
necel!llslllrily lt~£td to a ol:ur~nge tn the oorreot:t<~n. for the !iU!I!!m.ml!lt'l
l!lbllnd!llnce of calcium ill stellar atmo11phe:re111.
one mu"'t oonl!llude, the t re~&l llltella:t> atmosphere& fll'EI
not flllr frcm rmd1ative equUibriwn flind that the t:N.ns?Ol't
at energ;v due to ef!nvection cannot exceed Iii :rew P.llll"!Umt ot
the total eney•g;r f'lllltl. Furtl:uh• the I!UIUlllmed oalciwn abund111noe
in ll!tellar atmoat~h~n·ful! !UJ' be about 25" t.oo h1sh. 1\lince that
correction ot th• al:nmd!llnce, would reduce the lo&l!'!r1 thm· of
the oqulvdeat width b3f o.05.
9• Stellar temperature ece.le.
Asaum:tns thet the sero-pnint c.orreot:l.on to the Oreen
wioh - l'!ou.nt Stromlo I!I'lild1ent ll!Yilltl\!lfl is ... o. 95, one ·oan IU!e
the gr~u11ent - apectNl t3"t>e rel11tion to :rtnd the «~ffect:tve
tempenture of the ll!tsl:'s illorrel!!ponding to theh• epeotMl .
type and :lnminoa1ty. The gNid1ent - epootral t3ft>& rela t1oa
1& known from the 1nve!11tig~Rtlons ot s.o.B. <n.seoigne (Hi),
R.v.d. Woolle.;l, Gottlieb and A. P:rlil;rbyll!lki (17). From
th:l.s rcl11t1oa onl!l f:lndllt the gradient, eorreaponding to a
certain 111'!)11ll!lti'al type and lum1noa1. t;r a.nd then lUling the
gradient - reciprocal stu•f'aoe tsmperm.t~tre relllltion (ti.g. 3 } ,
known f'rt:llll the present inveetiglil.tiOD!i!, one ~'indlll th$ ;a~nrt'ace
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gr~:.t'lient, l!U:•& more sen t:l.ve to
absorption coef'f'toient.
in the mean
The true value the m•1o!ftn trbso:!"'£ltion coef'f'!.<~illlnt
is not known until the defi t.ive stl'Ucture <lf the! model
etellsr utr:·;o!lc;he:re ie d1~te:rm:tned; every V!lllue the
maar< abflnrpt ion coeffi e1.emt Ul!led as a basi a t'!'r the
col!mutationa must tlH!r•efore be reglll r-ded ae an ~tn:rproJ:t ...
m!lti.on eul:l;!eot to fu:rthe'l" oorrect'l.cmh If these
coritecttona au•e reBHonably sm!!ll the the<::n:et:! of!l gradient -
equivalent width .relati(')n 1$1 not !!ltlO.h affected by them.
aubatMtiall;v sM,tt thie rel1~tion on the diagr~m~ towarda
;:.t'!ll!lller value!ll of the gradient and eonv•!n'sillly a (leore&ae
would antft 1 t towa!'df!l lf•rger valuea. Since, however,
we <M~nnot e:&:J;H':Ot that this shift will be u.nU'orm f()I' all
pointe on the gr•a , the cu!'VtS will not only be ah1fte4
conllli(l!!I'N~Uol'UI ~!!hOW th!llt thili! would affeot the p:t>OViOUI!I
oonoluaion• th:!l.t the effect of oonveot:lon on t.he
et1•u.oture of stellar atmOIIlclheHs ia small, l!linoe it is
possible to aohteve an agreement of' the observed olu•ves
with the theoretical grmdient - t ''l'idth rl!llatic>:~'l,
:tf the later ia slig;htly <Ilstorted. Fig. t, show,;; the
theoretiol1ll r&ll1ltion for the K l1ne distorted in thrt~~e
different ways. curve !. r&'!)Pel!ents thil:l relation
wi 'il(l unifomly by 20;11:; curve !at compressed unH'orml;v
by and finally in Cllrve _g the diatof'tion Wl1ll1l gradually
onangedt the lef't end of: the o1u•ve was compressed by
:?Ojl!:, tho middle part h:f't unchanged and the :right end
a goo1l at;.reement of' these curvetl!l with the obfl!erved IP'SIJ!l.:Umt ·
equivalent width rel& tlon using f':t'en•nt zero-point.
- -
be l""C~t;"iCJl1~J.bly
llm1t, it cnn
tiH.II'l'I"Gti, C!'il
the observed
small.
of 1
,, ,·.;
small
be ,., ,,
ent
one ir;
(l.)
an,J w·Ul
thut the the
'iV!l on wi.th
any aoobt
me1m u.lH!Ol'ot:l.on coeffi.·~1ent l!l."!:'e lower; for modeln with.
e. bat;ween 0. ~~ c1.nd 1. 0 the V111 l.!.l\$EI Of p and l:H'E)
and its
annatonay la laao rtPik •
in Oil:!'
unpublished rasults on the EIQU:tvalent widtl'U'l E~n\1
epect:!'al t;nlll' daterrninations.
Om~~putationli hiive been llllidli!l ol' model sti!i!llnr
atll\os:pheriU!! co:rrt'!:sponding: to spectral t;nlel! trom about
0 0 to l•o• Tho atmospheres have .been oonl!ltrueted with
rad.lative gradients an<l. also with li!Hiililb',,tic gradi~W~.te
in thoee T:'!!!gi one wh<n•e they sr111 lese than rl!l.diatt ve
gradient~:~.
give the colour S"!'lU:Iients 4!Uld the et;j\11 val 1;n t widths
of the lines s (1,~27A) and l ( 39~•A) of neut·d and
iooi~eed calc!. um ref!peot1 vel;r.
Die..<' :rams conne<,ting. the equivalent widths of K
and s lin·aa witb the gradi~W~.ts w~Jrfll oon~>trlu.lted tor
radia t! ve and adi•! ba tic ll!Odli!h~ IU'l<l comi)l!l red with the
same r<i!!lb.:tionl!l E!fJt.lilblhhed from observ· tional data.
It was :tou!u\ th~:~t the aC~tual atell!:ir ·sp(:otra follow
elO!'!I'Il:V the thetrreticsl ent ... l'.l>qtl\v~alent width
relatione forjNd1!it1vEII models an.:i that, therefol'e, the
e:treot of oonveetl on on thtl et'l:"uotul"e of m tellar
atmospl'un'l!ls is slllll!ll.
- 31 -
l) a. stromg,,en, K. G;rld!1Jnkaullrnlll, Ill. <'ludk3Ji'bing and
2)
J:.A. Th~l"l'U>fH Tabllill!!l of' !!l!Hiel €1tell&11'
Publ ik~ tionf}l:' og min\lre Meddllllelii!er fr>a K~c.nhavns
Oburva t,orium Nl'. l3fl ( l9h'').
l'fr. Abao:rr) tion
Ano.J. lOh,
1i• llr111eru Oa the Continuoue n""""'"loi~mt or the N1111g'''tive rogen It>n.
!•30 (191!6).
3) ,J'<&!:!Sii ""• H. Mond L"ld E, Riehard Gohfll'H Our
Knowledge~ of' th., il.t•,mio Gonstantll! F, N, m, and n
in 19117, and of Ot.her Constanta Derlvl!l:bltll Tl'H!Iret?'om.
R!l!Vlews ot' Modern Phlf'dos !it (l9MH.
I!) Chlilrlotte li:. l\!ooMt lltom1(l lilnergy !.evele. C1ro. o:t the N$lt1Qni!l.l Btl.:!'ll!ft\1 of ':ttU'l!ardll h67 (191•9).
5) Ra;rmond. lHohal'•~'h Sur l•uel:lge dell ooeffh1ente d' 1!1 bsor!')tion r.la!lll 11'1 construction modele a
d'atmolllphln•ee shllail:'&lll~ An. d'Ap. li• 291 (l9h9).
6) Donllld H. ~enli:el and Oha !m L. P!!!l!:erbt Abs<:n';:)tion
ooef'tic1en1H!! and h7d.rogen line :!.ntene:!:Ue1;;.
M.N. 'i!~t 77 (1936).
7) ChlltulraGekhart on tht'll ''adhtivl!l Equ111b·ri.llm o'f Iii
6tell1!11' Atmo!J:~herll! Il'. Ap. J. lOOt 76 (l9Mr)
8) ll!. M,ndholnu :Oial:~<c:trt!ltion Ubl!lr d:!.e v~:rbre.1 tun1 und
VereohietHing von S:peotrall:!.ni!!lrn; Up·')ellla, 19!'2•
9} B. !'!tromg!'EilU The :!lcnmdar;r-V<~lll<'~ !"!•obllllm the
Tlleocy of Stellar Absorption I,1n''!~.
10} n.v.d.R. Woolle;y 1utd D."'l.rf. Stibbe: The outa:r Laye:re
ot a !~taP, Ch. VIII and IX; O:rford, 19'53.
ll) M "~'lll A Study of DoppleP 1md D~'<mping ~.fteots
in the Solall' Atmos't'here; fiarve;rd Uni'lfe:r<sity, 1951.
12} w. Allen: Fraunhofer Intenlil:lt;r 'l'~i~blen~ Mem. of the Co!11ll!onwealth ~olaP Obo;ervatory 51 Pa:r·t Two {193h ).
13) '"~•111'• r.::orgeul, P.:::. Keentul, E. Kellmn: .lin Atlas of:
8tellar SpeatN.
11•) J.A. lf.ll'l'Uskt Astroph;rdce, New York 19.'1 Oh.l.
Olllul&if1o$itlon of Stellal' l'l':.eatra.
15) Relative Gradients of stare D&t rmined at the
Royal Observator;r, Greenwich; N. 2!• 1~9 (1939}.
16) c.s. Ga<!oo1gne: Relative Gr1Hihnte for 166 '~outhem Stars; M.N. !!Q, 15 (1950)
- 32 -
( eontd).
Further 0bl'1111l'VIiitt ,~ne of' ~agn1tud.e with a ""'""''"'
Gr~~tir•!H lf,N, 112, 665 (1952).
lll) T. Du.'lham Jl't The Gonto111'a of Hydro.!\s&n Lim~e :!.n IH;ru~s
of Cl!iiss A; n.r. 858 (1921::). 19)
O,H, Payne: the Abi!<orpt.i.r>n LintH'l of:
Cll!le!um; H.B. 858 (1925).
20) K .• G. IVUlit<mtH The Equinlent 1Hdths o:f · tellsr
Gslehtm L:!.nea i Publ. A, '!W • .!J.§. 113 (1936).
21) ~: • 'r • R. \Ullillii'IIIU A flpeOti'O!lhOtometr:l.e Study o:t At 'c'\tliil'GJ H.O. Jll.!!, (1929).
22) L. !{, Aller: A-Typl! Stars v<ith abnormal aoeetra;
Jl:f!!.,J. 106, 76 (19''7).
23) G. P. lui per: ll!agni tude of the sun, the :;tell!iir
Temr;er~lture Scale, and Bolornetr•1o Corrections
/i;p •• r. J!, lt29 (1938).
'!'able 8.
&quivalent •ldtha or calcium lines •
lifo. IUl' 1-pootl:'UIII Gradient lo~t • ..
loil' llotellJ
1 ~J. An4 BlpHIMn ..0.10 1.51 a -I & HJ"a Af'Np -o.o6 1.29 a J a. Ola A1V .. o.os 1.cs2 a .. .. .. !.57 b b r UMa AOV -o •. 05 o.1b Ill
' 8 •• A1V ..0.02 1.98 a
' ~on B8Ia -0.01 1.83 0
1 "' L;rr Ao\f o.oo 1.78 4 ... " .. f.97 a .. " " "'·'2 Ill
8 "' Cyg Ali!% a (o.oo) ... ,. !.27 e It It .. o.oe 4 It .. .. 0.21< c
9 (,] wa (AOV) (o.oo) "·" a 10 Y Ono (AOV) (0.00) 0.08 Ill
11 l'*cm (AOV<) (0.00) f.SI a 12 6 H,-a (AOV) (0.00) 0.31 f
1:5 15 One (A.OVp) (0.00) 1.30 f 14 8!.1 Ula (AOVp) (o.oo) f.30 f 1S 17 00111. (AOVp) (0.00) 1.30 f 16 119 Ono (AOVp) {o.oo) 1.60 f 17 ). Ser (AOVp} (0.00) 1.90 t 18 m!h81Ei A OJ!) (0.00} 0.36 't 19 a OrB AOV 0.01 f.85 • 10 {iS on! (AIV) 0.02. 0.22 a . 1019 ,
'·'' 21 r Gem A1IV 0.10 o.o3 b .. It It 0.09 !.92 e 22 (1, Gem A1V 0.11 0.12 a 23 <& PeA . (A3V) 0o'f1 o.67 c 2h 78 Vir (.A2Vp) (0.11!) 1.78 t 25 lliArl (AUV) ( 0.1 h} 0.07 a 26 a too A3V 0.19 0.29 • 27 15 tllltA (A3Vp) (0.19) 0.11''1 a 26 tlr~~;) (A3IUp) (<>.19) 0.141 Ill
29 21C:cm (A3VP) (0.~9) 0.51 l!\1
30 {~~~5} ASp (0.28) f.oo t 31 " 0l)h A5III o. :?9 o. 31 4
'* .. .. o.6o I 31 6Jl '1'u (A!IIV. 0.}3 0.57 Ill
33 n Lyr A6Ia (O.hO) 0.59 L!2 h 311 a. Aql A7V o.112 o. 4 .. .. .. o.63 I
i i .. .. •• 0.78 Ill
Jlio. Star lpl!letrum GNdl. !llftt lo~.t w• loe;W lito tea
3!> ,ll era :JOVIl" o. ,., ' o.€37 lf'.l,~ • 36 113 'fll'A rav 0.61 0.90 '·'2 .. 0
.
31 Hy1 ~ o.65 0.93 ~ \:!. Cll '
38 \1, OIU. :r5IV o.85 0.96 It .. '* .. 1.07 L.1i h .. .. .. 'f.6; 1 .. .. .. .e...u c .. .. .. 1.02 f. "10 k
I .. .. .. f. '11 l .. " .. 0.96 Ill
' 39 II. i'IU' J'!')Ib o.sa 1.1} f.71 k .. .. " f.63 1 .. .. .. I f.52 ft
ttO j,,oep 11.t.tJ lrii o.S~o ! '. 7!! 0
h1 I~ Oyg (J'5V} (0.90) t:ro 0
hi It~ :lqtJ. (J'5Y) (0.90} 1'.75 0
hb !
Pup P5Ib ( o. <)b} f.63 1 If It 5 I~ Wa P6UI 0.97 '1'.55 1
h6 l(k mu. J'8Ib 1.01 f.91 l
1!7 C& li'6V (0.98} I
1.11 0,:?1 c 1!8 '{ Lop l!'6V 1.0] s..a c Itt lUll oov 1,1 I 1.2.0 o.os q
" If .. 0.05 1
" " .. 1.19 o.a h 50 r on wan 1.11 1. 31! 1'.95 k
" "' .. 1.118 o.o6 h
" .. .. (1.93) l
51 >. Ser GOY ( 1,15) 1.~1 r 52 II Ot.ta f8Ia 1,17 1,28 .2a11 0
53 !!' .!l.u.r !'Oiap 1.20 o.79 L.:l! h
5ll ~ tma (OOV)Vu 1.21 r.:n l
55 Ill B;vi GHV 1.21 .2:lll ;)
56 ~ lhto G5Y {1.:511) (Os 12) l
51 ;v Rer G!)V (1. }h) 1'.99 :1!'
58 I; Gem (l0Ib 1 •. 39 "·"e ~ h
59 {0fr~·} GSV (1.h5) o.o9' '!"
60 1:0 Oet G~V ('l.tx5) 0.()8 I'
61 • I'ICep M i:rJ 1JJ6 1,82 0
62 ~ :Peg G2U-II 1.53 ;.~1 :t" ,, .a Aul:" ®III 1. 511 .tb.U h
Not1111U a .. 'f. Dttaham Jr.; s.:s. 858 (192$).
b ... L.s. Aller; All!. J. Ji, 321.(19h2J.
• .. o.:~~. Pa;vn.o; u.:s. 858 (HJ21). 4 • O, Struve, O.». Higgs; Ap.J. J!, 131 (1929).
... 35 ..
lfoteat e ... W.,l'h.teom~~be; Ap.J,. JJ.lt.• 13 ( 1951 ). t "' r .H.All~»l'J t.p.J. 106,715 ( 1 9ll7). 1 ... A. dill vauoollleurl!ll; unpublhhet'l.
h • lil.G.Will1au; i'llol.A .• S.P• h!,113 (1936). 1 ... J.L.GreerusteinJAp.J.10l•~ 51 (H~M!).
k - K.O.Wt'ight ;Publ.lmll:l.n~on Obs. \"'II No.1. l - A;PI!UU'I.il!lkoek; " '• " VITI No.5. a .. J.L.GreensteiniW.A.fU.ltn•r; Ap.J • ..19!,265 (191t8). n .. H.n.s·teel; Ap •• J. 1.2!•'•3 (·19115). o - A .• Parmekoel!:,G.lil. VlllnAlbttd!l ;Publ. A11'18terdl!l.a, J!.lo. 6. p ... K.li!.Pett'ie; Pttbl.Doainioll Obll. V'II No.16.
q ... G.ll'.~f.Mulders; 'l'he!ll.18 1 Utrecht, 193h.
r • ..._.r. L. G rllllillnill tEt.in; unpubl:l shed.
- 36 -Table 9. Bt~!illlllt' I!! tl!l@l!lph!!i!'lll• e
0 .. o.6. log g " 3.5
T e log p leg p 111
o.oo o.6oo o.ol 0.598 1. t'l. 92 o. 0.596 1.50 1.07 0.1:)3 0.593 1.61 1.17 o.oh 0.591 1.69 1.2b o.os 0.539 1.75 1.29 o.o6 0,587 1.80 l •. :5h
o.o7 0.585 l.flh 1. 3!1 o.oe 0.583 l. l.hl 0.09 o. 5~l1 1.91 1.hh 0.10 0.579 1. ':lh 1.h7 o.u 0.576 1.99 1.52 o.122 0.575 1.99 1.52
r~Hlia the mo4E~1 a dill I:!!!! tt c model 7 a loB!;p log Pe 7:' e log p loa: Pe
o.lh 0.572 2.03 1.56 o.1h 0.572 2.02 1.56 o.u 0.569 2.06 1.60 0.16 0.569 2.06 1.60 o.u 0.565 2.09 1.61! 0.18 0.567 2.09 1.63 0.20 0.562 2.11 1.67 0.20 0.565 2.12 1.66 0.25 0.55h 2.17 1.71, 0.25 0.560 2.17 1.71 o. 50 o. 5t:7 2.21 1.79 0.30 0.556 2.22 1.77 0.}5 0.5h0 2.25 1.!"11! 0.35 0.553 :?.25 1.81 o.t,o 0.533 2.27 l.fl& 0.1!1) 0.550 2.28 l.Sit 0.1,5 0.527 :!!. :50 1.92 o.1,5 o. 5t•7 2.:51 1.87 o.so 0.522 2.32 1.95 0.50 o. 5ll5 2.311 1.90 o.6o o.51l 2. }6 2.01 o.6o o. 5ltl 2.38 1.95 0.70 0.501 2.M) 2.05 o.1o 0.538 2.M1 1.99 o.eo o. '•95 .21t} 2.09 o.eo 0.535 2•''' 2.02 0.90 o.t:i5 2.H5 2.13 0.90 0-5:53 2.!,8 2.05 1.0 o.h77 2. hi;~ 1.16 1.0 0.531 2.51 2.01 1.2 0•'' 61! 2.55 2.:u 1.2 0.527 2.55 2.13 l.h 0. 1 52 2.57 2.26 l.lt o.52tt 2.59 2.17 1.6 o.hh2 2.61 2 •. ~0 1.6 0.521 2.62 2.20 1.1'1 O.h.33 2.65 2.3h 1.8 0.518 2.65 2.23 2.0 o.M!It 2.68 ~~. 38 2,0 o. 516 2.67 2.26 2.5 O.h06 2.76 2,1!6 2.5 0.512 2.73 2.32 3.0 0.392 2.83 2. 52 3.0 0.508 2.17 2.36 3.5 0.319 2.89 2.58 3.5 o. 501! 2.51 2.hO ,,.o 0.369 2.9lt :11. 61! ,,.o 0.502 2.!'1h 2.1tll lt. 5 0.360 2.99 2.69 ••• 5 O.lt99 2.87 2.117 5.0 0.351 :;.oh '2.13 5.0 0.!,97 2. 2,50 6.0 0.337 3.11 2.81 6.0 0.1,93 2.9h 2.5h 1.0 0.326 3.15 2.88 7.0 o.M~9 2. ~58 s.o 0.316 3. 2'' 2.9h 8.0 o.M36 3.01 2.62 9.0 0 •. 307 3.30 3.00 9.0 o. "'~" 3.011 2.65
10.0 0.300 3.35 .3.05 10.0 0,1,82 07 2.68
- :.;u -
~?a.blli$ 11. '1.4odel !'lte:UIIIr !!t!!ll:llll]lh<!t>O. e0,.0.71 lo!'· g " 1,.5 '7::' e 1os P p
o.oo 0.700 0.01
0.02 0.03
0.01!
o.os o.o6 0.07 o.oa o.o9 o.1o 0.12
o.697 0.695
0.16 0,18 0,20
0.25 0.256
o. 0.690 (}, 688 0,685 0,66.3 0,680 0.678 0.676 o.672 o.667 0.665 o.659 0.656 o. 6116 o.6•dS
rt~.diatv1e model r e log p
o. 30 0.638 3.76 o. 35 630 3. 78 O.hO o. 3.1'11 O,h5 0.615 ;\,82
o. 50 o. 609 3. !lh o. o. 596 3. 86 o.1o o.585 ;.ee o.eo o.575 3.89 0.90 o. 3.90
1.0 0.557 3.91 1.2 0.5hl 3.92 l.h 0.528 3.93 1. 6 o. 516 .3. 9h 1.6 0,505 3.9h
2.0 0.'•95 3.911 2.5 o.~t71• ;,95 3.0 O.h57 3.96 3.5 o.l.h3 •~.96
h.O O.h30 3.97 ''• 5 O.ld~O 3. 91 5. 0 0,1,10 3. 97 6.0 o. 391! 3. 7.0 o. 380 3.99 e.o 0.369 ,,.oo
2 •. 31 2. 37 2.h3
2. 2.62 2,70 2.77 2.81!
2.96 3.05 J,lh 3~20
3.25 3.35 3.bh :;.;l .3.56 .3.60 3.62 :;.65 3.67 3.6&
9.0 10.0
0.359 '-.01 10 o. 350 h.02 3. 71
5.12 .':i. :;. 29
;. 311 3 •. 39
'·' :s },1·7 .3.50 .3. 55 :;. 59 3.62 :;.65
67 3. 72
3. 73
1,16 l •. 3h
1.h3 1.50 1.56 1.61 1,65 1,68 1,72 1.75 1,1!1
1.86 1.91
1.95 1.99 2.08 2.09
7:
o. o. 35 O,I,Q
(),I,; 0.'50
o.6o 0,70
o.eo 0,90 1.0 1,2 1,1!
11.6 1.1'1 2,0
2.5 ;.o 3.5 t•.O
''• 5 s.o 6,0
7.0 a.o
10.0
··•
ad is btl tic model e log P lol Pe
0.6313 0.628 o.62h 0.620 0.615 o.61o o.6o6 o. o.Goo 0.595 0.591 0.587 o. 'jJ;\)!t 0 •. 582 0.576 0.571 0.565 0.5611 0.562 0.559 0.555 0.551 0.51·8
').16 ::.16
3.1!1
3.83 3.fl5 .3.68 3.90 3. .3. 91• 3.96 3.99 h.02
!; • Oh , •• o6
h.01
"·ll h.lh
,,.16 ,,.19
''· ,, • :!'2
••• 25
''•28
'•·30
2.22 2.21 2. 31 2 • .3h ;
2.110
2.1t5 2.h9 2.52 2.55 2.61 2.65 2.69 2. 72 2.75 2.61 2.86 2. 2.93 2.96 2.99 .3.011
'I '"" ·"'· '!..J/1;;,}
o. ,,, 5 "· 3.111 0.51•3 h.Jh 3.11
- ./:1 -'fable 12 • • I!Hlel l!i tell a r a tmoaphm.-e. a
0 .. o.a, log g "" 3.5
'T e p log Pe o.oo 800 o.o1 0.191 :1'.66 '' I 3 ., ... '!
o.o2 o. 79'· 2. 56 03 791 5.01 o.65
0.01, 7M 3.09 0.71 0.05 o. 3.16 0.76 0.06 0.783 :;.n o.Sl o.o7 0.780 3.25 0.85 o.oe 0.778 ~.29 89 0.!)9 0.775 ,;. ,~,~ 0.92 0.10 0.773 '· 311 0.96 0.12 o.768 3.39 1.02 o.lh 0.763 3. ~13 1.07 0.16 0.758 5.t.tS 1.12 0.18 o. 751t 3.h8 1.17 0.20 0.7h9 3.51 1.21 0.25 0.739 3.55 1.31 0.30 0.729 3.58 1. ,,o o. 3'·6 o.721 ., 61 -· l. hl!l
radiatvi«t model llld1sba tio modttl T e log p log Pe "T e log p log p!ll .
o. 35 0.720 3.61 1.h8 o. )5 o.no 3.61 1.M! 0.110 0.711 3.63 1.55 o.ho o. 712 3.63 1.55 o.~>s 0.70.3 3.65 1.62 o. ItS 0.706 }.65 1.60
0.50 0.696 3.66 1.69 o.so 0.701 ~.66 l.65 o.6o o.61l1 3.68 1.80 o.6o 0.693 3.69 1.72 0.70 o.669 70 1.91 0.10 o. 3. 71 1.78 o.ao 0.657 3.11 ::>.oo o.ao 0.681 3.73 1.83 0.90 o. 61.6 3. 72 2.09 0.90 o.677 J.71t J..87
1.0 o.636 ::;.n 2.16 1.0 0.67.3 3.75 1.90 1.2 0.618 3.7h 2 •. 30 1.2 o.667 .3.78 1.96 1.11 0.603 3. 7'· 2.1.2 1.1. o.661 3.80 2.01
1.6 o.st9 3.75 2.52 1 .• 6 o.657 3.81 2.05
1.8 0.577 3.75 2.61 1.8 0.653 }.83 2.09 2.0 o.s66 3.76 2.69 2.0 0.650 3. 8!; 2.12 2.5 o. 51•2 3.76 2.86 2.5 o.6h3 }.87 2.18 ;.o 522 3.71 2.99 3.0 o.63S 3.89 2,21!
3. 5 0.506 3.17 3.09 .3.5 o.6:n ~.91 2,28
h.O 0,1,92 3.71 3.17 h,O 0,629 3.93 2.32
r...s 0.1179 3.77 3.23 "·5 0,626 3.95 ,.:35 5.0 o.H'59 3.77 :~.28 5.0 0,623 'I 96 ..,,-. 2.38 6.o 0,1!50 3.77 3.35 6.0 0,618 98 2.1!3
7.0 o. h31t 3.11 .3.39 7.0 o. 6111 It, 2.h7 e.o o.M?l 3.78 :3. h2 s.o o.6lo h. 2,51 9.0 o. h10 3.78 ;.1111 9.0 0,607 h.Oh 2.5h
10,0 0,1!00 3.78 3. 1t5 10,0 0,605 h.05 2.57
Tablf; 13. ( [,...... -
t#o4t~~l 11'1 tella :I' 111 tmoaph e:l'l!l• e0,.o.e, log g " ''·' T' e bgp log Pe
o.oo O,Ol 0.797 }.hl o.e2 o. 7'111 3.62 0.9h o.o:s 0.791 3.7:5 1.02 o.oh o. 788 ,,3.81 1.0& o.os 0.786 3.87 l.lh o.o6 0.7&3 .3.92 1.18 0.07 0.7&0 3.97 1.22 o.os o.na h. 1.26 o.og 775 ''·03 1.29 0.10 0.773 h.06 1.:n 0.12 o. 768 ,,.lo 1.38 O.lh o. 763 h.l!. l.hJI 0.16 0.755 ''•17 1.1<9 o.u o. 75h ''•19 1.5:3 0.20 o. 7H~ ·-!. 1.!)6
0.25 0.739 t..26 1.68 0.30 0.129 ''• 30 1.71
0.35 0.720 "· 32 1.1:15 0.1.0 0.711 11.35 1.92 o.1,39 0.705 lt. 36 1.96
!Jl!ildil!lt:l;l!'f.l IIIOt'if.ll aditlfiHl th model < a log p los Pe ?:' e log p l.Oilfl ·
O.h5 0.703 ''· 36 1.99 o.H> 0.703 ''• 36 1.99 0.50 0.696
"· 38 ".05 0.50 0.696 ''• 38 2.05 I
0.60 o.681 h.hO 2.17 o.6o 0,686 h.ltl 2.lh 0.70 o.669 h,h:? 2.2B 0.10 0.617 h.h3 2.21 o.so o.657
''· hh 2. 31 o.so o.61l tt.hl~ 2.27
o.go o.6Mi h.h5 2.Mi o.go 0.665 h.hEi 2.32
1.o 0.6;6 ,, • 1!6 2. 51! 1.0 0,660 h.b7 2.36 1.2 o.618 11.1!7 2.68 1.2 o.653 h.h9 2.1t3
l.lt 0.603 J~ohB 2.80 1.11 o.6Mi lt.5l :?.ht 1.6 0.589 '•·119 2.:n 1.6 0.61.1 h. 53 2.53 1.8 0.571 11.119 ;.oo 1.8 0,637 h.5!! 2.57 2.0 0,566 h. 50 3.09 2.0 o. h. 55 2.61 2.5 0.5h2 ''·50 3.26 2.5 o.625 11,58 2.68
;.o 0.522 11.51 3.111 ;.o 0.619 tt.6o :?.?fJ
3.5 0.506 h. 51 3.53 3.5 o.61h h.!!i:? 2.79 , •• o o.~t92 h. 51 3.62 h.O o.6lo , •• 6; 2.83 ,, . ' 0.1.19 ,, • 52 3.71
''· 5 o.606 h.65 2.86
5.0 o.h69 ~>.52 .378 ;.o o.60:? h.66 2.90 6.0 o.tt;o h. 52 J.89 6.0 0.597 ·~. 2.95 1.0 o.tt3h ''·52 3.91 1.0 0.592 11.70 2.99 e.o 0.1!21 "·52 lt.OJ s.o o.5tt h.72 .3.03 9.0 o.1110 h. 52 lt.08 9.0 o. 11.73 3.01
10.0 O.hOO '•· 52 h.ll 10.0 0.582 ''·75 3.10
- 1\1 -'fable 111. MOdfl!l llltellar a tmo~&t~h'lllrtl. e
0 .. o.9, log g .. b.
- ,12 -'l'abl~t 15. Model stellar •tmell!pheN. e
0 .. l.O, l'~S s .. 3.5.
7: e log 1> log Pe o.oo 1.000 o.o1 0.996 3.37 1.66 o.o:c 0.993 3.5h !.so 0.03 o. 3. 611 l.es O.Oh 0.986 .3.10 -1.95 o.o, 0.982 3.76 0.01 o.o6 0.979 .3.80 o.os 0.07 0.975 3.Bh o.ot o.os 0.972 3.l~1 0.12 0.09 0.969 3.90 o.15 0.10 0.966 3.92 0.18 0.12 0.959 3.96 0.23 O.l!. o. 9'5.3 1·.00 o.:n o.l6 0.9M~ ,, • 03 n.31 0.18 0.9!.2 '•·05 0.35 0.20 0.937 t..oe 0.39 0.25 0.923 t•.l3 O.h7 0.30 0.911 h.l7 t; 55 ..• o. 35 0.900 lt.20 0.62 o.t~o o.M9 11.22 o.69 0.~>5 o.879 h.25 0.75 0.50 0.869 tt.27 o.Bl o.6o 0.1'152 ... 30 0.93 0.70 o.s:;6 h. 1.01! 0.702 o.a~;5
''· 32 1.05
!'ad1&t1Te model adiatullt1o model 7: a hip log Pe 7: e log p lq Pe
o.ao o.e21 It. "'''
l.lh o.ao o.a22 '~. Jh 1.111. 0.90 o.sos !:. 36 1.21· o. 0.811 ''• 36 1.22 1.0 0.795 ''· .31 1.33 1.0 o.so2 11.37 1.29 1.2 0.;773 ''· 39 1.50 1.2 0.768 ". 3!il 1.110 loll 0.751· h.bO 1.61, l.l, 0.717 "·''l l.Mt 1.6 0.736 l~ohl 1.77 1.6 o.76f! h.M! 1,.55 1.8 0.721 ,, • ·~ 2 l.llg 1.6 0.760 ,, • hh 1.61 2.0 0.107
''· h3 2.00 2.0 0.75'• ''· h5 1.67
2.5 o.677 h.bf, 2.22 2.5 0.7hl ''·b1 1.17 :5.0 o.653 !t. "5 2.M) 3.0 o. 731 h.h9 1.85 3.5 o. ''·"5 2.56 3.5 0.722 1!.50 1.92 h.O 0.615 ''· h5 2.70 , .• o 0.116 I 5" ,t. ft. 1.98
"·' 0.599 h.Mi 2.81 ''· 5 0.710 ~..53 2.02
;.o o.;M ''· hb 2.92 ;.o 0.705 '•· 5h 2.01 6.0 562 11. M> 3.09 6.0 o.696 ~t.56 2.1!1
7.0 0.5h3 ~..Mi 3.2fl 7.0 o.690 '•·57 2.20
8.0 0.527 h.h6 .3.35 s.o o.6m, ''·" 2.25
9.0 0.512 ". h6 3.''' 9.0 o.679 '··60 2.29 10.0 o.soo h.MS 3. ,,, 10.0 o.67h ,,.61 2.33
- h3-Table 16. Uod~~tl t:!tel11u:• at~spb~re. 90 wl.o, loB 1
., "'·' '!" e l€11 p log 'Pe
o.oo 1.000 o.ol 0.996 3.92 0.12
o. o. h.08 o.::n 0.03 o. '·.18 0.35 o.o,, 0.985 1•.25 O. M:>
0.05 0.91'12 ''• ,30 0.1>7 o.o6 979 "· 3'• 51 0.07 0.975 h. 0.55 o.oa 0.972 h.hl 0.58 o. o. :~69 "·"'' o.61 0.10 0.966 h.M) o.6h o.12 0.959 ~>.50 o.6e o.1r. 0.953 '•· 5h 0.72 0.16 c.9h8 ''• ,57 0,76 0.18 0.9h2 ~..60 o.eo o.:co 0.9,37 ••• 0.1'13
0.25 o. '·.67 0.90 0.30 0.911 ''• 72 0.97 0.35 0,900 h.75 1.03 o.,.o o. ~ •• 78 1.08 o.h5 o.879 ''· 81 1.U o.so o.M9 ·~- ~.3 1.19 o.6o o.f11,52 1!.86 1.30 0.70 0.836 '•· 1'19 1.39 o.so 0.821 "·92 1.119
0.817 o.BU 11.93 1.56 radimtive mod0l adiabatic ~del
'T e log ·p log p6 7:' a lee p 101 1>
0.9 o.aoe h.9h 1.58 0.9 o.aoe '•.9h 1.58 1.0 795 ''· 95
1.66 1.0 0.797 1•.95 1.~
1.2 0.773 '•.98 1.52 1.2 0.779 ,, • :J8 1.78 l.h o.751• ;.oo 1.96 l.h 0.766 5.00 1.88 1.6 0.736 s.o1 2.08 1.6 0.755 5.02 1.96 1.8 0.721 5.02 2.20 l.S 0.7h6 ;.o:s 2.81 2.0 0.707 ~).()3 2.30 2.0 0.739 5.0h 2.03
2.5 0.677 5.05 2.53 2.5 o. 721• 5.()7 2.20
3.0 o.653 ;.o6 2.12 3.0 0.113 5.09 2.2~~
3.5 o.632 5.06 2. 3.5 o.70h 5.10 2. '51 1•.0 0.61'.5 5.07 .;.ol h.O o.696 5.12 2.113
'··5 0.599 5.07 3.13 '•·5 0.690 5.13 2.h8
5.0 0.586 !i-08 3.21' 5.0 0.685 5.111 2.53 6.0 0.562 ;.oa J.hh 6.0 o.675 5.16 2.60
?.0 o. 51,3 5.0!'! 3.56 7.0 o.668 5.17 2.67 e.o 0.521 5.08 3.68 e.o o.661 5.19 2.12.
9.0 o.su 5.09 3.79 9.0 o.656 5.20 2.11. 1o.o o.soo 5.09 3.88 1o.o (J.651 5.21 2.81
- IU~ -Table 17. 'MOd$l etllillt~~t< atll\os;pb(~H. e
0 .. a.o, lo~{ g ,. *·.5
'fl'i th blJJU'tket:l.ng e:t1'eet.
T a log '!) loii p9
o.o 1.000 0,01 0.996 3. o.t'9 o.o:<> 0.993 l· .O!t 0,2fl
0.03 o. !.t.l'• 0.32 o.Ol· o. ,,,21 0.36 o.os o. 1!,26 o.,<~
0,06 0.979 h.30 o. !til 0.07 0. r75 "· 311 0.52 0,08 0.972 h.37 0.55 o.og 0.969 h. 39 0.58 0,10 0.966 1•,1.:? o.6o 0.12 0.959 ".It!) 0.65 0.111 0,953 h. 50 0.69 0.16 0.9h8 h. 53 o.n 0,18 0.9h2 1!.56 0.76 0.20 0.931 ''·58 c.so 0,25 0,923 '·.63 O,lil7
0.30 0,911 *·.67 0.93 0.35 0.900 h.71 o. O,Ml 0.989 ,, • 7!l 1.011 0,1!5 o.H79 h.76 1,10 o.;o 0,869 '•.79 1.16 o.6o 0.852 h,82 l.l?7 0,70 0.~36 11.85 1.36 o.eo o.sn 11,87 1.h6 0.90 o.soa 'l· 1.55 1.0 0.795 1..91 1.61t 1.2 0,11?:. 1t.93 1.79 1,1, 0.751t 11.95 1.93 1.6 0.736 h.96 2.06 1.8 o. 721 1>.97 2.11 2.0 0.707 1!,91:'1 2.28 2.; 0,671 ;.oo :?.;o 3.0 0.653 5.01 2.69 3.5 o.E;15 5.02 2.98 h,O 0.615 5.02 2,91:'1
''•; o.599 6.02 3.10 5.0 0,586 5.03 3.22 6.o C:,562 5.03 .3 •. ~9 7.0 o. 511.3 5.03 .3. !Jh e.o 0.527 5.0h .3.67 9.0 0,512 s.oh 3.76
10.0 0.500 ;.o.,, 3.1:'16
