Transparency and stability of low density stellar plasma related to Boltzmann statistics, inverse stimulated bremsstrahlung and to dark matter

Transcription

1 Trasparcy ad stablty of low dsty stllar plasma rlatd to oltzma statstcs, vrs stmulatd brmsstrahlug ad to dark mattr Y. -Aryh Tcho-Isral Isttut of Tchology, Physc Dpartmt, Isral, Hafa, Emal: ASTRACT Th rat of stmulatd vrs brmsstrahlug s calculatd for low lctro dsty stllar plasmas ad th codto udr whch th plasma bcoms traspart s prstd. Th stablty of low dsty stllar plasma s aalyzd for a star wth a sphrcal symmtry qulbrum btw th gravtatoal attractv forcs ad th rpulsv prssur forcs of a dal lctro gas whr th aalyss s dvlopd by th us of oltzma statstcs. Fudamtal ad surprsg rsults ar obtad by whch th radus ad th total mass of th star ar vrsly proportoal to th squar root of th lctro dsty th star ctr. Th total gravtatoal forcs of th star wth vry low lctro ad mass dsts ar vry larg (!) du to th xtrm larg star volums. Th absorpto ad msso of radato for xtrmly low dsty star plasmas vashs ovr all th tr lctromagtc spctrum. Th prst rsults ar supportd by umrcal calculatos. Smlar ffcts ar prdctd for low dsty stllar plasmas whch hav dffrt structurs ad th proprts of such plasmas mght show crta smlarts wth thos of dark mattr. Ky words. Stllar plasma wth low lctro dsts-vrs stmulatd brmsstrahlug-dark mattr 1. Itroducto Th stllar plasmas statstcs udr o-rlatvstc codtos ca b aalyzd for two dffrt asymptotc rgos: a) Th Ruthrford-or approxmato s vald for vry fast lctros gvg classcal collso frqucy whch scals as 1 / v wth lctro vlocty v [.g. Gzburg 1961; Pl 1; Ch 199; Ftzpatrck 15]. Th us of or approxmato s vald udr th codtos: Z / 4πεħv 1, m v / ħω 1, whr s th lctro charg, ħ th Plack costat (dvdd by π ), m th lctro mass ad ω th frqucy of th lctromagtc fld.(.g. Marcus 196; Kraov ; Myr-tr Vh 8). (I th prst work w us th MKS uts ad trat th plasma wth o oc compot but th aalyss ca asly b gralzd to mor oc compots). Vry ds stllar plasmas satsfy Frm-Drac statstcs whr th avrag lctro vlocty s dpdg oly o th lctros dsty ad s dpdt of tmpratur [.g. Ashcroft & Mrm 1967; Sgr 1991]. For ths cas by avragg th dpdc o 1 / v ovr Frm-Drac dstrbuto w obta strog absorpto ad strog rflcto for lght cdt o such plasmas. Such codtos ar vald for fast lctros wth hgh dsts. b) I th prst work w ar trstd th cas of rlatvly slowr lctros ad smallr dsts for whch Z / 4πε ħ v > 1. Th absorpto ths cas s qual to brmsstrahlug stmulatd absorpto mus th stmulatd brmsstrahlug msso. I zroth ordr approxmato th total rat of stmulatd radato for o lctro vashs [Kraov ]. W 1

2 calculat th rat of stmulatd vrs absorpto frst ordr approxmato by takg to accout th dffrc th umbr of lctros for stmulatd msso wth thos for stmulatd absorpto ad fd th codtos udr whch th plasma bcoms arly traspart. Vry hgh dsts of plasmas producg wht dwarfs, whch ar rlatd to Frm-Drac statstcs, wr tratd qut log ago [Chadraskhar 1967; Edgto 1959; Kppha, Wgrt &Wss 1]. I th prst work w trat th trasparcy of plasmas wth rlatvly low dsts at ough hgh tmpraturs at whch oltzma statstcs s vald. Whl th mass dsty of a star wth th prst plasma codtos s rlatvly low ts stablty codtos, drvd th prst work, show that t ca xtd o xtrmly larg volums so that ts gravtatoal forcs bcom vry sgfcat. Du to th trasparcy of such stars w ras th da that such star plasmas mght hav mplcato to dark mattr. Dark mattr s thought to b o-baryoc, possbly bg composd of som as-yt udscovrd partcls [.g. rto ad Hoopr 18; Trmbl 1987]. Its prsc s mpld astrophyscal obsrvatos, du to th xstc of gravtatoal ffcts that caot b xplad ulss mor mattr s prst that caot b s. Th am dark mattr rfrs to th fact that t dos ot appar to tract wth lctromagtc radato ad thus t s vsbl ( dark ) to th lctromagtc spctrum. Usually stars ar obsrvd thr by radato mttd by thm or by rflctd ad trasmttac of lght cdt o thm from othr sourcs. It s trstg to ot by followg th prst aalyss that stllar plasmas wth vry low dsts do ot mt ay sgfcat radato ad thr trasmttac s arly qual 1 (rflctac arly zro). W fd that th stablty of a star udr sphrcal symmtry, whch s basd o th prst plasmas codtos, ca xtd o xtrmly larg volums so that ts gravtatoal forcs bcom vry larg. Whl w dvlopd th prst aalyss for stars wth sphrcal symmtry smlar rsults ar prdctd for star plasmas whch hav dffrt structurs. W study th possblty that such plasmas mght b rlatd to dark mattr. Th prst papr s arragd as follows: y usg th thors o stmulatd vrs brmsstrahlug, w trat scto th trasmttac of moochromatc lctromagtc fld through stllar plasmas wth rlatvly slow lctros ad low dsts. W show that udr crta codtos aalyzd th prst work ths plasmas hav xtrmly low trasmttac ad rflctvty. I scto w study th stablty codtos of such plasmas showg that that although thr mass dsty s qut low thy ca xtd o xtrmly larg volums ad thrfor thy produc vry sgfcat gravtatoal forcs. Du to ths proprts w fd that such plasmas mght b rlatd to dark mattr. I scto 4 w dmostrat our rsults by umrcal calculatos. I Scto 5, w summarz our rsults.. Optcal proprts of low dsts stllar plasmas rlatd to vrs stmulatd brmsstrahlug ad to plasma trasparcy Th dffrtal cross scto for stmulatd msso or absorpto for o lctro-o collso (glctg quatum corrctos) was gv by [Kraov ; rsttsk, Lfshtz & Ptavsky 1998 ; Maakov 15]: 16π z v dω dσ = 4πε ħc mv 4πε c ω. (1)

3 Th trm 1 4πεħc 17 Eq. (1) rprsts th f structur costat. Eq. (1) s a good z > approxmato udr th codto 4πε v ħ trasformato [Kraov ; Fdorov 1997]: 1 for slow lctros. I Eq. (1) w apply th ε E ω d π c E π c ħω = ħω dω =. () εe ω dω Hr s th rgy dsty of th lctrc fld E, s th umbr of photos pr ut π c volum for th arly moochromatc lctromagtc fld wth frqucy trval dω ad ħ ω s th rgy of o photo [Kraov ; Fdorov 1997]. y substtutg Eq. () to Eq. (1) w gt 6 1 Z E dσ =, () 8 ε ω ħ p 4 whr p s th lctro momtum. Th rat of th stmulatd msso, or absorpto w, a, of th photo th Coulomb scattrg of a rlatvly slow lctro s gv by multplyg dσ by p whr s th umbr of os pr ut volum [Kraov ]: 1 Z E 1 Z E w C C 6 5, a = = ; = ε ωħ p p ω. (4) W otc that w, a s proportoal to 1 / p comparso to th dpdc o 1 / p for fast lctros. W otc also that w for stmulatd msso s qual to stmulatd absorpto wa so that th total stmulatd radato w T for o lctro-o collso zroth ordr approxmato vashs,.. [Kraov ]: wt = w wa =!. (5) I ordr to gt th total rat of vrs stmulatd absorpto of th plasma w d to multply w, a of Eq. (4) by th umbr of ozd lctros of stmulatd msso from th uppr lvl mus th umbr of ozd lctros of stmulatd absorpto from th lowr lvl ad avrag th trm 1 / p ovr th oltzma dstrbuto. Th dffrc btw th lctro rgy E th uppr lvl for ay stmulatd msso ad th corrspodg rgy of th lctro th lowr lvl E dow for th vrs trasto of stmulatd absorpto s gv by E up E dow =ħ ω. So accordg to oltzma statstcs w gt for ach stmulatd trasto th rlato up, dow Eup Edow ħω = xp xp, up kt = kt, (6)

4 whr k s oltzma costat, ad T th absolut tmpratur. Summg ovr all trastos w gt ħω ħω ħω ħω, dow =, up xp, up 1 + ; 1 ;, dow, up, up. (7) kt kt kt kt Hr ñ, dow ad ñ, up rfr to th total umbr of lctros for stmulatd msso ad for stmulatd absorpto, rspctvly, ad w assumd hr th approxmato that th photo rgy s small rlatv to th thrmal rgy: kt. W avrag th trm 1 / p ovr th oltzma dstrbuto obtag approxmatly (8) T ( 1/ p) ( 1/ k Tm ) oltz ħω W multply Eq. (4) by, dow, up ad assum avragd valu for 1 / p accordg to Eq. k T (8). Th, w gt for th avrag lctromagtc rgy U pr ut volum, absorbd by th plasma pr ut tm du ħω Z = C = E = E dt ab T k T T 6 1, up / β. (9) ω I Eq. (9) w usd th approxmato, up, dow. Th proportoalty costat β gvs th part of th lctromagtc rgy rprstd by ad ut tm. Udr th codto β Z ω T 1 = / E whch s absorbd by th plasma pr ut volum 1, (1) th plasma bcoms traspart. Ths codto dpds o th paramtrs:,, T ad ω whr for th prst lmtg cas ths codto s mprovd for lowr o ad lctro dsts ad hghr tmpraturs. O should otc that th absorpto of th plasma s proportoal th multplcato ( ) so that th absorpto of th plasma dcrass vry much for smallr valus of th lctros dsts. Th abov calculatos ar mad for slow lctros for whch For oltzma statstcs w us th approxmato of avrag vlocty codto ca b wrtt as Z 5 Z / 4πε ħ v > 1. k T v = ad th ths m > T K. (11) 5 So, for Carbo plasma ( Z = 6 ) ad Oxyg plasma ( Z = 8 ) for T = 1 K t s a good approxmato but for Hydrog plasma ( Z = 1) t s oly a far approxmato. 4

5 . Th stablty of stllar plasmas wth low dsts udr oltzma statstcs W study hr th stablty proprts of a star whch cluds plasma wth rlatvly low dsts. For smplcty of dscusso w trat a star wth sphrcal symmtry whr ts stablty s rlatd to gravtatoal forcs wthout ay othr xtral prturbatos.g. magtc flds tc. Whl ths topc was studd udr hgh lctro dsts usg Frm-Drac statstcs [Chadraskhar 1967; Edgto 1959; Kppha, Wgrt &Wss 1] w study hr th problm for low dsts usg oltzma statstcs. Th gravtatoal forc at a dstac r from th star ctr s du trly to th mass tror to ths dstac: = r /, (1) g GM r whr G s th costat of gravtato. Assumg that φ s th gravtatoal pottal th g = dφ / dr ; φ = GM / r. (1) Accordg to th hydrostatc quato r dp = gρdr. (14) whr P s th star plasma prssur ad ρ th star dsty, both ar fuctos of dstac r from th star ctr. Isrtg Eq. (1) to Eq. (14) w gt: dp = ρdφ. (15) Ths quato dscrbs th cras of th plasma prssur as w mov to largr valus of r balacg th attractv gravtatoal forcs. Assumg that th star plasma bhavs as a dal gas th th prssur P s gv by P = k T. (16) Assumg also that th gradt of tmpratur s small rlatv to th gradt of th lctro dsty (sothrmal procss) th w gt: dp = k Td = ρdφ. (17) If thr ar κ uclos for ach lctro, th th mass dsty s gv approxmatly by Hr m ρ = κ m. (18) Kg s th mass of th uclo ad w assumd hr M r = whr th umbr of ozd atoms s qual to th umbr of ozd lctros. Isrtg Eq. (18) to Eq. (17) w gt: 5

6 d κm κm κm = l ( ) + C = φ, C = l ( ) l = φ dφ kt kt kt κm ( r) = xp φ ( r) ; φ ( r = ) = ; ( r = ) = kt Ths quato dscrbs th dcras of th plasma dsty ( r) ( r) of th xpotal fucto of th pottal φ ( r). Hr φ ( r). (19) = as a fucto of th chag s chagg from zro to larg gatv valus, as fucto of th dstac r from th star ctr. Our am th followg aalyss s to fd r = r as fucto of th dstac r from th star ctr. I ths calculatos w th chag of ( ) ( ) assum that s th dsty of ozd lctros th star ctr whch s tak as xprmtal paramtr. I ordr to fd th dpdc of th lctro dsty ( r ) o th dstac r from th star ctr w d to tak to accout th Posso quato for th pottal φ whch for a star wth sphrcal symmtry has th form: φ( ) φ( ) + = 4 πgρ( r). () d r d r dr r dr Hr ρ ( r) s proportoal to th ozd lctro dsty ( r ), as gv by Eq. (18), ad G s th gravtatoal costat. Substtutg th rlato ρ = κm from Eq. (18) to Eq. () ad usg th rlato (19) for ( r ) w gt φ φ κm d ( r) d ( r) + = 4 πgκ m ( ) 4 r = πgκ m xp φ ( r) dr r dr kt. (1) W ot that o th rght sd of Eq. (1) appars a xpotal fucto of th pottal φ( r) wth a κm vry small coffct gv by. O should otc also th coffct bfor th xpotal kt s a vry small umbr for low lctro dsts. It s dffcult to gt xplct solutos to ths quato, as srs xpaso of ths xpotal fucto covrgs vry slowly for larg valus of r. W prfr r. It wll gv aftr som th followg aalyss to trasform Eq. (1) to dffrtal quato for ( ) calculatos th chag of ( r ), from ts tal valu ( r = ) = at th star ctr (tak as xprmtal paramtr), to smallr valus as a fucto of th dstac r from th star ctr. Accordg to Eq. (19) w gt: 6

7 ( ) m ( ) ( ) 1 ( ) r κ φ r φ r r kt = ( r) = r kt r r ( r) r κm ( ) 1 ( ) 1 ( ) φ r r r kt = + r ( r) r ( r) r κm. () Isrtg Eq. (18) ad th pottal drvatvs accordg to Eq. () to Eq. () w gt: ( ) ( ) ( ) 1 r 1 r 1 1 r kt G m = π κ ( r) r ( r) r ( r) r r κm. () Multplyg Eq. () by ( r ) ad rarragg th trms: ( ) 1 ( ) 1 ( ) ( κ ) r r r m + G = π r r r ( r) r kt 4 ( r). (4) W df ( ) ( r θ r = ), ad dvd Eq. (4) by th w gt: ( ) ( ) ( ) ( ) ( ) θ ( r) θ r 1 1 θ r κm 46 κ θ r + G r = π θ = r r r θ ( r) r kt T W df 4 ( ).54 1, ( r) θ ( r) = ; = ( r = ) (5) ( ) κ κ m 46 ξ = 4π G =.54 1 ; x = r ξ r.71 1 κ. (6) k T T T W dvd Eq. (5), for θ ( r) by ξ. Th, ths quato ca b wrtt as: ( ) θ ( x) ( x) θ ( x) θ = θ x x x θ ( x) x ( x). (7) Eq. (7) dscrbs th chag of th ormalzd lctro dsty ( ) / ( ) x = θ x as fucto of th ormalzd dstac x = r ξ from th star ctr. Ths quato dscrbs also th chag of th mass 7 dsty ρ( Kg m ) = κm ( m = Kg ) as fucto of th dstac from th star ctr. Hr m s th mass of uclo ad κ s th rato btw th dsty of uclos ad that of lctros. 7

8 (For compltly ozd plasma: for Hydrog plasma κ =, for Carbo κ = 1 ad for Oxyg κ = 16 whl for partal ozd plasma ths umbrs should cras vrsly proportoal to th part of ozato). y glctg th thrd trm o th lft sd of Eq. (7) w obta a dffrtal quato whch was solvd umrcally ad gv as th La-Emd quato wth = [.g. Moha & Al-ayaty 198]. It shows th chag of th valu of θ ( x) = 1, for x =, at r =, to th valu θ ( x) = for x = 1 = ξ R whr R = 1 / ξ s approxmatly th star radus.. Such bhavor occurs also for th solutos of th full Eq. (7) but sc th thrd trm, o th lft sd of Eq. (7), caot b glctd th θ x s dffrt from th La-Emd quato (wth xpot = ). Lk th form of chag of ( ) La-Emd quato th rag 1 x = dscrbs approxmatly th rag of valus of θ ( x) 8, ad by substtutg Eq. (6) th valu x = 1 = R ξ (whr ξ th prst codtos s xtrmly small umbr) w fd that th radus R of th star strtchs ovr a vry log dstac whch s of ordr T R = 1/ ξ ( m). Lt us gv mor xplct stmatos for th ordr of magtuds to th star radus ad star mass: Ordr of magtuds for th radus ad mass of th low dsty star udr oltzma statstcs: Th radus of th star s th gv approxmatly accordg to Eq. (6) by ( ) R star radus k T T. (8) ( m ) ξ 4π G ( κm ) κ W gt hr th trstg rsult that th radus of th star R crass vrsly proportoal to th squar root of ts lctro dsty th star ctr. Th total mass M star of th star s proportoal to ts volum, ad to th avrag mass dsty gv accordg to Eq. (18) as κ m whr s th avrag lctro dsty gv approxmatly by. Th w gt: / / / π k T 9 T κ = = m πg κ κ 4π M star ( star mass) R m ( Kg). (9) 4 Hr w substtutd th valu of th star radus accordg to Eq. (8) ad srtd a avragd valu for th lctro dsty whch s proportoal to ts dsty o th star ctr. W fd that th star / mass s crasg wth tmpratur as t s proportoal to T. W fd also th astoshg rsult that th star mass s vrsly proportoal to th squar root of th dsty of th lctros ts ctr du to cras th volum proportoal to R. Th abov ffcts ca b summarzd as follows: for low dsts plasma th balac btw th dal gas radato prssur ad gravtatoal forcs lads to xtrmly larg valu for th star radus so that although th mass dsty s qut small th star volum s xtrmly larg so that t

9 lads to vry larg star mass wth strog gravtatoal forcs. W dmostrat ths ffcts th xt 6 scto by umrcal calculatos for thr xampls whch: a) 1 ( m = ), b) 1 ( m ) c) = 1 ( m ) ad for all ths cass wth th tmpratur =, 5 1, K. Ths rsults ar dcrasg somwhat for lowr tmpraturs as th star radus s proportoal to T ad th star mass s / proportoal to T. W dmostrat our rsults by xplct calculatos oly for xampls. Th prst ffcts bcom v much strogr for lowr lctro dsts. 4. umrcal Calculatos W dmostrat our rsults th followg xampls. 6 Star wth lctro dsty 1 ( m ) Accordg to Eq. (8) w gt: =, tmpratur 5 1, K R ( star radus) 1 ( m) = 6 ( m) = ( km). () κ 1 κ κ For compltly ozd Hydrog plasma κ =, so that th radus of such star s gv by: 9. 1 km. It has a radus whch s largr approxmatly by a factor rlatv to th Solar ( ) radus! Th radus of th prst star bcoms smallr for Carbo or Oxyg plasma for whch κ = 1 or 16, rspctvly. Th radus s also dcrasd for lowr tmpratur as accordg to Eq. (8) t s proportoal to T. W hav to tak to accout that for partal ozato th paramtr κ crass vrsly proportoal to th part of ozato ad thus th star radus s dcrasd furthr. Substtutg (9) T = 1, = 1, th w gt : / M star ( star mass) ( Kg) = ( Kg ). (1) 6 κ 1 κ Ths mass, for compltly ozd Hydrog plasma ( κ = ) s largr approxmatly by factor / rlatv to th solar mass. For smallr tmpratur ths mass dcrass as t s proportoal to T. It also dcrass for largr valus of κ for Carbo ad Oxyg plasma ad also for partal ozato. Th abov ffcts bcom much strogr for smallr dsts as aalyzd th followg xampls. For compltly ozd plasma w assum Eq. (1) th approxmato = / (glctg utral atoms) ad for plasma bcoms: 6 5 T K = 1 ; = 1, th trasparcy codto for th Z 1 β 1. () κ ω 9

10 For hydrog plasma whch s th most commo plasma w hav Z = 1 ; κ =, ad th codto 5 for trasparcy s gv by ω So, for th optcal rgo ad ar frard th Hydrog plasma s traspart but for th far frard th absorbac bcoms larg. Also th absorbac bcoms much largr for plasmas for whch Z 1 ad dcrass for plasmas for whch: κ >. Th codto for trasparcy s mprovd vry much for lowr dsts, as tratd by th followg xampls. Star wth lctro dsty 1 ( m ) =, tmpratur 5 1, K Substtutg Eq. (8) = 1, 5 T = 1 w gt: R ( star radus) 1 ( m) ( m) = ( km). () κ 1 κ κ For compltly ozd Hydrog plasma κ =, ad such star has a radus whch s largr 5 approxmatly by a factor 1 rlatv to th Solar Radus! Ths radus bcoms smallr for plasmas for whch κ > ad for lowr tmpraturs. Substtutg Eq. (9) th valust = 1, = 1, th w gt: / M star ( star mass) ( Kg). (4) κ 1 κ So, for compltly ozd hydrog plasma for whch: κ =, th prst star mass s largr 4 approxmatly by factor 6. 1 tha th solar mass. For plasmas wth valus of κ largr tha ths valu ca b rducd but ay cas t gvs vry larg mass for low lctro dsty star du to th xtrmly larg volum. For compltly ozd plasma w assum Eq. (1) th approxmato = / (glctg utral atoms) ad for plasma bcoms = 1 ; = 1,, th trasparcy codto for th 5 T K Z 1 β (5) ω For hydrog plasma w hav Z = 1, ad th plasma bcoms compltly traspart for 1 1 ω > 1 (sc ) whch cluds th tr sgfcat lctromagtc spctrum. Th absorbac bcoms largr for plasmas for whch 1 Z. Star wth lctro dsty 1 ( m ) =, tmpratur 5 1, K Substtutg Eq. (8) 5 = 1, T = 1 w gt: R ( star radus) 1 ( m) ( m) = ( km). (6) κ 1 κ κ 1

11 For compltly ozd Hydrog plasma: κ =, so that th radus of such star has a radus whch s 6 largr approxmatly by a factor.16 1 rlatv to th Solar Radus. Ths radus bcoms smallr for plasmas for whch κ > ad for lowr tmpraturs. Substtutg Eq. (9) th valust = 1, = 1, th w gt: 5 M star ( star mass) 9 15/ (7) κ 1 κ So, for compltly ozd hydrog plasma for whch: κ =, th prst star mass s largr 6 approxmatly by factor 6 1 tha th solar mass. For plasmas wth valus of κ largr tha ths valu ca b rducd but ay cas t gvs larg mass for low lctro dsty star du to th xtrmly larg volum. For compltly ozd plasma w assum Eq. (1) th approxmato / 5 = (glctg utral atoms) ad for = 1 ; T = 1 K, th trasparcy codto for th plasma bcoms 6 Z 1 β (8) ω For ths cas Hydrog, Carbo ad Oxyg plasmas bcom traspart ovr all th tr sgfcat lctromagtc spctrum. W dmostratd th prst das by aalyzg xampls. Th prst ffcts bcom v strogr, f th dsty of lctros s rducd furthr. W assumd our aalyss a star wth sphrcal symmtry udr th opposg forcs: gravtatoal forcs ad radato prssur for plasma bhavg as a dal gas. Th assumpto of a star wth sphrcal symmtry s assumd hr maly du to mathmatcal covc. I xpct that smlar ffcts wll occur low dsty star plasmas whch th plasmas wll hav dffrt varous structurs. It sms that th prst aalyss s rlvat to dark mattr. 5. Summary Dscussos ad Coclusos I th prst work w hav show that th absorpto of low ozd stllar plasmas wth o oc compot wth Z atomc umbr, satsfyg oltzma statstcs, whch th codto Z / 4πε ħ v > 1 s satsfd, ca b tratd approxmatly by thors about stmulatd rmsstrahlug. Th stmulatd msso mus stmulatd absorpto lads to vrs stmulatd absorpto whch ca b glctd udr th codto gv by Eq. (1). As ths codto s proportoal to multplcato of th dsty of lctros tms th dsty of ozd os assumg (spcally corrct for Hydrog ozd plasma) th plasma bcoms traspart for small such dsts. Ths codto s mprovd for smallr valus of th atomc umbr Z (Z=1 for Hydrog) ad for hghr tmpraturs. Th mass dsty ρ of a star basd o th prst codtos s gv by Eq. (18) as ρ = κm whr m s th mass of uclo ad κ s th umbr of uclos pr o lctro (for Hydrog compltly ozd plasma κ = ). W fd that for th prst traspart plasmas th mass dsty s vry small but, as aalyzd th prst artcl, th stablty codtos for such plasmas lads to xtrmly larg volums so that gravtatoal forcs bcom vry larg. I th prst work w assumd that th star plasma bhavs as a dal gas whr th prssur 11

12 P = kt was gv by Eq. (16), whr k s th oltzma costat. Th stablty of th star wth low lctro dsts s obtad by cosdrg t to b qulbrum btw two opposg forcs: Th rpulsv dal gas lctro prssur ad th attractv gravtatoal forc. For a star wth sphrcal symmtry th balac btw ths oppost forcs lads to Eq. (19). I ths quato th dsty of lctros s gv by a xpotal fucto of th pottal wth a small coffct th xpot. I prcpl, ths quato ca b solvd by takg to accout th Posso quato for th pottal gv Eq. (). Sc ths quatos ar dffcult to solv w prfrrd th prst aalyss to trasform Eq. () to a dffrtal quato for th dsty of lctros ( ) r as fucto of th dstac r from th star ctr. Aftr som calculatos w obtad th dffrtal quato for ( r ) Eq. (). Aftr som addtoal trasformatos w obtad th dffrtal quato (7) for th ormalzd lctro dsty θ ( r) = ( r) / as fucto of th ormalzd dstac x = r ξ of Eq. (6). W stmatd th star radus R as 1 / ξ whr ξ s a vry small radus gv by Eq. (6). Usg ths stmat w mad umrcal calculatos for th star radus ad mass o scto 4, for xampls for whch th lctro dsty th mass ctr s gv by 6 a) = 1 m, b) = 1 m, c) = 1 m. W dmostrat by ths calculatos th ( ) ( ) ( ) trstg rsult by whch th radus R of th star crass vrsly proportoal to th squar root of th lctro dsty at th star ctr as gv by Eq. (8). Th mass dsty of th star crass proportoal to th lctro dsty as gv by Eq. (18). Th star volum, howvr, crass proportoal to R ladg th total mass of th star to cras vrsly proporto to th squar root of th lctro dsty at th star ctr as gv by Eq. (9)! Usually o would thk that brmsstrahlug phoma caot b gord wh lctros ar colldg wth os but w should tak to accout that, udr th codtos prstd th prst work, brmsstrahlug stmulatd msso volvs also brmsstrahlug stmulatd absorpto so that th summato of ths two opposg ffcts lads to vrs stmulatd absorpto as gv by Eq. (9). Sc ths absorpto s proportoal to th multplcato of th dsty of lctros ( r ) by th o dsts ( r ) (whr ( r) ( r) ) ths absorpto s rducd vry much for lowr dsts ad vashs ovr all th tr sgfcat lctromagtc spctrum, for ( ) 1 ( ) r m, as dmostratd by th umrcal calculatos scto 4. Tryg to rlat th prst work to crta phoma about dark mattr w d to us th followg cosdratos. I th abov aalyss w obtad xtrmly larg volums for plasma wth low lctro dsty, whr w got th rsult that as w dcras th dsty of th lctros th volums crass vry much. Th astoshg rsult from th prst aalyss s that a star wth low mass dsty xtds ovr xtrmly larg volums so that th total mass s vry larg ad gravtatoal forcs bcom vry sgfcat. Ths phoma whch sm to b qut strag follow from th fact that ordr that th gravtatoal forcs of low dsty star wll b qulbrum wth th radato prssur of dal lctro gas thy should xtd o largr ad largr volums as th mass dsty s dcrasd. Th prst aalyss of a star wth sphrcal phoma was mad for smplcty of calculatos but t dos ot mply that such plasmas wll hav sphrcal symmtry sc smlar ffcts wll occur o othr dffrt structurs of th plasmas whch th xplct calculato should b dffrt. Fally, followg a artcl [-Aryh 1967] whch was publshd qut log ago, th mssvty (th rato btw th xprmtal radato ad th black body radato) s qual to th absorbac so that for traspart plasma th mssvty vashs ad w wll ot obsrv ay radato mttd from such plasma. I dd ot tr th larg ad complcatd fld of dark mattr but oly potd out crta smlarts btw th proprts of th prst plasmas ad thos of dark mattr. 1

13 Rfrcs Ashcroft,. Mrm,. D Sold Stat Physcs ( Phladlpha: Saudrs) -Aryh, Y. 1967, Appl. Opt. 6, 149 rsttsk, V.., Lfschtz, E.M. &Ptavsk, L.P. 1998, Quatum Elctrodyamcs (Oxford: uttrword- Hma). Eq. (9.) rto, G. & Hoopr, D. 18 Rv. Mod. Phys. 9, 45 Chadraskhar, S A Itroducto to Stllar Structur (w York: Dovr) Ch, F. F. 199 Itroducto to Plasma Physcs ad Cotrolld Fuso (w York: Plum Prss) Eddgto, A.S Th tral costtuto of stars (w York: Dovr) Fdorov, M. V. 1997, Atoc ad Fr Elctros a Strog Lght Fld (Sgapor: World Sctfc): Ftzpatrck, R. 15. Plasma Physcs: A Itroducto (w York: CRS Prss) Gzburg, V. L Proprts of Elctromagtc Wavs Plasma (Lodo: Gordo ad rach) Kppha, K., Wgrt, A. &Wss, A. 1. Stllar Structur ad Evoluto (w York: Sprgr) Kraov, V. P., J. Phys. : At. Mol. Opt. Phys., 1595 Maakov,. L., Krylovtsky, A. A. &Marmo, S. I. 15, J. Exp. Thor. Phys. 11, 77, Eq. (1) Marcus, D. 196 ll Syst. Tch. J. 41, 1557 Myr-tr-Vh, Tror, A. &Cag, Y. 8, 5 th EPS Cofrc o Plasma Physcs D, P-.1 Moha, C, & Al-ayaty, A. R. 198, Astrophyscs ad Spac Scc 7, 7 Pl, A. 1 Plasma Physcs (rl: Sprgr) Sgr, T Smcoductor Physcs (rl: Sprgr) Trmbl, V A. Rv. Astro. Astrophys. 5, 45 1