Fermi Gas. separation

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Sarah Campbell
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1 ri Gas Distiguishabl <---> Idistiguishabl Classical <----> dgrat dd o dsity. If th wavlgth siilar to th saratio tha dgrat ri gas articl h saratio largr traturs hav sallr wavlgth d tightr ackig for dgracy to occur dgrat lctro xals - coductors ad sicoductors - rssur at arth s cor (at last so of it) -aids i iitiatig trasitio fro Mai Squc stars to Rd Giats (allows T to grow lctro rssur is iddt of T) - whit dwarvs ad Iro cor of assiv stars Nutro ad roto xals - ucli with ri otu = 5 MV/c - utro stars 1 1/

2 Dgrat vs o-dgrat

3 Coductio lctros Most lctros i a tal ar attachd to idividual atos. But 1- ar fr to ov through th lattic. Ca trat th as a gas (i a D box) or lik a fiit wll but rgy lvls (ad dsity of stats) siilar (ot boud stats but vibratioal stats of lctros i box) dth of wll V = W (rgy dd for lctro to b rovd fro tal s surfac - hotolctric ffct) + ri rgy at T = all stats u to ar filld V illd lvls W

4 Coductio lctros Ca th calculat th ri rgy for T= (ad it dos t usually chag uch for highr T) N ( ) D( ) d T= D( ) d 8V ( h h N 8 V x. Silvr 1 fr lctro/ato / ) 1/ 1/ d or N V atos/ ol 1.5g 18 g / ol c fr lctros / c 4 (6.61 joul sc) kg (14V) 8.511MV ato / 8 / 5.5V 1V J 4

5 Coductio lctros Ca dtri th avrag rgy at T = for silvr. V ca coar to classical statistics T.V k Pauli xclusio forcs lctros to uch highr rgy lvls at low traturs. (why s ot ivolvd i scific hat which is a lattic vibratio/hoos) D( ) ( ) d D( ) ( ) d 1/ 1/ d d.v 8.61 V / K 5 5/ / 5 o 4, 5 K 5

6 Coductio lctros 6

7 Coductio lctros Siilarly, fro T-ddt 1 Nh kt1 5/ /... V (kt) th trs aftr th 1 ar th dgracy trs.larg if dgrat. or silvr atos at T= K ot util th dgracy tr is sall will th lctro act classically. Has at high T Th ri rgy varis slowly with T ad at T= K is alost th sa as at T= You obtai th ri rgy by oralizatio. Quark-gluo lasa (covrd latr) is a xal of a high T ri gas kt 1 ( ) () 8 7

8 ri Gass i Stars quilibriu: balac btw gravitatioal rssur ad gas (ithr oral or dgrat) rssur total gravitatioal rgy: dsity varis i oral stars (i Su: avrag is 1 g/c but at r= is 1 g/c ). Mor of a costat i whit dwarvs or utro stars will hav ithr oral gas rssur of P=kT (P=<>) or rssur du to dgrat articls. Noral dds o T, dgrat (ostly) dos t = articl dsity i this cas GM ( r) Mass( r r) r R ( r) r G 4r dr 4 r ( r) dr r M G if cos ta t rssur 5 R V 8

9 Start with = <> o-rlativistic N P 5 1 P dds ONLY o dsity Dgrat ri Gas Prssur dsity stats : D( ) D( ) 1/ A 1/ / K 1/ / d 1/ d K 1 d / 5/ 4 K rlativistic both A d 1/ d K cass d 1/ 1/ 4 / 9

10 o-rlativistic P 5 1 / K / Dgrat ri Gas Prssur K 1 / 5/ 4 K 1/ rlativistic K 1/ 1/ 4/ P dds ONLY o dsity Prssur dcrass if, for a giv dsity, articls bco rlativistic if shrik star s radius by dsity icrass by 8 gravitatioal icrass by if o-rlativistic. <> icrass by (N/V) / = 4 if rlativistic <> icrass by (N/V) 1/ = o-rlativistic stabl but rlativistic is ot. ca aid i collas of whit dwarf 1

11 Oldr Su-lik Stars SKIP Dsity of cor icrass as HH. H irt (o fusio yt). Cor cotracts lctros bco dgrat. 4 r H ucli. lctros hav logr wavlgth tha H thral quilibriu lctros ov to highr rgy du to Pauli xclusio/dgracy. No logr i thral quilibriu with, H ucli rssur bcos doiatd by lctros. No logr dds o T P total P P H allows T of,h to icras raidly without oral icras i rssur ad chag i star s quilibriu. Ost of HC fusio ad Rd Giat has (hliu flash wh T = 1,, K) P H H P H P H H P H H H 11

12 Whit Dwarvs Lftovr cors of Rd Giats ad (usually) fro C + O ucli ad dgrat lctros cors of vry assiv stars ar ucli lus dgrat lctros ad hav siilar rortis gravitatioal rssur balacd by lctros rssur which icrass as radius dcrass radius dds o Mass of star Dtri aroxiat ri rgy. Assu lctro dsity =.5(+) dsity N V M Su 1 1 volu arth lctros ar i this rag ad oft ot coltly rlativistic or o-rlativistic d to us th corrct = + rlatioshi h N ( o rl) 8 V. MV N 1/ ( rl) hc.8 MV V 8 / 51 5 / 1

13 Whit Dwarvs + Collas If th lctro rgy is > about 1.4 MV ca hav: Thrshold 1. 4MV ay lctros > T disaar. Th lctro rgy distributio dds o T (avrag ) # s T th lost lctros caus th rssur fro th dgrat lctros to dcras th rgy of th utrios is also lost as thy sca cools th star as th ass icrass, radius dcrass, ad ubr of lctros abov thrshold icrass 1

14 Whit Dwarvs+Surovas SKIP aothr rocss - hotodistgratio - also absorbs rgy coolig star. Siilar rgy loss as + cobiatio H H 4 At so oit th ot vry stabl quilibriu btw gravity ad (ostly) lctro rssur dos t hold Whit Dwarf collass ad so fractio (-5%??) of th rotos covrt to utros durig th collas givs Surovas L L SN SN ( light ) 1 9 L Su ( light ) ( utrios ) 1(1?) L SN ( light ) 14

15 Whit Dwarvs+Surovas 15

16 Nutro Stars-arox. ubrs SKIP Surovas ca roduc utro stars - radius ~ 1 k - ass about that of Su. always < ass Su - rlativ :: ~ 99:1:1 gravity suortd by dgrat utros lug ito o-rlativistic forula for ri rgy 14 MV (as ass =94 MV, o-rl OK) look at wavlgth saratio ca dtri radius vs ass (lik WD) N V ca collas ito black hol M Su /.6 / 4 (1k) h 1. h 14MV 94MV 14MV 16

17 Nutro Stars SKIP sarat ri gass: :: + ar i th sa ottial wll du to strog uclar forc assu iddt ad that / =.1 (dds o star s ass) ( ) ( ) so d to us rlativistic for lctros but ot iddt as <---> ( ) ( ) lus ractios with virtual articls N / 145MV V 5 6MV N / 145MV 94 1,MV ( wrog) V fr utros dcay. But i a utro star thy ca oly do so if thr is a availabl ufilld lctro stat. So sursss dcay 5 N 1/ hc MV ( ) 5 V

18 Nutro Stars SKIP Will d u with a quilibriu btw -- which ca bst b s by atchig u th ri rgy of th utros with th - syst utros with > ca th dcay to --u (which raiss lctro dsity ad its ri rgy thus th balac) d to iclud rst ass rgis. Also dsity of lctros is qual to that of rotos ca th solv for / ratio (w ll ski algbra) givs for tyical utro star: 18 / / 1/ ) ( ) ( ) ( c h h c hc / / 1 1 / kg

How many neutrino species?

How many neutrino species? ow may utrio scis? Two mthods for dtrmii it lium abudac i uivrs At a collidr umbr of utrio scis Exasio of th uivrs is ovrd by th Fridma quatio R R 8G tot Kc R Whr: :ubblcostat G :Gravitatioal costat 6.