SPHERICAL WINDS SPHERICAL ACCRETION

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Emily Henderson
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1 SPHERICAL WINDS SPHERICAL ACCRETION Spheical wins. Many sas ae known o loose mass. The sola win caies away abou M y 1 of vey ho plasma. This ae is insignifican. In fac, sola aiaion caies away eg s 1, which euces sola mass by abou M y 1. Howeve, vey luminous, supegian sas ae losing mae a a vey high ae, fequenly in he ange 10 6 M y 1 o 10 4 M y 1 (cf. Lae Sages of Sella Evoluion, Poceeings of he Wokshop hel in Calgay, Canaa, fom 5 June, 1986, Eios: S. Kwok an S. R. Poasch, 1987, D. Reiel Publ. Co., Volume 13 of Asophysics an Space Science Libay. A supegian wih a luminosiy 10 4 L mus bun abou 10 7 M y 1 o compensae fo he aiaive enegy losses. In his case a mass loss ae a he level exceeing 10 7 M y 1 is moe impoan fo he mass balance hen nuclea buning is. In spie of common pesence of mass loss fom bigh sas hee is no goo quaniaive heoy ha coul explain i. I is believe ha aiaion pessue in amospheic lines, i.e. he boun boun ansiions ae esponsible fo mass loss fom blue supegians, while aiaion pessue on us is esponsible fo mass loss in e supegians. The sola win is ue o vey high empeaue of he sola coona, above 10 6 K. We shall consie hee wo vey simple moels of mass ouflow, none of which is iecly elae o any of he eal, an complicae objecs. In spie of hei simpliciy he moels offe a goo qualiaive picue of he chaace of a seay sae ouflow. We shall consie spheically symmeic mass ouflow, wih he aial velociy of gas v (/ M being a funcion of aius, bu no of ime. Tha is we shall consie a case of a seay sae, ime inepenen ouflow. This is possible when he amoun of mae wihin he ynamically impoan flow is much smalle han he amoun of mae wihin he conaine fom which he mass is flowing ou. Typically, he conaine may be he sella envelope, i.e. his pa of a sa which may be los in he paicula mass loss pocess. The assumpion of a seay sae coniion has a vey well efine mahemaical meaning. Le us consie any physical quaniy, say q, which in geneal may be a funcion of ime an aius. We may always wie: ( q ( q + ( ( q ( q + v ( q. (w1.1 In a sic seay sae we have ( q/ 0. In pacice, we shall assume ha hee is a seay sae when ( q/ ( q/ M. In his case we may wie he elaion (w1.1 as whee ( q v ( ( q M q, (w1. Ṁ 4π ρv cons, (w1.3 is he ae of mass flow acoss a spheical suface wih a aius. In a seay sae moel his ae is consan in space an in ime. Thee ae wo equaions of sella sucue which conain ime eivaives: he equaion of moion, an he equaion of hea balance: ( P G 4π 4 1 4π ( L ǫ T win 1 ( ( S., (w1.4 (w1.5

2 These equaions may be simplifie in case of a seay sae ouflow by noicing ha he oal amoun of mass wihin he seay sae pa of he flow mus be much smalle han he oal sella mass, ohewise he flow woul no be saionay. Theefoe, once we ecie o suy a seay sae ouflow we shoul aop M cons in he em escibing gaviaional acceleaion. Also, in he oue pas of a sa hee ae no nuclea enegy souces, an we have ǫ 0. The wo ime eivaives may be wien as follows: 1 4π 1 ( v 4π 1 ( 4π v M v 4π ( ( S S T TṀ T ( 4π M ρ S ( ( v, (w1.6. (w1.7 Now, ha we eplace ime eivaives wih space eivaives, he equaions of sella sucue become oinay iffeenial equaions. We may wie hem as follows: 1 P ρ G v v ( GM v, (w1.8 ( L S u TṀ Ṁ P [ ( ρ ρ Ṁ u + P 1 ] P. (w1.9 ρ ρ The las wo equaions may be combine o obain whee [ L + M (u Pρ v + + G Ė L + M (u Pρ v + + G ] Ė 0, (w1.10 cons, (w1.11 is he oal enegy caie wih he flow acoss a spheical suface wih a aius. The equaion of moion an he equaion of hea balance gave us enegy consevaion in a seay sae flow. Togehe wih mass consevaion as escibe wih equaion (w1.3 we have wo consevaion laws. Theefoe, we nee jus wo iffeenial equaions o escibe he flow, hese may be he equaion of moion (w1.8 an he equaion of aiaive equilibium, which shoul hol in he opically hick pa of he flow: ( a 3 T 4 κρl 4πc. (w1.1 These wo iffeenial equaions, (w1.8 an (w1.1, ogehe wih he wo consevaion laws, (w1.3 an (w1.11, allow o calculae he vaiaion of T, ρ, L, an v wih aius. Of couse, hey have o be supplemene wih pope bounay coniions in he eep ineio, an a he sella suface. This geneal poblem is ifficul o solve, because he iffusion appoximaion fo he hea anspo, (w1.1, is no vali a small opical eph, above sella phoosphee. Nomally, in a sa ha is in a hyosaic equilibium, an opically hin amosphee is also geomeically hin, an Eingon appoximaion give easonable esuls. In he win case he opically hin amosphee is geomeically exene, an hee is nohing as simple an as goo as he Eingon appoximaion. Thee is one vey geneal popey of a seay sae win moel: in he eep sella ineio he sa shoul be in a hyosaic equilibium, i.e. we shoul have v v s, whee v s is spee of soun. A vey lage isance fom a sa we expec he flow o escape fom gaviaional poenial, an hence we expec v v s. Theefoe, somewhee in beween hee shoul be a ansiion fom a subsonic flow o a supesonic flow. I uns ou ha he poin a which v v s is vey special. I is calle a sonic poin, o a ciical poin, an is exisence is a common popey of all win moels. We shall emonsae is exisence is he following way. The equaion of moion (w1.8 may be wien as 1 P ρ 1 ( P lnρ GM ρ 1 + v ln v, (w1.13 win

3 an his gives whee he spee of soun is efine as lnρ Gvs v lnv vs, (w1.14 v s ( 1/ P. (w1.15 ρ Taking a logaihmic eivaive of he mass consevaion equaion (w1.3 we obain + lnρ + lnv 0. Combining equaions (w1.14 an (w1.16 we fin (w1.16 GM lnv vs vs v v esc v s v s v, (w1.17 wih he coniion v v s a small aii, an v v s a lage aii. Noice, ha GM/ v esc, whee v esc is he escape velociy fom he gaviaional poenial GM/. A he sonic poin we have v v s. The soluion of he iffeenial equaion (w1.17 may be smooh a his poin only if GM/ vs a he same poin. This is a non ivial coniion on he flow, an i is as impoan as any bounay coniion in eemining a unique soluion of he iffeenial equaion. Of couse, he igh han sie of equaion (w1.17 being of he 0/0 ype a he ciical poin, canno be calculae iecly. Insea, we can use e l Hopial s ule, accoing o which f/ g f/g if f 0, an g 0, simulaneously. Remembeing, ha a he ciical poin we have GM/ vs v, we obain ( lnv GM vs (v s (w1.18 v v s G ( ln vs 4vs ln vs v ( ln v 1 ( ln v s ( ln vs ( ln v. This may be wien as a quaaic equaion fo ( lnv/ln : ( lnv ( lnvs Thee ae wo eal oos of his equaions, povie This inequaliy is saisfie when o ( lnv 1 ( lnvs 0. (w1.19 ( ( lnvs lnvs > 0, (w1.0 ( lnvs > , (w1.1a ( lnvs < , (w1.1b The logaihmic eivaive of he spee of soun wih espec o aius may be expesse as 1 ( lnp lnt lnv s 1 lnp 1 lnt 1 ρ [( lnp lnρ ln ρ T ] lnρ 1, (w1. win 3

4 The inne bounay coniion equies he win moel o be in a hyosaic equilibium a small aii, so i coul be mache wih a hyosaic equilibium moel of he whole sa. Thee ae also wo oue bounay coniions. We expec he flow o expan ino empy space, an heefoe, ensiy an pessue shoul be falling own o zeo a vey lage aii. Also, a he phoosphee a hemal bounay coniion mus be saisfie, i.e. L 4π σt 4, (w1.3 a opical eph τ /3. This is only appoximae coniion, because Eingon appoximaion is no goo in he exene amosphee of he win, an he coniion (w1.3 may be saisfie a some ohe opical eph. As ou equaion (w1.1 an he ineacion beween gas an aiaion become complicae a small opical eph, hee is no simple an accuae way o fomulae he hemal oue bounay coniion. We shall consie now a vey simple, isohemal win moel. The moel is so simple ha we shall fin analyical soluion fo he flow. A he same ime he isohemal moel eains all he mos impoan chaaceisics of he geneal, seay sae ouflow. Fo he isohemal flow we have ( P vs kt cons. (w1.4 ρ T µh Le us efine imensionless aius ξ, an imensionless velociy u : ξ c, u v v s, (w1.5 whee c is he ciical aius, i.e. he aius whee he velociy of ouflow is equal o he spee of soun. Accoing o equaion (w1.17 his coespons o c GM v s The equaion (w1.17 may be wien in imensionless vaiables as The vaiables in equaion (w1.7 can be sepaae: ( 1 u u u an he equaion (w1.8 can be inegae o obain GMµH kt. (w1.6 ln u lnξ ξ 1 u. (w1.7 ( 1 ξ 1 ξ ξ, (w1.8 lnu u lnξ + C, (w1.9 ξ whee C is he inegaion consan. Fo he soluion o pass hough he ciical poin we nee C 1.5. The whole family of soluions coesponing o vaious values of he consan C is shown in figue 1. The wo hick soli lines ha coss a he ciical poin: u 1, ξ 1, ae he wo ciical soluions. The line ha saisfies he win bounay coniion is he one ha goes fom he lowe lef cone o he uppe igh cone, i.e. i coespon o a subsonic flow a ξ < 1 (i.e. a small aii, an o a supesonic flow a ξ > 1 (i.e. a lage aii. Spheical acceion We may consie a poblem opposie o he win ouflow, an his is acceion of mae ono a sa. In his case we nee some finie ensiy meium a lage aii, gaually falling ono he sa. The inwa flow woul be subsonic a vey lage aii, an woul become supesonic fee fall a small aii. The equaion ae he same as fo he win ouflow, jus he iecion of he flow is evese. In he acceion flow hee is also a ciical poin, whee he infall velociy is equal o he spee of soun. Fo an isohemal acceion he elevan ciical soluion coespons o a hick soli line ha goes fom he lowe igh cone of figue 1 o he uppe lef cone of ha figue. win 4

5 If he acceing objec is a black hole, which oes no have a ha suface, he supesonic acceion flow may go igh ino he black hole. Of couse, he elevan equaions of moion have o be elaivisic, bu his oes no change he opology of soluions hey emain he same as hose in figue 1. Howeve, if he acceing sa has a suface, he supesonic flow has o be soppe a some poin, an he infalling mae mus come o es. A shock wave is fome a aius smalle han ciical aius, an he infall velociy is euce fom supesonic o subsonic. I may be shown ha he soluion fo a saionay acceion flow below he shock wave coespons o one of he cuves in figue 1 ha go o he lowe lef cone he inflowing mae comes o a hyosaic equilibium, an meges wih he sa. A possible soluion is inicae in figue 1 wih a hick boken line. win 5

ENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 9 Solutions [Theorems of Gauss and Stokes]

ENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 9 Solutions [Theorems of Gauss and Stokes] ENGI 44 Avance alculus fo Engineeing Faculy of Engineeing an Applie cience Poblem e 9 oluions [Theoems of Gauss an okes]. A fla aea A is boune by he iangle whose veices ae he poins P(,, ), Q(,, ) an R(,,