hc3.seikyou.ne.jp · pp.45'479(19 g) l. Introduction W sball investigate the Cauchy problem On a...
67
On Mathematical Models of Gaseous Stars Tetu Makino (Prof. Emer. at Yamaguchi Univ.) November 7, 2019 / Mathematical Science Workshop in Yamaguchi 2019 1
Transcript of hc3.seikyou.ne.jp · pp.45'479(19 g) l. Introduction W sball investigate the Cauchy problem On a...
On Mathematical Models of
Gaseous Stars
Tetu Makino (Prof. Emer. at Yamaguchi Univ.)
November 7, 2019 /
Mathematical Science Workshop in Yamaguchi 2019
1
(EP) Euler-Poisson equations
∂ρ
∂t+
3∑k=1
∂
∂xk(ρvk) = 0, (1a)
ρ(∂vj∂t
+
3∑k=1
vk∂vj∂xk
)+∂P
∂xj+ ρ
∂Φ
∂xj= 0, j = 1, 2, 3, (1b)
ρ(∂S∂t
+
3∑k=1
vk∂S
∂xk
)= 0, (1c)
Φ(=
3∑k=1
∂2Φ
∂x2k
)= 4πGρ. (1d)
2
(1d) ⇐ Newton potential
Φ(t,x) = −G
∫ρ(t,x′)
|x− x′|dx′ (2)
3
(A0) Equation of state for ideal gas
P = (γ − 1)CV ρT, (3a)
P = ργ exp[ SCV
]. (3b)
1 < γ < 2 (4)
γ = 1 +2
f, where f = degree of freedom of a molecule, = 3 for
monoatomic gas, = 5 for diatomic gas, e.g., air, etc.
4
(A0E) - isentropic: S = Constant, P = Aργ
(A0T)- isotheormal: T =Constant, P = Aρ.
(A1) - general barotropic:
P = P (ρ), 0 < P, 0 <dP
dρfor ρ > 0,
P = Aργ(1 + Υ(ργ−1)),
Υ is analytic near 0,Υ(0) = 0.
5
Equation of state for white dwarfs
P = A
∫ y
0
q4dq√1 + q2
, ρ = By3
⇒
P ∼ A
5B5/3ρ
53 as ρ→ +0
P ∼ A
4B4/3ρ
43 as ρ→ +∞
6
pp.45'479(19“ )
l. Introduction
W sball investigate the Cauchy problem
On a Local Existence Theorem for the EvolutionEquation of Gaseous Stars
By Totu MerrNo
,160 t,. M^ur,
(4-i) ttl,..n-:r,'t ti ,,7,\.Harc K, 7, f, orc positive collliltDtJ. l.irc rrlkoowns arc rlrrl'(t, x\, u '(t,, u.-, t'"1 , t(r, rl, p ilt, \\ kt,lt, \) , ,/, Olt, \l I
'(rr,r,,ru)€nr, whilc po-p.(r) aDd 2,r ,i(i) are initial Arr,,eslablish thc existcncc of a solution p(t. r), n1,, xl, ?1, x), $(t,10, fJxnr-((/,r)lOat=7, xe /t,l for givcn pd anrt zo.
Equations (l), (2), anri (3) dcscribc thc evolutioo of a sr,rriscntropic idcat gcs with sclf-gravitarion. Ihc varialrlc n rnc.of.thc gas, ? tho prcssure, .y' the Newtonian gravitational potcvclocity. EquatioD (t) is the cquation of continuity, arrexpress the conservatiotr of momeotum. ln this papcr tlre sysiDg of (1-0), -l), -2), -3) wiil be called thc Eulcr cquariou.thc equation of state, I bcing the adiabatic cxponcnt. wc kcf--513 it the stcllar matcrial is treated as a monostomic Aaradiation prcssure is taken into account and supposcd to beother valucs of f havc significanccs of their own i*c Il], Cha;or [6], Section 53)" Equation (3) is poisson's
"q,,rrio,,, , ,
constant of gravitation. 'I'he solution of (l) of physical inrcr,by Ncwtonian potcnrial:
( 3 )* ,1,(t, x\--rl t:!!:lt-,1r,)EA lr-yl
Fo|a dctailed discussion in astrophysical contcxts o[ thc (.:ru((4) we refer to P. Lcdoux and'fh. Walravcn [6].
An existence thcorern o[ thc problem will tc cstlll]lishcil ,ing linc: First, taking an arbitrory p(r), oot neccssarily a r,gratc Poissoo's cqrrarion (3) by (3)* to got rhc potenrial ,!r(r)we intcgratc tbc Iiulcr equrtio. (l) togcther with (2) with rclgiven potential q1(l) undcr thc inirial coiditio[ (4) to ohuirdistributioo /(/) (Scction 3)t il l(r) coincidcs with rhc lirsl ptrc a solution of rhr: problcor ( t ) (4)_ Thc final work wiil bc rilhe fixcd pQint lh(:r)rcrn (Sccli(tn 5). A rnotlcl ol.this;rroccrir,in thc study ol Vlusov's rqulrion by S. Ukai and '1.. Ohatr
Whcu wc pr()grcrj in this wly, wc ucejl with tlillicultv rrol tlrc Iirrlcr r:(llr,rtr,,rt. I lrc :.trrrrr|rrrl rnrrllrcrtrrlielt trcrtrrr,cqriti()r iri lo Irtn,il'o'(t it t, a symnlrtric hypcltrolic s1lrrrctlriclr's tltcrrry is rlrlrliralrlc. Wc rrc tc(lurintcd wilh sr(Klairrr.rrrr;rrr io(l A. Mirlrlr i5l, ll thcit:itrr(ly, !lrl in illo::lrr;ttlrcrnrrlicirrrrt r:lrlrli:lrrtrl, rrgororr:l llrcoric:; on Jlilitl tivnrrrlllr (l( r'.rtv /, ,)l tlr. llrrr,t lr.nr,Ilr.. t.\ ilr.rt,,rr/. :t lxr.,ltrvc (,,iltltlrrlglr,,rrl Ilic wlrol|i11f,.a ,lIfi0& thc rrrotiorr. I Iowcvct, in rr'i,rl.il tlr( (lrr!.i!t,r' l.i rrl!ftlrrl l0 lrav,1.(rIllx(r1 Xil|1lr)rt
Abslrct. Thc equatioo ofthe hydrodyoamical evolutiotr ofso adiabaticgascous star, which coosists of the compressible Euler cquatioq coupled withPoissoo's cquatio[, is waitiog for matbematical treatises. Alfhouih uoyintercstiug studics from thc ouocrical poitrt of vicw supported by the devclop.ocat of big computers caa bc fouod aooog asftophysicists, rigorou Dathe_matics fiavc oot 6ught up with theo. Thc euthor h6 heard rotbiDg oboutevon the proofofthe existo[ce ofsolutio[s for thc associatcd Cauchy prcblcE.Now this article is devoted to cstablishitrg thc existcoce of local aolutio[surder a suitable cotrditioo oo the iritial data. Thc discussio[ is along astaodard lioc cx@pt for the crucial diffiqlty ofiotcg(atiDg ths Eulcr equatioofor solutiotrs with coopact support, which does aot alpear io the hydro_dyoamics ou the Earth. flowevcr thc r6ult is too rcstrictiw so thai tucauthor hopes that hc has prsetted a tcntativo trcatise iaitiatilg the furthcrdcvelopmeqt of oatbcoatical tficory on this iotcrcstitrg equatio,
, Key rmds; Cauchy probleG, quasi-liucar hypcrbolic systems, hydto-dyoamics, self-gravitatiog systems, astrophysis
(1‐ 0)
(1-i)
(2)
(3)
(牛 0)
子+Σ″[券十Σttρ=0,
ρ[争
+』:ク=警]+争
+ρ欝=0(′ =1,2,3),
ハ″‐4F″″,
―
鰤醐螂 A",,t鰯 ,
0l,.reF .
7
(EE) Einstein-Euler equations
for the metric ds2 = gikdxidxk:
Rik − 1
2gikR =
8πG
c4Tik, (5a)
T ik = (c2ρ+ P )U iUk − Pgik. (5b)
⇒
∇iTik = 0 (6)
8
″
844s饉 uig、、.1流=sl譲
軸 .口
"轟e餞腐摯唸inl K議薇,vnl 1,V“
““r i916
醜()FoldgLeichn昭嘘 繭 G轟鸞t雌.o麗。
trroa A. th.lsrutr.
11:zw(i vor k、戦霞ぬ く,ぉ轟ienenell ttt幅 1,・ )騨 :ょ:LJ“ lol騨協・
:綽,wi・
mm 2u relaJCidil 伊ヽ1lCr O鵬蟻tauOn鮮lm撃ユ:am,dtt dem P"m‐
1麟 3雛meinO Reb i宙機t entw" hen,d,卜・ale l :h腱=誠:『mChCn
F"駅m3bdiebigeli SttbStttu爺 onal der Rユumttit,優 戯ュbein gegetiber k`ン
vtt"飩、t smと、
Der Eatwiよ 111 3響"gw=dttd郎 伊11 為 l:回:st価(1越 1
(議 eicI“ in.。11,17elche die N`w蟹 ,N8,“ .Thα〉ri。 :』s Nttlenm=enthalmll
w,d bdiebigen suヽ"極
ti |、ul vOn der]発饉 腱in“れ :絆騨nibeF k‐
マ鑽抽餞 Wttn.菫 ie腱ピ 伽ユd1011,遇 凛い0“ Oleidl, 組1ポ雌0111
ovユ五離 範おpFeChen,1 1騰 t101 SL 1‐ des EnerJe“ nsて,郵 (lel・ 。監ト
"He“ve"。1,wildet. Iλ s Koo薇nぁ糠nwstOm wtt a“ln鱒h`bF ett・
伽山ol hgd zu叩“
:虚饉榊m,姦3/ニラ 劉 :騨鯉 hC輌“1,WdⅢ Ch
こc Gleio悪齋ge,1(lo‐ rheOri`ヽ eme et減 m :l機 VOFOittdiing C,偽 Ll“n.
D3湊i mtiStO tter,宙 erWttnち die IIypOtll‐ in蓼)豊tt WO過 en,
(la (ler shtt aes hlergic崚"澪
ょor ttaterle vo"chwin(1・ .
Nee贅h=1=s ndo icll nu".(1 31腱 oLlle Hyp tllese tlbcr山 :l
ド.citic“I"ur deri瞼轟.:ne ausL。 1嶽inc:l kmm,wenil t額l(len ttlergi゛
海:“rd"Matc轟。 :.。静 as mdemr wcis。 温 と.rd(聴LiChun3( 1
01餞鳳 ,:ふ i_11l ineitirn bd`lぬ 鋼 leru mttim響 l騨曇 elt籠
,ot. Di(,Fcld81old〕 頓18〕i ftF磯 Vtttu|, :山f Wdeb 綺]l diO E"
kttrはgお,PeHhelhweg叫lg des Mer ur g螺壼ndet l遷秘 ,1)teibm vo,t
il艶
“r扇引田鵬搬o bo戯11お loh絆
“
hier ix"よ 画山戴3 dio騨 l口e13‐
嫡 chtu鴫,`hmitれ ,I`“er nicht ge,1輸 辞 i轟,a。 無le“::菫ittitunttn
un3 ,3α“'tzt ll
mzl:五 e1lo,1,
Aus acr i"瓢 ntt・L"嬌ヽヽのNK゛、′an譴 :鮨 ,vi針
"n路珊響い lelttt
u瑯1 6genれ K。_lmm zw:抽 :]LI弓 ,ぃ b:
GFぉ =ム.+S. (:)
f'*mt Db frei:Uenngo akt Hhff6 8aitlio allgueln tovrritntan zeh Oletolrungm ilee Grrvltrilonrtrd*
in &lomen, ln ilenem ,l[ehto. fbhltr er]dter v&, fnd6 sir rorc&a61. = o. (tl
Dteac GlaicLunpn lasBon doh aln&ohor gc.tdte, v€ml @ ilrrEcrugecyataur po *thlrt, 6"6 |E-g: I iet Droo vcrrchrinrfut &-ropn (rl), eo daB um strlt r) ortn&
*.:Tl#*]rrfrrr=o (s)
/ニラ=:.
・1=・―fr
く3→
」場一ゴII+幕 :l絆
島"=事智―尋勢‖
Dabd ist
軽)
gosctzt, welchrr GnlB€n wlr ds ilio 'Kompmentenr dos Gravitatio*e-feldes bezeichnen.
Iet ln dem bctrmhtahn Raume ,l[a*erta. vorhanden, !o trlttilaEc
oder
L`哺enSOr tt derコ饉 1機n難毎 VOn(2)b".(3)鳳.1職 ,c籟
1弓[=IIπ=ll瓦
~｀~~1,ぶt
ll ヽ “ ノ 、w`,bci l.、、__ ______一 ――・――
―
‐―ゃ
Fr"4.-!t: r G)
gcretrt lsti ? tst dor Strlar dsE Eae*gietolsotE iltr .ilataie., itto rrohteSoito yotr (ea) eln Teueor, Spedrlldoeo rrlr rt3tlor dag foodfarfto-dyrtnm tn iler gevoSnten Wdr6, so €Ualteil wlr en Stdle ro (ra)ille firlulvrlenten Oleio]rungur
a.u;$$*1r,r,rs,: -*r^-;*-r) ru,
-g = t. (srlfle gt6tt nehmen wlr ao, daB dle Dlvergenz il* Unctglot nrorr
iler Materle lm Shne iles dlgrmeinen Diftrn$dtelhub vuetUvtne(tmpulaenergieaa&). 8et iler Speztaltalenug ilcr Kooafrutenrrbl go-rni0 (3*) kommt dt* tl*auf hineur, ala0 rlie 4. rte Bdtn5egm
事:響 =― :F31雀:・ ,
早絆=-lr就
”
↓
13レ“"
1“ 1・.ヽ日V・ ド・778〔“ld Xl '1,S・ 7",1,`5・
(1→
erfrllen sollen.
Fir
9
r
ハ
(
t.R 、てん
1‐
|「 ・1・ | || : :.|‐ 1 111
ｒト
ト
´ イ・
工風ЛAH′理y凛島色ЛИΦΠIИЦ
彗
畿 `ヽ
L
手
ふ一壕一
酸 |,
2 絲Gら,0
辱 ioltr*rq* *rt
「
＝ヨ
Ｆ』Ｌ
1 Ъ f
10
Physikalische Zeitschrift der SoⅥ ′letuniOn/Soviet Union Vysshil sovetinarOdno90kh°Zra｀IStVa() ノ′や・イ′ ″。 2 rイ 932) _,ャ と:ゞ
`= 二.、:‐
ヽ
ON THE THE()RY o「 sTARS.
β〃L 五“
7r742′ r
(Recoived 7.Jalluaよぅ.1932)
Fl'o lt, 七lle thcol.Otical pO11lt of 、工le、、 the l)]lVSical llaturc Of Stcllt、1ctltlilil,1ltti11 lS COnsidered.
rrhe astr。1、1lysical nlethois usllally apl)lied ill attacking
tll° pl'ObleluS °f Stellar StruCture are CllaraCteriSed D`y lllalく illg
l)h5'Sical assunlptions cllosel1 0nly fο l' the sake of inatlle―1llatiCal COllVeniellCc. 35r tllls is cllaracterised, for iustatllce,
Nl'. N11lne's p oOf of tlle illlpOssibllity Of a star coslstjng
througllolたt Of classical ideal gasi this poOf rests on tl,e
aSSertloll that, 10r[trbitrary_ι al)d if, tlle funda11〕 ental e(lu a_ti° nS °f a Star C01lSlStin3 0f classical ideal gas adll,lt, 1,1
Sellert｀ 1, 110 rC3ular sOlut101). さ.lr. Iゝil n e seellls tO lla、 アe Ovel'―100ked tlle fa.Ct that this assertion results
。1lly fronl tlleaSSu lllptlon ()f opacity・ belllg cOustallt thTolig110ut tlle star,
・「hlcll assunll｀ tiOn is inarle Only∫ 01, 11latl〕 elllatical purpOses
a‐ncl has i10tlling tO (lo with reality. only in the case Ofthis asstlllュ i)tion the radius _R cllsappears frolll tlle
〕'elatiolll'Ct｀ Veell _こ,lf alld fi〕 necessary for regularity Of tlle solutiOD.iゝ 1ly reaS°nable aSSunlptiOnS abOut the Opacity
、vOuld le′ adtO a l'Clati° n betWeen L'ユ √ and P'WhiCll l'01at10n WOuld lDeqllite exelupt frolll the pllysical criticisllls pllt lor、
vard agalnstIEladington,s lnass_1.nlinOstty_TelatiOn
lt seenls reas01lable tO try tO attac]t the pl.01)lel1l of stellar
Sll. cture by nlethods Of tl100retica1 1)hysics, 1. e t0 1vesti‐
gate tlle p ll y Si C(｀ l llaturC Of Stellar equilibriuln. FoT that
,u]・pOsO、 ve lllust at first i〕 lvestigate the sta′ tistical cquilibriunl()f a givell lnass、、・ithout 801lerati011 01.energy,the cOnditiollifoT、ァ11lch equilibl'1llnl l)eing the intnilnum Of fl'ee ellolヽ
gy 「
(f01・ glvell tempel,ature). Tlle part Of free ellersy dlle t。gravitatioll is negative :Ind lllversely pro10ortional to some
11
(A1R)
P = P (ρ), 0 < P, 0 <dP
dρ< c2 for ρ > 0,
P = Aργ(1 + Υ(Aργ−1/c2)),
Υ is analytic near 0,Υ(0) = 0.
12
Equation of state for ideal neutron gas :
ρ = 3Kc3∫ y
0
√1 + q2q2dq,
P = Kc5∫ y
0
q4dq√1 + q2
P ∼ 1
5K2/3ρ
53 as ρ→ +0
P ∼ c2
3ρ as ρ→ +∞
13
ヽ
「
ヽ
___´~
´
‖ GtC
14
15
158 fa. 6. XonodHoe aeilLecr7c
$ 8. Ilgealsutrri nefirpouulrfi raa npH csepxssrcorori flrorHocrlr
B xacecrse nepBoro lpu5.,rrrlreHln, e nrouepcr;oii pa6ore On-neureriuepa a Bo.lxoea (i!135) ntr.a paccrrorpeH Herirpouuuii raaE npenno,rorieHtilt, uro ueiirpoHbt Hrir(ati He ssai,ltr,to;lelictBvtor r,te-xry co6oft. FlerirpoHsr rn.lircrcg Qepxuorialru;,ur,tr* o6paso",,BCq Teoplrq BblpoxleHHoto Hefirpouuoro ra3a oKa3btBaeTcg noAo6-uori reopi.ru BbtpoxleHiJolo 3rerTpoHHofo ra3a. llogo6ne :roI]]\eeT MecTo nocTo.rlbii\,, noclio.Ibty Mbt 3 KaqecTBe segasncullofinepeueuuori IroJ:b3yei\1c9 nJorHocrbn cacrriu lz il.!u, qro ?o xe,tr,,IorHocrbro noKofl p0. B uacrnocru, cel:l Qepuu-ulrnyJtbca I nB ToqHocru coBlraraer r",lq 3.eItTporoB H uerirpoHos:
). pi rln:2. s .a"rtr:3:; (6.8.1)
Haii,4eu xapaxrepHJ,io fl.tcrnocrb noroff, npii xoropori trporc.\oi1nlnepexoA or Hepe.irrrrBt.lcrcrioto K perqrrrEllcrcroxy uelirpouHourcpeplru-rasy: ]'cJloBtle't )
ДaeT
(ρ O)`~ 3
2ル3~ 5,25・ 1015 2ε ,`3.
06 o3HaqrlI 6e:pa:lrepnr ro n.rorHocrb norioq
__L__ャ(ρ O)`~~″ (6S.2)
Bse;lelr 6e:pasuepHuri Qepuu-rixfl \'r'tbc
`=子争=χlll (683)
14MeЯ B BИ Дy peЛ ЯTИ BHcTcKOe BЬ Ipaxく eHne ЭHep「 I11, HeЙ TPOHa
Fn=1/″櫂+7=″Or2/再:蔦訂ユerKO nOЛ yЧ ИNI Bbipa,KeHIIЯ ДЛЯ ΠЛOTHOCTИ ρ II ДaB.l e HHi彎 :
ρ=3(p。 )θ 」/1平 929,″′=
(pO)` [1zt, :_r1t tf F +1 - tn(t yyVlt)] (684)3
*)'4Ы3Д eCЬ He pa3Л IIЧ aeM ″20(c祗 _ cHocKy Ha cTp 156)11 卜Iaccy HeЙ TpOHa
Pa3Л ИЧИe OKOЛ O l% npeHe6peЖ ИMO ManO no cPaBHeHliЮ c B.lИЯHueM Дpy「ИXy口 pOЩaЮЩИX ΠpeAΠ OЛ OЖ eHИ Й.
ρ=(ρO)珂 ≠寺 =,(6.8.5)=:(p。 )〆
2[/ /「平フ l十+ln/2_1
9- I41eattaati eae c 159
1 `α3B=7丁 °
ア戸雨(6.8.6)
o qacrHocl!1, npII X:l:1, r. e. npu IipHTliqeclioii r'qorHocrH tro-
loq p0: (po)., uueeu p:1,26(ps)", P:0,154(p6).c'.Jlerxo ualiru acltltrrorllliy npll I+@:
3 , I I --. - (p- 3rpf,7r1. P -- -rp,t,r:.y.t ':Tp.2. a "--11. 16.8.7r
llrax, e npe.4e,rle p )pe uaccotl Ioloq Mor(Ho npeHe6perarl.I,laea,rlHuii -v.r'rbrpape.rrrttBllcrcxufi cpepun-la3 IcNIeer cxopocrb3Bl,ria, acrl I\i nrorrq eciitl n pu 6.1HiI aHlurllrccq K
-i- -- o.;8r.l3
3ror pear,.lrrar roiKHo cqfirar6 ecrecrBeHHllN'1, npattTIrJecIH Bce
caCTLIubt J.Blrtli)'TCs co cliopocTblo cBeTa, llo no BCe[4 HanpaBr'le-
HnsM.3a6eraq enepel (ypasHeHlie cocroflHllfl npH BL,ICol(oli TeM-
[epar.ype c11. B rr'L 8). oruerult. qro A,']9 .r]o6oro 5trr61p2psJls111-Bucrciioro fa3a acllM[Tcrilcecht, (a nptt urlesoti ['Jacce noxoq qa-
CTHU, T. e.1.ll nefirpitlto ll IiBaHToB TOtlHO), XI\ielOT NIeCTO COOT-
HOHEII]fl
p-]p.,- ^ - .-
16.8.81' 3"'-l lr'
S 9. Haearsnuil ra3 c yqeroll 93aHMHoro npeBpaueHlts qacrl'!q
Bsrne 6u.r paccrorpeH rag, cocroqtqttil Torbxo u3 neiirpouoe.Ocgoganr'telr aJ'Ir SToto 6u",trt pacuelut ueiirpor:u:autlu, T. e. o6-paraoro 6era-npoecca:
e' + P-nr r'
Hanou:rrrl,r, qro ilpil nlepHoii fl.rlorHocrll pacqer npuBoluJ I Bbl-
BOIy, qro paBHoBecHafi ItoHueHTpauug 3,lelirpolloB I1 IIpoTCTHoB co-craBJrffer mesrue I %. 3ror pe3ll"rb'rar ecrb c,!eACTBHe roro Qaxta,ЧTO ΠpИ paccヽlaTP:IBaeMOi ΠoT1locTII ЭЛeKTpoHЬ I ЯBЛ ЯloTcЯ pTliBИ CTCKHヽ 111. 1lX OHeprHЯ lrHoro 6o,,tbille 3Heprur'1,
Щ en 、lacce noKoЯ , И B TO ,Ke Bpe、 4Я HellTPOliЬI H
ЯBЛ ЯHDTcЯ peЛ ЯTIBИ CTCK:;MH. HX ЭHep「ИЯ
ЧaeTcЯ oT ЭHeprИ 口,cooTBeTcTBさ ЮЩ eЙ liacce
2
16
Spherically symmetric equilibrium
ρ = ρ(r),v = (v1, v2, v3) = 0, S = S(r)
dP
dr+ ρ
dΦ
dr= 0,
1
r2d
drr2dΦ
dr= 4πGρ
⇔− 1
r2d
dr
r2
ρ
dP
dr= 4πGρ
17
Polytrope gas sphere (A0E)
u :=
∫dP
ρ=
Aγ
γ − 1ργ−1
− 1
r2d
drr2du
dr= 4πGρ = 4πG
(γ − 1
Aγ
) 1γ−1
u1
γ−1
18
Lane-Emden equation –Lane-Emden function θ(·, ν):
d2θ
dξ2+
2
ξ
dθ
dξ+ (θ ∨ 0)ν = 0, θ = 1 +O(ξ2) as ξ → +0 (7)
ν =1
γ − 1
(=f
2=
3
2,5
2, · · ·
), r = aξ, u = uOθ,
a2 :=1
4πG
( Aγ
γ − 1
) 1γ−1
u− 2−γ
γ−1
O .
Note: θ(ξ, 5) =(1 +
ξ2
3
)− 12
19
J. H. Lane, On the theoretical temperature of the Sun: under the
hypothesis of a gaseous mass maintaining its volume by its internal
heat, and depending on the laws of gases as known to terrestrial ex-
periment, American Journal of Science, Ser. 2., 50(1879), 57-.
R. Emden, Gaskugeln: Anwendungen der mechanischenWarmetheorie
auf kosmologische und meteorologische Probleme, Teubner, Leipzig-
Berlin, 1907.
20
~~~「 ~~「~~~~~「 ~~~
――+―
・ ― ― ― ‐|―
― ‐
―
~‐―
~― ~
――――――・――――――・
―――十一―す一
| | . | | | |
____上__L___L__ ____L_| | . | | | |
| | | | | | |‐一十一 ―――‐―一 ¬――――
「
――
「
――――
| | | | | | |――¬―――+―― L―・+――」―一――一―
GASKIJGELNANIVENDUNGEN DER
MECIIANISCHEN WARMETHEORIEAUF I(OSMOLoGISCH]Iin,D IETEOROLoolsClIE PROBLEl江 :E
DE R EMDEN
メ1,,1,``OR,N.1,Dl`GRA,`XEN OND' ,,ヽ1ヽ 0(
LEIPZIG UND BERLIN
DRUClモ UND VERLAG VON 3 0 TEUBNER1907
~~十 ~
「
」__L
ヽ
田
21
IT――十二一+=―――
――― ――十 ‐― ―~ ―― ― ―― ¨ ―~―――‐一 ―-4-― ― ―‐ ―― ―
―― ― ― ―
_― ――
―― ― ― ―
‐ ― ―
「
― ―一 ――――二,一―|―――――一
__― 一ユ ーーーー―トー―| . | | |
vie.is Kapitel Die DittreⅢ LlglcFhung d3 1,0 t,Open幌 skuge1 39
Vヽi]be1landeln in folgenden ntl〕 P01ytlope Caskugeh]Dit einem
ExP。 .enten た,l Jede diesei Polvtropen easkugel1l k6n.01 il
dann l。 .ma glelch bel.andeln w10 0he adibatische GRsLugel eines
ingicrten Gases,dessPl κ―たlst
ヱ),ι P()し′,り,´ ,tク 7θ i`卜`,t,"1 7′
ι,ら れ&:π″ "五 i“′,` ` ,r'μ 2,“ 10'
§2にル`,`:``β '
′J`,P'71′
′""を
'aSた “′α (λ >1,″π l),Sο α'“
λ
。.,2`, 1,,α`,′ `/3`,7`″ `saら
,Iθ ,〕クじ,l,,"ss,, p una r_o bestim〕 〕eu
aie ol]e,ticLe delselben
A aer Glenze た-l ge t dic POlytlope Casku8し 1 1ber in
eine lsothe.me caslngei rlulch die salize Masse hindluch ist,'一 oコ st
§3.Wi haben lD erSter Ll ie das Cesetz anfzusuぬ en,llach
Telthel.in ein=.11。 lytropen Gaskugd P,pu.arlょ lgs,v=H ere・
、lT ethaltcn dasseibe in FO m elneF Dire entlalgleichung,lndelΥ Tir
at嘔elen.。 )l de.Oleichung(58b),die wir i aer Fo1ll sch eiben
,2∬ 一-4嗅rO′ 'ar
41 ″ ル 〕面 Bt〔lie biS Zum RadiuS'eingeSCllloSSい・ Masse Besitat
dL(lasktwl eheu f,ste.Ker.マ 。n der Masse y:ulid“ mR血 us
lt,so hutet die lechte selte der eleiclbung_`エー4,9″′,
In beide1l Fユ lleu abol erhalten Wlr, illdCnl Wir uo(. mals naCh′
direrentieren, fur iede beliebige CaSkugel
(59, 手(/2洲)― -4だ。デ
Den AusdF″
sPeZinls]eren Wl fm poly ope,astugel ,
mtlem wi3 ausgellon vor de.ole〕 chung(44ヽ P-O I t und erhalien
(60〕 ″-1″ C,~専 ―1夏
0メキ:
Setz3 1「 ir
(61) α―ず缶g amm~・
und ihren lr die Di“ reniatlon in(59)aus,So ertalten wir
(62) 等≠_|÷」ギⅢ-0
,i0 ZF.,il.r Teil. Differeniiailexiehrrs€!
und mit IiDftiDtrLF cier l3ezerchulgen der iileichugel l45l .or, r: 2,
r :,.-4, erl:altel rir die
Li.fl er eitlioklleic'trqtlt .ler .i)alat aDetx Gailiqtei
il) ".! -, -"-*+ o;,r:0,ii': ' I it
rn relcber a urri ;u gegeLen siad, rud rrelche lei Beacbtlng der aufSeit. 2'! gecebelerr RerIe*regel in bezrig auf Ilinietsron stimmt.Aus dcr Gierchlrg (3i) 'J - tt, sered *irr riall die -urki,icn u clieTerr.ar:.:u' miili. wenr als llirheil rlerselber die oolytrope Tem-r:eraiur €1, 3_ere;iLlt nrrd.
llie lliil'erenrialgteichuag (I) gilt Dur tiir die Fnlle I> t, r*.lie Dif;erelitaiglr:irhruiE tler isothermer Gaskug.el (ri: 1j werden wirspbtef a'olert.o tld LehauJeln.
~丁丁下 丁 T
|―」-1§4 Dle
(1)
kduor wir irRadl油 、・ 1
… '~T elniLren wobei wir durch lelchte
帰:十 二:手
_0
Fiir n > 1 ist die Ditrerentiaigteiehuag nichtLdsung durch Quaclraturen rlarzustellen ist mir uicht
(Ia)
eder,llteu wil t‐ ― ′λ:setzeD,=]balten ivl
(Iい 寡+率 :-01st durch Auf16sttn3 ‐0]1(Iヽ 1` gel lLat . 3o kann a.1ls deu B[)‐
五eLinrnゃ =r'.I― ′・′1,P― tr1 1″ り1,Dthte,TeШ pelatur tlnd
Diり ka jedel Stelle,in absoluten Malen a1lsgealickt Terdeコ
Das alige cine lnteg,ai dtt dielchuり g(I)1に 3t sic,1 r di3 Fユ lle
,7-01′ た― CC.o～ にonstl und,t-1,(ト ー 2,leiCht hdou ManelLAlt i‐ r i=― |
(63)
寡+:守 +′ -0,
I`― q+争 ―宅ll
戸~ア
″ T″%=°,
7-q ll計 4+α 号「:
fiir 2: 1
(64)
wir den reziproken
Umforoung erdaltol
mehr lirear; ihre
÷穿+α2″ -0
||
一 ――
scheinlich wird dureh dieselbe eine
WttΥ―
22
′
ユ_
0
9
′
νくり(⇔ ÷くυリ
′ ｀ヽ
すくり←←>y.《‐登ノ
0=― /1(
1ヽ~=lマノ
0 難つ》
し
,1
ミ∵
ツ 上 _リ
ヽノ
ｅ望ヽ
0
υ
c(R_n‐
ハ
し
´`
―rヽ ぃ ~
エつ
(ジミリ
済(5<リ
ヽ
23
U := −4πGrρ2(dPdr
)−1
= −ξθν(dθdξ
)−1
, (8a)
V := − r
P
dP
dr= − γ
γ − 1
ξ
θ
dθ
dξ. (8b)
rd
dr
[UV
]=
(− 1γV + 3− U
)V(
γ−1γ V − 1 + U)V
. (9)
24
|一―
「―‐」―― 一 ――+――+―一
―
| | | |
十 一
―
―~― ―― i , 1 1 , | | .
| | | | | | | |___L___」 __4___トーー1-――__」 ――‐_― ―――――‐・―――
――| ――
―――‐――す
~~す ~
| 1 1 1 ‐ | | ‐ ・ I
I I I I I I I 1 1‐―――~―
|―
~~―
―
~― ~~~‐ ~~~~~~~~‐・
~~~~~~~~‐ ―――+――十
‐―十一
に
「
十一―
~T
4
―十‐計
―――
―――
―――十一―L_」_____
‐―― ¨―…
―一― ―――――,一―=――=――‐―
| |__→ ――
_| | |
――______|――二_1_二 十十寸‐ | . . 1 1~T~~,F~~T~~ 十
|
↓コ L―――‐「
_――――+
. 1 ~~~~~=~~~
| | . | |
――+―= L― ―――――+―――-1-――
」 __上__十一一|
1 1 1 1 1 11 1 1 1 1
――――+――+――――――
――――‐――斗 一 ―トーーー
′ )
0
___L_ ___+― ―=
_」_=__|_
|~=T
1~~―|
¬ | |十一十
|
上
+__L
ヽ
V
′
‐:|<ノ
ヽ0
びθ
25
ヽ
ヽ
:``
:・・
AN ON″・OF .'°
i
S.CHANDRASEKHAR
26
1`4 STUDY OF STELLAR STRUCTURE
The roots are imaginary, and the solution (.+zg) can therefore be
written in the form
,L: Aet,1.o" ft, * u1 , (+;,)I 2 )'
where 6 is a constatrt. We see that (432) is exactly the limiting formof our earlier equation (264) as ir-o, z+ -. From (432) we
have
n :d1':lilr,,lcos('1,+a) - \ 7,in (rr+a)l . raljl' dt ' ' L---\, -t \ 2 tJ
We see that the sirgular point (o, z,) is approached spirally as
t - - -, t - .. The general run of the solution-curves is illus-trated in Figure r9.
From (+ro) we have
-g: z^ zlogf: str 11. (+g+)
From (+z-5), (+g:), and (+3+), we have
「、/´ ¬
―ψ=2′ +10g2+″たCOS t子 ′+可 ;
or, since : e-t, qe can also write
Finally, since
P : |'e-l '
we have for the Law of Densit)' Distdbution:
一％イＴ
二″
十２一ξ（
_1 ρ=ヽまexpl許 COS[平 1。gt-11(ζ →∞)
the exponential and rctain only the llrst two termS WC ind in this
way that
ρ=ヽ:|二
十鼻COS[キ多10g ξ―δ]|(ξ
~'∞)・ (438)
Since the exponent tends to zero as + - , we can further expand
μｌ″―
|"|
FIc 19-The imcliical QⅣ es fol theisothermal equa on n the(夕 )Pianc The
dirSram is reproduced from Emden's 6asl?6elz.
27
The first mathematically rigorous proof using the Poincare-
Bendixson theorem:
D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problem
driven by positive sources, Arch. Rational Mech. Anal., 49(1972/73),
241-269.
————————
H. R. Beyer, The spectrum of radial adiabatic stellar oscillations, J.
Math. Phys., 36(1995), 4815-4825.
28
Tolman-Oppenheimer-Volkoff equation
dm
dr= 4πr2ρ, (10a)
dP
dr= −(ρ+ P/c2)
G(m+ 4πr3P/c2)
r2(1− 2Gm/c2r)(10b)
J. P. Oppenheimer and G. M. Volkoff, On massive neutron cores,
Phys. Rev., 55(1939), 374-381.
—————
T. Makino, Kyoto J. Math., 38(1998), 55-69.
29
Stability/unstability of equilibria
Pioneering work:
V. A. Antonov, Most probable phase distribution in spherical state
systems and conditions for its existence, Vest. Leningrad Univ.,
7(1962), 135-146.
Introduction to the Western world:
D. Lynden-Bell and R. Wood, The gravo-thermal catastrophe in
isothermal spheres and the onset of red-giant structure for stellar sys-
tems, Mon. Not. Roy. Astr. Soc., 138(1968), 495-525.
30
1962 BFar″ ″人 `″F〃 r′″′P`″ cκ 07・0 ざ″″BrPc″ FF7, ,ヽ7 腎′′
ACTPOHO,MИЯ
8.,, AIroHoB
HA148EPCЯ THEЙ lIIEE(わ A3030E PACnPEД Eユ じHIIEBCじ
'111'tCКИX 3RE3,lH卜 lX C C rEAヽ AX,i yCЛ OBliЯ ErO
Cy[ЦECT30BAH,1,
l laて(a.151:bi e 3レ l:i 3]′01 1'l Clel)H ЧCC:(On 3110e ド01t Cl]Cl eヽ 1ム | 卜〔ヽДpCACia31ne｀ 【 CC6e CneД y:[)I IHi( 06pa3い 1 33e3ЦЫ 6epゝ I C300 111 1n
111 1aЛ Ll:[IX C,(OpOCteii na、 : lo HO lle n3Berle11 1'pHel, e卜 [“N, C 10｀ iК H3pel::IЯ COnpe｀ :eH:iOrO COCrO'IIHЯ 3Be3ス i10H ′[「
Pて,HO,11il, :(aH(eTCt ДO |‐
1(aIム0'tt)ue島 110n3BC3/( li薫 c):lP,:Я ′l ecTlt cltie、(L:Дp"Ot B03ヽ 10X
t:〔 ta｀ l B31) IB:10:0 よ`つaK lepa 10 3 '0)〔 ![ム p)'「 Oヽ[ C」 Vae nep30111Ч alH,HOeC Op H:LI′ ゅ. xHЬl no l::Γ ()、 liO HpO、 0 1'お :10iい a Mel,a、〔
0'ua:(O CDflい ::OCIC o611i303a1ll夕 I clepい iecKoti 3Be3ユ ,0漱 CHCiい IЫ
l'CTV口 1:。 I B `1 ltC]l]l【 e Hつ OCIVl'pHLle CrЛ :メ I輸 :(0 ΠpOЦCCC paCC,、`〔
,rpCl:3 palole l.A Alel( la 111,]Д e 01(113a10,コ 1。 B Ha vnЬ li。重め′13e6ン1
. :|'lel卜 :`eC'0 `OJI:,菫 101( 01p'3 3BeSД :13 1∫ e´ 1]lPa,:Ь :lo': o6.lacTH lヽH "、 c‐ pateli。lp
'( Cチ
le′ζy ltう tЦ yt。 中`13y,
0「Д[l B ЦeHlpaЛ し1101t ttaCT:l ドXe flBCy‐Πl[』 O Coct()Я :I‖ e c 、
`aKcB(,′〔
`,Lci::1〔 pac pcДこlc::Heゝ 〔cК opocic:: DTo p(Ic‐
. :]CF'こ γ。工 ()l ЦCIF『 )8 :〈 『lCp"ゃ cp:IH ム01,Kel: 61,:,Ь HeIっ e,118'メ
, 110′ !′け:
(1 III:'C. C躍1,110 3al[CД OICi). 33ユ 03Ц HOi CICTellC ;ヽ0′ l(11(]Da3 u tll:,
|〕
憎
1
:;1:逮ll:;i'IT:lC:出 ::::||:Ti[器躍 ll口 1::L::lil:]:胤)1:Ilil:P:::
311a`lil'. ::]0 01'6 lH.[IPO｀ 0Я HiHt: nc'HHく o、 I Blic ocHo3Ho10 1eltl, oltyr‐
И3」 eCIキ I reop=1: 30 bЦ ya:la 121 yIBcPXД aeT.t「 10 1'pery潤 ,lde゛:i 0',T『 0¨
“'yPC↓口H[L itpOЯ iHOC'1,1,a30」 (HO p`C,pOrerellHЯ :.
Cle 0,a PC・ 1ち HO, eC 10「 aP14ハ:(DHT,o Πll Ю) Teope)la 3o』 ЬЦ、allj ДQ[(a。`1:"tB ΠpeA 0,10 XCH H, ITO npOl`CXOAlT TO LKO nap::鼠 e C10 1ヽ10 3e H口 : 1、
′
FOrO Xapaklepa ::'IH )it 「
DeД cr8B Я:0:11“ e c000A B3al:MOAlne,BIl e :iCl:―lpaЛ
“
Ь:卜:Hこ H lヽlH B 3EC3AHЬ:X CHCTeMaX Ha A3HXQHa[:(aXД 01 0,lenЪ
ユe1 0八 HaК 0 010 00CiO TelЬ C'BO ne CTO L CyЩ eCTSCIHO Д e.10B10M i31.'I'0 1eCHHC I(pa1lIЫ C c6.lli〕 1(ct]1 : 3Be3ュ ー ,E leliИ ,i t:t〕 ()30b卜 la l::lo pe ki:こ
,
2 КpaTHHe i11う 。XO,【 ェell,コ ,lo 6o4卜ぃぃ: p:lec( ЯHl[Я x nOД HヽH'lЮ Iし
'3010`∫)'
1131,鷲 :‖ I:Ъ。:ill::::鮒Ь:lei:寧Д:箱 e:li『
l[跳|:サギ
|]Iざ ,111::∬lil
“
|`ム 十raゎふld“→
| (Or 8.:vest ien grad Oniv. ,135, 1902)
The laitlal stages of the evolutlm of a s?herical star systen canbe picrured io the followirg say. The stars have thelr be8inning 1n aeompaxatirely small tegimt the aechanisr of lhe orlgin of iritialvelocilies is flot l(nom exactly. Ftom the polnt of vieB of the conteDr-pola.) oL<Lp ot "Lel,ar dBcrmomy! iL is an arcepruble view th.r c\e jn'-Eial velocllies have rhelr6tar toards the center ofassunPtlon is to atl!1bute
each individuaisecond possible
lo be alrrost e\^ccly radial.,E.\rewer) irmediatell after rhe formatlon of a sphexi.al star syEteh,
irregllar forcEs enter rhe picture. Such a process is investlgated iilth" paper by T, A. Agekid! irhere it i3 shoLr that in the lnltlal lhase
stars qi11 escape fron the central resion. ue sba1l consider rhephase, i. e. rheo in the central part a Uawelltan distriburion of
velocities has already been estab116hed. This dlstributlon graduallyplapagate$ torards the periphery, In reality, the transitlon fron lhecenrer to the perilhery has to be conrlnuous, blr in oxder ro sihpulyth6 coilputaElons, Fe sha11 Bupaose.har Ehen rhe evolurion stQs (ooreprecisely, s1o6 dom etrorgly) it ts possible ro dlsrlnguish the oalnbdy of the star systen lron the eorofla." In *e nain body a Haxwelliandlstrlbution rs establlshed (iie neglecr rhe truncarion of rhis disrribu-tion). In the corona, the oxlAlaal disrriburion of velociries is ?re-served. This neans that there are so orbi.s \ihich entlrety avoid rhe
Soltzoan.'6 !re11-knorm theoren2asserts that lrregular forces can
relocities ro forces ofm eaplosive .haracter. But ln etther case, init1a1Iy' all orbi.s hare
assmptions that only blaary
¨,i… ヽ.無
“`“
嵐 (entroPy)_
' of the !h4e dlsrribution and, consequenl-BoltzMnn's thearen ls proven uader the
colltsions of an elasti..haracler takel pace and that the interacこ
・ ons exhibited arc centr31 f(〉r ces. Hovever
l th・ S C rCttStmCO iS nOt that eSSentia . The reaSOn iS・ that COSe
i mu t.っ le mcOunters of st“ s“ e extre漣 v rare p“ nmena,a“ mltiple「 enCOttt∝ S at工 筆 ge d'St“ C‐ are Subjeet tO the l評 。I SuperPOSiti",
i.e. are equivale!c tc several consecutive binary passatses.On the other hand, regular torc.s e*hibir no lnlluence on the
I J. Coadeat ntd ? illt pd\ L D) R.nil.t al St.1 Chslet \, 5)5 i4t)
31
Boltzmann-Poisson equations for f(t,x,v):
∂f
∂t+ (v|∇xf)− (∇Φ|∇vf) = Q[f ],
Φ = 4πGρ = 4πG ·m∫fdv.
M :=
∫ρdx = m
∫ ∫fdxdv, E = T +W,
T := m
∫ ∫fv2
2dxdv, W :=
m
2
∫ ∫fΦdxdv,
H := −k
∫ ∫f log fdxdv.
• Assume a reflecting wall at r = R such that f = 0 on r = |x| ≥ R.
32
Equilibrium: mf = (2πA)−32 ρ(x)e−
v2
2A
ρ is the isothermal gas density with
γ =5
3, (γ − 1)CV =
k
m, P =
k
mTρ = Aρ,
− 1
r2d
dr
r2
ρ
dP
dr= 4πGρ,
H =
∫Sρdx+Const., T =
∫CV Tρdx = CV TM.
33
Isothermal Lane-Emden function:
ρ = ρOeθ, r = aξ, a2 =
A
4πGρO,
− 1
ξ2d
dξξ2dθ
dξ= eθ, θ = O(ξ2) as ξ → +0.
U = −ξeθ(dθdξ
)−1
, V = −ξ dθdξ
ξd
dξ
[UV
]=
[(3− V − U)U(−1 + U)V
]
34
Number of equilibria when E,M,R given: T, ρO should be chosen,
or A,Ξ = R/a should be chosen
M =AR
GV∣∣∣ξ=Ξ
,
E =A2R
G
(− 3
2V + UV
)∣∣∣ξ=Ξ
α :=ER
M2=
1
V
(− 3
2+ U
)∣∣∣ξ=Ξ
A =k
mT =
GM
R
1
V (Ξ)
ρO =A
4πGa2=
M
4πR3
Ξ2
V (Ξ)
35
V|
Ⅸ=一 ′
d=αλ
0く
α≫上
“
ぬ
fi121ノ
′ rV
4´ヽイ、ノ
´
ノ
３
一又
36
Antonov’s criterion of stability:
H is maximized at the equilibrium f , or,
δH = k
∫ ∫(−f log f + f log f)dxdv < 0
for ∀ admissible f near f
37
0ゝ
D。0
０‘
″く
二
上ワヽ
/ヽ/ ノし/
38
ノソ鳥‐ノゾμ 弓
“
″ル7″′″/ガ /'Iφ/ふクだ 4.′ 7′.為 F′/′絆/77.｀レン′r″′'ぶ"佛専ン'
ガ′iたt ・″れノ,'み`え :`′ 摯携″r4/′`ん '′Z嫁 瘍ヽ二七ヴ′｀
'、
んノ
`′
=ヽ,1直″疲iメどイカ
t:,. '-dE;:ri, ri'
′′々≒r.、レ′ν`″・・ 'んィム・.
4′
`′
′_アメf′ケ
211, '
・・′
お′=藤
″凛_′′
`´
.=`
4′″;´
``晟:I
F,´tι
4:=′・´A,,″ .:.ゃ
―
―
―
・１
”ｒｌ
39
Stabe/unstable in the sense of Ljapunov
Thesis to Kharkov University A. Liapounoff, Probleme general de la stabilite du mouvement,
Annles de la Faculte des Sciences de Toulouse, 2eme Serie, Tom.
9(1907)
40
zro
Soicnt
\
q,n:/,(t,) -t €i qro =/,(tu)+ e
',g'to- l"lto\ + it, ,1lo*./,lto\ + 2','
,tu:,ft(tol + ct,
q'm: lLltc\ 'r e'o
PROTLtTD Ggltf,At DE Ll srlEtllri DU llouviltNT' xll
oosilifsEr, E,, .'., Er, El, El' "'' El,
tek quc, lct inigalitls
lerl<Er, l!il<Ei (') Q=r't' "''k\
Caana rcmplies, on @il
lQ,*F,l(L,, lQr-F'l<L', lQ.-I'l<L"'
poar toui,ct lctaalcurs de t qui dlPw.renl t., le otoureilent non ltoubli' sera
dil stable prr !^D!ort lsr eu^rrltas Q,, Qr, .'., Qr; dare le cas cotltaire,il scra dit, par rappoil ar,i m€mcs quanticds, instable.
Gtons des eremplcs :
Si un point matdriel, attird Par un ceDtrc 6re en raisou iuverse du carri de
la distanee, ddcrit une trajectoirc circulaire, son mouvement, PrrraPport au rayon
vecteur, tr.cd A partir du centre d'.ttraction' et d8alement Psr raPPort e sa vitesse,
est rtable. Le mdme mouvem€nl' Per raPPori aur coordonndes recunguhires du
Point, est. insrsblc.Si le nrGme point ddcrit une trajectoirc ellipriquc, lon mouYemetrl est insrable,
aon lerlement por repPort aur coordoundes rectangulairesr mais encore par rap-
por! au myon v€cleor et A l8 vitesse. Mais il est stable, par eremple, Par rapPorl-
I lo quantitd
t-,-t-#@sr'
ot p et e soot lc param0lre et I'ercentricitd de I'ellipse ddcrite par le point dans le
mouvcment non tmubld, ct ,'et? le raJro[ vecteur du point dans le mouvetnent
troubld et I'aagle fait par cc Bron vecleur avec le plus petit myon tecteur dans le
mouvereeqt non troubld.
Quend un corps solide a ut point firc ct n'est soumis I aucune force, et qu'iltouroe suiour du plos grand ou du plus petit der a:es de I'ellipsoide d'inerrie re-
latifi ce point, son Douvqmenl est stable par mpport i la v;tesse an$ulaire el our
rrrgles quc fait l'are instnntsnd avec des sres fir€s ou arec ceur invarirblelnenl
lids au corps, Au contraire, quand il toume autour de I'are tnoyen de l'ellipsoide
d'inertie, ron Eouvement per mppo.t i ces rn6mes quantitds est insuble.ll peut irrircr qu'il soit impossible de ttouver des lirnites E7, Ei satirfaisant i
le dCGnirion prdcddeqte, quand les perturbations sont quelconques, maisqu'il soit
( t) On 6anarrl, noo! cotrv.non! d'cnrcnilr prr I c I la vrleur gbroluc dc lc quaotitd r ,,u
ron modulc, qurnd clla ..r irnrgiftrirc.ror. d. r..1'5., t\.
oir e;, ej sont des constantds rCelles.
Ces constantes, qtre nous appelleronsperlt' baliotts, dcfiniront un llrouvelnent
trouble, Nous ,upioru"on, qtt'ol peut leur attribtter toutes les valeurs suffisalr-
rrena petites.
En parlent des mouvments lroublds uolslls d'nn nrourernent non troltbli' nous
errtenirons par li ,les moulertreills pour lesquels les perturbations sont assez
pctites en raleur at solue,
Ceia posC, soient (]r1 ar, .., a, des fonctions donnies, rdelles et continues'
rlr:s ,ltranti Lesjr (tt, .-.,ltt q\, q,, ..,, q'r
Pour le mouvernent non troubld elles deviendront des folctions connues de l,
(lre nou6 ddsigncrons respectiriment par F,, 81, .', F,,' Ilour un lnoove'|ue't
trorrbld elles serotrt des foncrious cles <luantitds
l, fr, €r, ...' E*, e'r, e'r, ''" al'
Qrantl tous lcs e7, ei solt 6gatt i ziro, les quantitds
Q,-l'" Q'-Ft' '' Q'-F'
serolt nullcs pour clraque raleur de t, Mais si, sans rendre les conslnntes €i' Ei
nulle-r, on I"r rrpporu totrtes infinitoent petites, la gues'iotr se pose dc satoir s'il
est possible d'asiig.er atrx quarrtitds (.1, - F, det Iirnites infiniment petites, telles
(lue ces guantitCs ne les surpassentjarnais en valcur absolue'
I-a sohtion de :ette question, <1ui fera l'objet de nos recherches, depentl dt
caracti.c rlu mouvc[rent non troul)ld considerd, ainsi tltre tlrr ehoir dcs lonctions
Q,, Q", , Q,, ct du rnortren!da ternps l0 l)onc, r'c choil tlLarrl fir! la rt1'onse
i c"tte q.,"rti.^ curactdrisera.'.rous trn ce'tain rapport, le lnouYement non trortbl€'
et c'e.t elle qrri cn exJrrirnetr la proprioti, que rrous appellerons slabilili' o\ la
proprirtd conl.,rire, qui sera appelCe iattabiliti'- Nous nous occuPerons excltrsivernenL dcs uirs oi la solutioo de la question con_
sidirie ne depcnd 1,as du choir dc I'instant lo tlans leqtrel se produisent les per-
torbations. C'est pourquoi rous adoplerons irri la rldfinition suilante :
Soient L,,1,,, ..., [,, r/cs rtontbres posirql's donnis' Si pour tt'ttles las ullcurt
rle <'cs nortl,rc,, qualque petitt's ,1t'tllts soitrrt, rtn feut ehoisit' tlrs nonl'rcs
"8
41
Dynamical system:
X: topological space, Ω: open subset of R × X, φ : Ω → X:
continuous, ∀x ∈ X : Ω(x) := t ∈ R|(t, x) ∈ Ω is an interval,
0 ∈ Ω(x), φ(0, x) = x;
(s, x) ∈ Ω, (t, φ(s, x)) ∈ Ω ⇒ (t+s, x) ∈ Ω, φ(t, φ(s, x)) = φ(t+s, x);
(tn, x) ∈ Ω, tn → τ, φ(tn, x) → b⇒ (τ, x) ∈ Ω.
• a ∈ X is an equilibrium ⇔ Ω(a) = R, φ(t, a) = a ∀t ∈ R.• an equilibrium a is stable
⇔∀neighborhood V of a ∃neighborhood W of a :
x ∈W ⇒ [0,+∞[⊂ Ω(x), φ(t, x) ∈ V ∀t ∈ [0,+∞[.
42
Spherically symmetric time evolution
(EP), (A0E) with6
5< γ < 2
Fix an equilibrium
ρ = ρO(θ(r/a),∨0)ν , r+ = aξ1(ν).
r = the Lagrangian coordinate
r(t, r) = r(1 + y(t, r))
43
∂2y
∂t2− 1
ρr(1+ y)2
∂
∂r
(PG
(y, r
∂y
∂r
))+
1
ρr
dP
drH(y) = 0, (0 ≤ r ≤ r+)
(11)
where
G(y, V ) = 1− (1 + y)−2γ(1 + y + V )−γ = γ(3y + V ) + [y, V ]2,
H(y) = (1 + y)2 − 1
(1 + y)2= 4y + [y]2.
44
Thus the linearized problem :
∂2y
∂t2+ L
( ∂
∂r
)y = 0,
where
L( d
dr
)y = − 1
ρr
d
dr
(γP
(3y + r
dy
dr
))+
1
ρr
dP
dr(4y).
L is self-adjoint in L2([0, r+], ρr4dr),
the spectrum consists of simple eigenvalues
λ1 < · · · < λn < λn+1 < · · · → +∞
λ1 > 0 ⇔ γ >4
345
Applying the Nash-Moser theorem formulated by R. Hamilton, we
have
Theorem 1. (T. Makino, OJM, 2015) Suppose 6/5 < γ ≤ 2, γγ−1 ∈ N.
Then for ∀T ∃ϵ(T ) 0 < ∀ε ≤ ϵ(T ) there exists a solution y(t, r; ε) ∈C2([0, T ]× [0, r+]) such that
y(t, r; ε) = ε sin(√λt+Const.)φ(r) +O(ε2)
Here λ is a positive eigenvalue of the linearized operator L and φ is
the associated eigenfunction.
46
We note that the free matter-vacuum boundary is
r = RF (t) = r+(1+y(t, r+)) = r+(1+ε sin(√λt+Const.)φ(r+)+O(ε2)),
r+ being the radius of the equilibrium, φ(r+) = 0, and
ρ(t, r) =
C(t)(RF (t)− r)
1γ−1 (1 +O(RF (t)− r)) r < RF (t),
= 0 RF (t) ≤ r
where C(t) > 0 is a smooth function of t
47
゛――
E=2F`ヤ)こ==r十十
・ε?“ )SIn
+θ(ap
(肝斗鏑 )＼ ~~ /
Pン 卜0
r,ハ
三十
48
Theorem 2. (T. Makino, OJM, ) Suppose 6/5 < γ ≤ 2, γγ−1 ∈ N.
Then there exists a number r such that for ∀T ∃δ(T ) ∀ψ0, ψ1 ∈C∞([0, r+])∥∥∥( d
dr
)j
ψ0
∥∥∥L∞
,∥∥∥( d
dr
)j
ψ1
∥∥∥L∞
,≤ δ(T ) ∀j ≤ r
there exists a solution y(t, r) ∈ C2([0, T ]× [0, r+]) such that
y|t=0 = ψ0(r),∂y
∂t
∣∣∣t=0
= ψ1(r).
49
But the condition 65 < γ ≤ 2, γ
γ−1 ∈ N restricts γ to 54 ,
43 ,
32 or 2.
On the other hand, for (EP), (A1), applying the Nash-Moser the-
orem formulated by J. Schwartz, we have
Theorem 3. (T. Makino, JDE, 2016 ) Suppose the equilibrium
−r2 ddr
( 1
r2ρ
dP
dr
)= 4πGρ, ρ = ρO +O(r2)
have a finite radius. Suppose 1 < γ < 54/53. Then the conclusions of
Theorems 1, 2 hold.
50
Another approach to the same problem was done by Juhi Jang, Anal-
ysis & PDE, 2016 , which is based on a sophisticated use of Hardy type
inequalities, without use of the Nash-Moser theory.
51
Unstability result by Juhi Jang, Nonlinear instability theory of Lane-
Emden stars, CPAM, LXII(2014), 1418-1465 :
(EP), (A0E) with γ <4
3Let ϕ1(r) be an eigenfunction for λ1(< 0). Consider the initial data
(y, ∂ty)|t=0 = (ϵϕ1,√−λ1ϕ1).
0 < ∃ϵ0 < ∃τ0 : ∀ϵ < ϵ0 :
sup0≤t≤Tϵ
E(y, ∂ty) ≥ τ20
52
where
Tϵ =1√−λ1
log2τ0ϵ,
E(y, v) = ∥y∥2X + ∥y∥2X1 + ∥v∥2X,
∥Y ∥2X =
∫ r+
0
|Y |2ρr4dr,
∥Y ∥2X1 =
∫ r+
0
∣∣∣dYdr
∣∣∣2P r4dr.
53
Spherically symmetric evolution in (EE)
Consider (EE), (A1R) withγ
γ − 1∈ N or 1 < γ <
54
53
A co-moving spherically symmetric metric
ds2 = e2F (t,r)c2dt2 − e2H(t,r)dr2 −R(t, r)2(dθ2 + sin2 θdϕ2) (12)
such that U ct = e−F , Ur = Uθ = Uϕ = 0 with spherically symmetric
density distribution ρ(t, r) is considered.
54
Equilibrium satisfying the Tolman-Oppenheimer-Volkoff equations
is fixed.
m =4π
3ρOr
3 +O(r5), (13a)
P = PO − (ρO + PO/c2)4πG(ρO/3 + PO/c
2)r2
2+O(r4) (13b)
as r → 0 and ρ(r) 0 as r r+(<∞).
55
Putting
m+ = 4π
∫ r+
0
ρ(r)r2dr, κ+ = 1− 2Gm+
c2r+> 0
we have
u(r) =Gm+
r2+κ+(r+ − r)(1 + [r+ − r, (r+ − r)
1γ−1 ]1)
56
We consider spherically symmetric perturbation from this equilib-
rium:R = r(1 + y(t, r)), V = rv(t, r) (14)
governed by
e−F ∂y
∂t= (1 + P/c2ρ)v, (15a)
e−F ∂v
∂t=
1
c2(1 + y)2
P
ρv∂
∂r(rv)− 1
r3G
(1 + y)2
(m+
4π
c2Pr3(1 + y)3
)+
−(1 +
r2v2
c2− 2Gm
c2r(1 + y)
)(1 + P/c2ρ)−1 (1 + y)2
rρ
∂P
∂r(15b)
Here m = m(r) is given and ρ is the function of r, y, r∂y/∂r given by
ρ = ρ(r)(1 + y)−2(1 + y + r
∂y
∂r
)−1
(16)
57
Linearization of the system (15a)(15b) at y = v = 0:
∂2y
∂t2+ Ly = 0
is given by
Ly = −1
b
d
dr
(ady
dr
)+Qy,
a =ΓPr4
1 + P/c2ρeF+H ,
b = (1 + P/c2ρ)−1ρr4e−F+3H
with Γ :=ρ
P
dP
dρ.
58
As the non-relativistic problem (EP) we can prove that L can be
considered as a self-adjoint operator in L2((0, r+); b(r)dr) whose spec-
trum consists of simple eigenvalues λ1 < λ2 < · · · < λν < · · · → +∞.
59
Theorem 4. (T. Makino, KJM, 2016 ) Given T > 0, there exists
a positive ϵ0(T ) such that for |ε| ≤ ϵ0(T ) there is a solution (y, v) ∈C∞([0, T ]× [0, r+]) of the form
y = εy1 + ε2y, v = εv1 + ε2v (17)
such thatsup
j+k≤n∥∂jt ∂kr (y, v)∥L∞ ≤ C(n).
Here
y1 = sin(√λt+Θ0)φ(r),
v1 = e−F (1 + P/c2ρ)−1 ∂y1∂t
,
while λ is a positive eigenvalue of L and φ is an associated eigenfunc-
tion.
60
Note
R(t, r+) = r+(1 + ε sin(√λt+Const.)φ(r+) +O(ε2)), (18)
φ(r+) = 0, and
ρ(t, r) =
C(t)(r+ − r)
1γ−1 (1 +O(r+ − r)) (0 ≤ r < r+)
0 (r+ ≤ r)(19)
with a smooth function C(t) of t such that
C(t) =(γ − 1
Aγ
Gm+
r2+κ+
) 1γ−1
+O(ε)
61
The Cauchy problem (CP):
e−F ∂y
∂t= · · · (15a), e−F ∂v
∂t= · · · (15b),
y|t=0 = ψ0(x), v|t=0 = ψ1(x)
was also solved.
62
The solution metric can be patched to the Schwarzschild metric on
the vacuum region:
ds2 = K♯(cdt♯)2 − 1
K♯(dR♯)2 − (R♯)2(dθ2 + sin2 θdϕ2), (20)
where
K♯ := 1− 2Gm+
c2R♯, t♯ = t♯(t, r), R♯ = R♯(t, r). (r ≥ r+)
63
Theorem 5. (T. Makino, KJM, 2016 ) There are t♯, R♯ ∈C∞([0, T ] × [r+,+∞)) such that the coefficients of the patched
metric are of C1([0, T ]× [0,+∞)). But then
∂2R♯
∂r2
∣∣∣r++0
− ∂2R
∂r2
∣∣∣r+−0
= A(∂R∂r
)2
,
A = −V2
c2
(Gm+
c2R2+
1√κ+
1
c2∂V
∂t
)(1 +
V 2
c2− 2Gm+
c2R
)−2
does not vanish if V = 0. In other words, the patched metric cannot be
of class C2 across the vacuum boundary, unless the static equilibrium
is concerned.
64
Further Studies
(1) Now we are studying axially symmetric rotating stationary solu-
tions and evolutions near them. Especially the matter-vacuum match-
ing problem for the axially symmetric rotating metric in (EE) is still
open.
Juhi Jang and T. Makino, Arch. Rational. Mech. Anal., 225(2017),
873-900.
Juhi Jang and T. Makino, J. Differential Equations, 266(2019),
3942-3972.
T. Makino, J. Math. Phys., 59(2018), 102502.
T. Makino, ArXiv: 1980.10639.
65
(2) On the other hand, the study of the time evolution of non-radial
perturbations from spherically symmetric equilibria in (EP) is very
interesting , and is far from being easy to deal with.
Juhi Jang and T. Makino, 1810.08294.
T. Makino, arXiv:1902.03675
66
THANK YOU
FOR YOUR ATTENTION!
***
Please visit my Homepage:
“Arkivo de Tetu Makino”(http://hc3.seikyou.ne.jp/home/Tetu.Makino)
67
![A CAUCHY–DIRAC DELTA FUNCTION - arXiv · But did Dirac introduce the delta function? Laugwitz [52, p. 219] notes that probably the first appearance of the (Dirac) delta function](https://static.fdocument.org/doc/165x107/5ac33aab7f8b9a220b8b8e19/a-cauchydirac-delta-function-arxiv-did-dirac-introduce-the-delta-function.jpg)
