Problem 1: CARBONATED WATER

Transcription

1 Problm 1: CARBONATED WATER Suppos you hav a rgular 1-litr factory sald bottl of carbonatd watr. You hav turnd th bottl cap slightly to unscrw it (so a hissing sound was hard) and scrw th cap tightly again. Now you would obsrv bubbls of carbon dioxid (CO ) rising upwards larg, at first, and thn smallr ons. Lt us study th procss of bubbl surfacing. On can asily s that th shap of a small bubbl is much closr to sphrical than that of a biggr on. 1) Estimat th maximal siz of an immobil bubbl such that th bubbl shap approximats sphr with an accuracy of 10% or bttr. Watr dnsity is 1 g/cm 3, th surfac tnsion 0,07 N/m, and th fr fall acclration is takn to b g 9, 8 m/s. Th numrical answr should b in mm. Considr a bubbl so small that it can b rgardd as almost sphrical. For instanc, lt th bubbl initial diamtr nar th bottl bottom b d 0, 0 3 mm. ) Figur out th bubbl acclration right aftr it has dtachd from th bottom. Th CO dnsity insid th bubbl at th givn watr tmpratur is 0, v 00 g/cm3. Th numrical answr should b in m/s. Drag forc xrtd on a bubbl, whn it is moving in watr, is linarly proportional to its cross-sctional ara, watr dnsity, and th bubbl vlocity squard. W assum th proportionality factor for this problm to b 0,. 3) Suppos that th bubbl volum rmains constant. What is th trminal vlocity th bubbl can rach? Writ down th quation and valuat th numrical valu (in m/s). 4) Estimat th tim it taks th bubbl to rach th trminal vlocity aftr dtachmnt, i.. whn its acclration bcoms much lss than g. Writ down th quation and valuat th numrical valu (in sconds). Actually, th dnsity of CO molculs dissolvd in a liquid is much highr than that in a gas bubbl. Thrfor, th dissolvd gas diffuss into th bubbl and its radius grows. It is rasonabl to assum that th growth rat of th bubbl volum du to th diffusion is proportional to th bubbl surfac ara and to an xcss of th dissolvd gas dnsity and invrsly proportional to th thicknss of th liquid layr through which th gas diffuss (calld th «dpltion layr»). Th fastr bubbl is moving th thinnr is th ffctiv dpltion layr (du to circulation of watr surrounding th bubbl). A kintic thory calculation givs for th dpltion layr thicknss: v is th bubbl vlocity and d is its diamtr. 1 d const, whr v 5) Suppos it taks tim T for a bubbl to surfac. Figur out th bubbl diamtr as a function of tim t, providing it has incrasd by a factor k 3 4/ 5, 4 should b an quation for d (t) in trms of T, t, d 0, and dnsity of CO dissolvd in watr during th bubbl surfacing. during th surfacing. Th answr 4 / 5 k 3. Nglct a chang in th

2 6) Dtrmin tim dpndnc v (t) of th bubbl vlocity (th answr should includ a formula for v (t) xprssd in trms of th paramtrs listd abov and th trminal vlocity v 0 ). 7) Dtrmin th law of motion of a bubbl, i.. tim dpndnc of its lvation h abov th bottom (th answr should includ a formula for h (t) in trms of T, t, and v 0 ; tak th sam 4/5 valu k 3 ). 8) Suppos th hight of watr column, which th bubbl travrss on its way upward, quals H 30 cm. Evaluat th tim of bubbl surfacing (th answr should b th xplicit formula including th paramtrs givn in th problm and calculat th numrical valu in sconds). During bubbl surfacing som hat is bing rlasd (th drag forc dos a work by incrasing th kintic nrgy of turbulnt flow which vntually dissipats as hat) and at th sam tim som hat is bing absorbd (du to CO vaporation from watr into a bubbl). 9) Assum that all th hat rlasd during th surfacing is convrtd into hating th «column» of watr which cross-sction quals th avrag cross-sction of a rising bubbl. Using this assumption stimat (by th ordr of magnitud) th tmpratur incrmnt of watr in th «column». Th spcific hat capacity of watr is c 400 J/kg K. Th answr should b givn in Klvin. 10) Evaporation into bubbls of 1 mol of CO dissolvd in watr rquirs approximatly 0 kj of nrgy. Estimat (by th ordr of magnitud) th cooling of th watr «column» (s 9)) du to this ffct during th bubbl rising and compar to th hating ffct (s 9)). Th answr should b givn in Klvin. What will th nt rsult b? Th answr should b ithr «+» (th tmpratur incrass) or (th tmpratur dcrass). Assum that th prssur and tmpratur in th bottl chang slightly and rmain clos to p 10 kpa and T 90 K. w 0 0 Problm : RADIOGRAPHY In 1943 thr was foundd «Laboratory» of th USSR Acadmy of Scincs, a Sovit physicist Igor Vasilyvich Kurchatov was appointd as its dirctor. Th laboratory mostly concntratd on th dvlopmnt of nuclar ractor and nuclar wapons. Sinc thn many yars hav passd, th laboratory turnd into a larg rsarch cntr and its nam has changd svral tims. Now it is calld th National Rsarch Cntr "Kurchatov Institut", on of th largst Russian scintific cntrs which dos rsarch in a varity of filds. Th main ara of th institut rsarch includs solid-stat physics and matrials scinc. Industrial radiography is on of th basic modrn mthods of matrials scinc. A studid objct is placd in a cohrnt monochromatic bam of X-ray photons of high intnsity which scattring pattrn is thn analyzd. Somtims mthods of spctromtry ar usd, i.. variation of intnsity of th bam passing through a matrial is masurd as a function of radiation wavlngth. Howvr, th most common mthods of study of atomic structur ar diffraction mthods. Thy ar basd on analysis of th diffraction pattrn rsulting from lastic scattring of X-rays by atoms of th sampl. Notic that radiation wavlngth rmains constant in lastic scattring. A common sourc of cohrnt X-rays of high intnsity with a wid wavlngth spctrum is synchrotron a larg ring storag of chargd particls travling at a spd clos to th spd of light. Such particls ar calld rlativistic bcaus thir motion is no longr dscribd by Nwtonian laws of classical mchanics, instad on must us th spcial thory of rlativity (STR). Th oprating principl of synchrotron is basd on th fact that a chargd particl mits lctromagntic radiation at trajctory

3 turns. Dirction of particl vlocity is changd by spcial bnding magnts. A particl (.g. lctron) path in th synchrotron ring consists of straight sgmnts, whr lctrons rciv kintic nrgy, and sgmnts of almost constant curvatur in a strong magntic fild of bnding magnts. In th curvd sgmnts lctron motion is highly acclratd, so thy mit lctromagntic radiation in th X-ray rang. A powrful synchrotron is oprating at th Kurchatov Institut in Moscow. Rlativistic quation of motion of a chargd particl in magntic fild is whr th particl momntum ), th quantity is th particl invariant mass and mass. Th nrgy of rlativistic particl is masurd in lctronvolts: 1 V (1, C) (1 V). Th factor «Lorntz factor». If th particl spd is clos to с,. [ ],. Hr с is th spd of light in vacuum ( is th particl rlativistic. Usually th nrgy of a microparticl is calld th 1) Evaluat th Lorntz factor for an lctron of nrgy E =,5 GV (th invariant mass is kg, th rst nrgy is MV). By how many prcnt is th spd of such an lctron lss than с? Th lctron charg is C. Th answr should includ formula and numrical valus. ) Dtrmin th curvatur radius of lctron trajctory in th fild of bnding magnt if lctron nrgy in th synchrotron storag ring is maintaind at E =,5 GV and th induction of th fild of bnding magnt is B = 1,7 T. Elctrons ar travling in a plan prpndicular to th magntic fild lins. Th answr should includ th formula and th numrical valu (in mtrs). Any acclrating chargd particl mits lctromagntic radiation. Th important fatur of synchrotron radiation (i.. radiation of rlativistic particls with travling along a curvd path) is its «sarchlight» natur: almost all th nrgy is radiatd «forward» along th particl vlocity in a narrow con with half an aprtur of (s Fig.1). Fig. 1. 3) Suppos an «obsrvr» О rsids at th circular orbit plan and can b rgardd as a point. In this cas h (or sh) dtcts radiation flashs corrsponding to th short priods whn sh is insid th «sarchlight» con of orbiting particl. Dtrmin th lngth of th arc travrsd by 3

4 lctron on th orbit whn its radiation is dtctd by th obsrvr. Th lctron nrgy and th magntic induction ar givn in ). Th answr should b th quation. 4) Dtrmin duration of th radiation «flash» dtctd by th obsrvr. It must b takn into account that du to lctron rlativistic motion it taks diffrnt tim for th «initial» and «final» portions of th «flash» to rach th obsrvr. Th answr should includ th quation and th numrical valu (in sconds). Thus, radiation of rlativistic particl travling along a circular path is obsrvd as a bright short flash. Th flash spctrum (frquncis and wavlngths) turns out to b vry wid: th width of th frquncy rang corrsponds to «charactristic» (or «synchrotron») frquncy wid rang of wavlngths provids a lot of opportunitis for using synchrotron radiation in radiography. Th charactristic wavlngth of a particular sourc is an important quantity for practical applications. 5) Dtrmin th charactristic wavlngth of th sourc dscribd in ). Th answr must b th quation and th numrical valu (in mtrs). Th main mthod of dciphring th structur of a crystal matrial is X-ray diffraction. Th radiation is bing diffractd (lastically scattrd) by atoms of a sampl. Th crystal srvs as a diffraction grid for X-ray bam bcaus its wavlngth is of th sam ordr of magnitud as a spacing btwn atomic plans. Whn th radiation is incidnt on th crystal at som angl, th rflctd radiation is dtctd not only in th dirction dtrmind b th laws of gomtrical optics but also at th angls for which th wavs rflctd by adjacnt plans hav optical path diffrnc qual to an intgr of radiation wavlngth. Ths rflctd wavs mutually amplify at a rmot dtctor rsulting in a significant ris of intnsity in th corrsponding dirction (diffraction maximum). 6) Using th condition of diffraction maximum driv th xplicit formula for th dirction at a diffraction maximum of X-rays rflctd by a crystal which lattic consists of a singl st of paralll plans. Th bam is incidnt at th angl to th plans, th intrplan spacing quals d (s Fig.).. A θ d Fig.. In a ral crystal structur it possibl to introduc diffrnt sts of quidistant paralll plans. Such a st can b dfind by a vctor prpndicular to th plans whil dirctions at th diffraction maxima dfind by th condition drivd in 6) can b spcifid by diffraction angl (th angl btwn th incidnt and diffraction bams). In what follows th obsrvd maximum of a givn 4

5 ordr for a spcific st of paralll plans is calld «rflx». Any atomic plan ncssarily passs through th nods of crystal lattic, so coordinats of th vctor prpndicular to a particular st of paralll plans can b givn by intgrs providing th coordinat axs ar alignd with th lattic principal axs (dgs) and a distanc is masurd in lattic constants. Thus, a rflx can b dtrmind by a st of thr intgrs. Suprconductor is a vry intrsting objct for th modrn matrials scinc. On of th most common and widly usd low tmpratur suprconductor is triniobium-tin Nb 3 Sn. For instanc, this suprconductor is usd in lctrical circuits of th Larg Hadron Collidr. A unit cll of its lattic structur (i.. a cll which rptitiv translation along principal axs rproducs th whol crystal) is a cub of sid L = 5,9 Å (1 Å = 10-8 cm). 7) Find th diffraction angl (th angl btwn th incidnt and diffractd bams) for rflx (110) at th first ordr of diffraction using th valu of charactristic wavlngth calculatd abov. Th answr should b th formula and th numrical valu (in radians). Fig. 3. In th Cartsian fram, which coordinat axs ar alignd with th dgs of crystal lattic, th coordinats of Sn atoms (in th units of L) ar: (0; 0; 0), (0,5; 0,5; 0,5), and th coordinats of Nb atoms ar: (0,5; 0; 0,5), (0,75; 0; 0,5), (0,5; 0,5; 0), (0,5; 0,75; 0), (0; 0,5; 0,5), (0; 0,5; 0,75), (s Fig.3). Atoms in th unit cll ar numbrd from 1 to 8 in th ordr thy ar listd in th txt. A pattrn of X-ray scattring is dtrmind by th distribution of lctrons in a crystal lattic (lctrons ar so much lightr than atomic nucli, so thy rspond much strongr to th lctromagntic fild of incidnt wav), i.. by distribution of atoms of various lmnts. Th ability of an isolatd atom of a crtain lmnt А to scattr radiation is spcifid by a quantity f(a) calld atomic scattring factor. This quantity spcifis th diffrnc of wav scattring by lctronic shll of a givn atom compard to that on by fr lctrons. Atomic scattring factor is a complx quantity (i.., whr is imaginary unit) and its absolut valu squard dtrmins intnsity of scattrd radiation of an isolatd atom. Th intnsity of an obsrvd diffraction pak for a crystal lattic is calculatd as th squard absolut valu of th rflx structur factor, which in turn is valuatd as: 5

6 whr sum is ovr all atomic positions in th unit cll. Hr (x, y, z) ar coordinats of atom at th n-th position; is th atomic scattring factor of lmnt А which atom rsids at th n-th position, and is occupation of th position by th lmnt. Occupation of a position is th avrag (ovr th whol lattic) numbr of atoms of a crtain lmnt at th position. In idal crystal (i.. whn all atoms of crystal lattic rsid at thir nods) ach position is occupid xactly by a singl atom of a givn lmnt and thr ar no «xtra» atoms, i.. an occupation is ithr 1 or 0. For xampl, in th unit cll of Nb 3 Sn at th 1-st position: and. Howvr, a ral crystal structur has dfcts distorting idal lattic, so occupations can b diffrnt. On of th most common dfct of this sort is th so-calld antinod disordring whn atoms switch thir positions. For instanc, if in som clls atoms of Sn at position 1 switch to position 3, and atoms Nb in ths clls switch from 3 to 1, occupation bcoms lss than 1 and bcoms non-zro. Nvrthlss, th gross occupation of any position rmains qual to 1, i.. any atom laving its position switchs to position of anothr atom and vic vrsa. 8) Suppos thr is an antinod disordring in th considrd structur, so th occupation of Nb positions by atoms of Sn bcoms qual to δ, i... Dtrmin othr occupations in th structur. Occupations of atoms of any lmnt at positions n = 1 and n = ar th sam, as it is th cas for positions n = 3,, 8. Exprss th occupations via δ. 9) Assuming and to b known valuat structur factor of rflx (110), using th occupations calculatd in 8). Th answr should b givn as quation. Hint: according to Eulr s formula complx xponntial is valuatd as. 10) Dtrmin th condition of qunching th rflx (110) (whn its intnsity vanishs) for th sam structur. Th answr should b givn as a numrical valu of δ. Problm 3: Nutrino Nutrino is on of th most pculiar lmntary particls. It has no lctric charg and dos not participat in th strong intractions (which ar rsponsibl for stability of atomic nucli). Physicists us th word «flavor» to spcify a nutrino typ. Thr ar thr nutrino flavors known to dat: lctron nutrino, muon nutrino, and tau-nutrino. A nutrino of ach flavor has its antiparticl (antinutrino). Th symbols usd for th lattr ar th sam as for nutrinos but with an uppr bar:,, and. Nutrino participats only in th wak intractions, th most famous procss mdiatd by th wak intraction is -dcay. In this procss a singl nutron in atomic nuclus dcays into a proton, an lctron, and an lctron antinutrino: n p (howvr, it would b a mistak to think than nutron is composd of ths particls, thr is also a procss p n!). A nutrino is always cratd togthr with its antinutrino or a chargd antilpton (positron (an lctron antiparticl), antimuon, or antitau-lpton cratd togthr with its nutrino or th corrsponding chargd lpton. 6 ). An antinutrino, in turn, is always

7 Nutrinos ar xtrmly lightwight, thir masss ar svral ordrs of magnitud lss than masss of othr mattr particls. Th prcis valus of nutrino masss ar still unknown. Du to thir small masss all nutrinos participating in nuclar ractions ar ultrarlativistic, i.. thir vlocitis ar vry clos to th spd of light in vacuum. Th nrgy of such a nutrino of mass m and momntum p is almost indpndnt of its mass: 4 E m c c p c p. A nutrino, lik many othr lmntary particls, has spin, i.. th propr angular momntum, which is non-zro vn in th nutrino rst fram. A spcific fatur of all dtctd nutrinos (antinutrinos) is th ngativ (positiv) sign of its spin componnt projctd on th dirction of nutrino (antinutrino) momntum. Loosly spaking, a nutrino dos not hav a «mirror rflction». Othr lmntary mattr particls can hav both signs of th spin componnt. Physicists xplain this fact by saying that nutrinos with othr spin componnt ithr do not xist, or do not participat vn in th wak intractions (so thy cannot b dtctd). Physical constants and data (can b usd in any part of th problm) spd of light in vacuum gravitational constant Planck constant proton radius Avogadro constant 8 c 310 m/s; 11 G 6,7 10 m 3 /(kg s ); J s; r m; p 3 N 6 10 mol 1 ; hydrogn molar mass g/mol; Solar mass Solar radius A 30 M 10 kg; C 8 r 7 10 m; man radius of Earth s orbit С 11 a 1,5 10 m; ccntricity of Earth s orbit 0, 017 ; radius of «activ» solar cor whr nuclar fusion procds and nutrinos ar cratd 8 r 1, 10 m; a rang of lctron dnsity n insid th Sun from th activ cor to outr layrs: from m 3 to 9 10 m 3 ; parsc (pc), an astronomical unit of lngth, 1 pc 16 3, light yar 310 m. 5,9 10 lctronvolt (V) is th unit of nrgy qual to th work don by lctrostatic forc moving a singl lctron across potntial diffrnc of 1 V. 31 Part I: nutrino masss and oscillations. Th Nobl Priz of 015 was awardd for th «discovry of nutrino oscillations indicating that nutrinos ar massiv». Nutrino oscillations is a procss of intrconvrsion of nutrino flavors. According to modrn thortical modls th possibility of nutrino oscillations is indd closly rlatd to thir masss (masslss nutrinos cannot oscillat). 7

8 It should b notd that nutrinos lik othr lmntary particls ar not som «immutabl» ntitis, rathr thy ar quanta of a nutrino fild (similarly to photons which ar quanta of lctromagntic fild). Thrfor, in diffrnt physical situations thy can appar in th stats with diffrnt proprtis. For instanc, nutrino stat of a crtain flavor (a stat in which nutrino is cratd or annihilatd in nuclar ractions) dos not coincid with nutrino stat of a crtain mass. To b spcific, considr oscillation (i.. w nglct th third nutrino flavor). An intnsiv flow of nutrinos can b rgardd as «almost classical» radiation of a givn wavlngth (hr th analogy with lctromagntic wav, an «almost classical» flow of a larg numbr of photons, applis again). Th xistnc of svral nutrino stats can b dscribd by introducing a «polarization»: u t, r ) u cos( t kr ) u cos( t k ). Th quotation mark indicats that this polarization is ( 1 1 r not a polarization in th «rgular» spac, this is polarization in th «spac of nutrino stats» although for our purposs this is almost insignificant. Th flux of nutrinos is proportional to u. Notic that th frquncy and wavvctor k of th wav ar rlatd to th nrgy and momntum of nutrinos by th common quantum formula: E and p k, whr is Planck constant. Th diffrnc of frquncis is du to diffrnc in masss: for th sam momntum 4 E m c c p. Obviously, an orthogonal «polarization» u 1, corrsponds to th 1, 1, 1, nutrino stat of crtain mass m 1,. Notic, that stats of dfinit flavor ( and anothr pair of orthogonal «polarizations» u, which do not coincid with u 1,. Th polarizations u 1, corrsponding to crtain masss and polarizations, to crtain flavors ar rlatd as: u1 u cos u u u sin u Angl is calld th «mixing angl» of and sin, cos.. In this cas and ) corrspond to u corrsponding indd do not hav «crtain» masss and do not hav a crtain nrgy for a givn momntum. For instanc, a masurmnt of th nrgy of lctron nutrino would «on avrag» yild th valu E E1 E E E1 cos E sin E cos, whr E c p is th man nutrino nrgy and E E E1. Such an outcom could b intrprtd as bing du to intraction of th stats and, whr th intraction nrgy is V E sin cos. Th quantitis introducd abov can b xprssd in trms of nrgy E and paramtrs m 1 m m, m m m1, and. It has bn alrady mntiond that accurat valus of ths paramtrs ar not known yt but to solv th problm it would suffic to adopt th following approximat valus: 3 mc 4,0 10 V, 3 mc 3,0 10 V, and 10. 1) Evaluat «man» masss of lctron and muon nutrinos for th givn valus of th paramtrs. Th answr can b givn ithr in kg or V/c. Now considr nutrinos radiatd from som small rgion and propagating along x-axis. Lt th nrgy of a nutrino cratd in this rgion b E MV, all cratd nutrinos ar lctron 8

9 nutrinos and hav th sam crtain momntum (this mans that nutrinos ar cratd with diffrnt masss and, thrfor, nrgis). Clarly, th nutrino wav propagating along x-axis is a mixtur of nutrino wavs corrsponding to nutrinos of diffrnt masss (hnc, a mixtur of wavs with crtain frquncis at a givn wavlngth). Th phas shift of th wavs varis with distanc x. Sinc th phas shift varis th contributions to th rsulting wav du to lctron and muon componnt would vary as wll. Thrfor, at any particular position x on would dtct not only lctron nutrinos but muon nutrinos as wll. Th intnsitis of th corrsponding nutrino fluxs vary priodically in spac. This phnomnon is calld nutrino oscillations. ) Dtrmin th oscillation lngth, i.. th priod of spatial variation of th tim avragd flux of muon nutrinos. Th answr should b givn as th formula and th numrical valu (in mtrs). Part II: nutrinos and th Sun. Oscillations dscribd in Part I occur in vacuum. At first glanc, it would b rasonabl to assum that mattr dos not altr th pictur significantly sinc nutrinos vry wakly intract with any mattr which dnsity is much lss than th dnsity of atomic nuclus. Thr is a powrful nutrino sourc clos to th Earth. It is th Sun. Nuclar ractions procd in th cntral rgion of th Sun suppling it with nrgy and crating nutrinos and antinutrinos, mostly, lctron ons. Howvr, th obsrvd flux of lctron nutrinos turnd out to b only a half of th flux prdictd from th Solar luminosity. Is it possibl to xplain this «dficit» by partial convrsion of lctron nutrinos to nutrinos of othr flavors on thir way from th Sun to th Earth (vacuum oscillations)? 3) Try to giv a justifid answr by using th data and th rsults from Part I of th problm. Do ncssary calculations to support your judgmnt. In particular, it is rasonabl to assum that nrgis of nutrinos cratd in nuclar ractions in th Sun ar not vry diffrnt from MV. Tak into account th fact that a nutrino dtctor accumulats data for a long priod of tim, up to -3 months. Th answr should b givn as «+» (ys) or (no). A dtaild analysis must tak into account that th nutrinos travl a part of thir path insid th solar substanc. It turns out, absorption of th nutrinos by th substanc dos not chang much th stimat for th flux of lctron nutrinos but thr is som additional circumstanc which is quit ssntial. Th solar substanc contains a lot of lctrons (s th problm data) and lctron nutrinos intract with thm much strongr than muon nutrinos do. Du to this fact th «man» nrgy of lctron nutrinos incrass by E, th quantity which is proportional to th lctron n 1 dnsity: E 1,7 10 V. At th sam tim th «man» nrgy of muon nutrinos and m th nrgy of intraction btwn th stats and rmain practically th sam. 4) Evaluat «nw» valu of mixing angl ~ (by taking into account th solar lctrons). Th answr should b th quation. 5) By how many prcnt can th solar lctrons chang th xpctd «dficit» of th lctron nutrino flux? Evaluat in prcnt (%) th maximum incrmnt of th flux of muon nutrinos du to oscillations (compard to th flux in th absnc of mattr). 9

10 6) Plot an approximat dpndnc (i.. show only th main faturs) of th mass diffrnc of and as a function of th distanc r travld insid th Sun from th cntr outwards. m μ m r/r C 0 0,5 1 Part III: nutrino and Suprnova xplosion. Aftr an «ordinary» star has xhaustd its nuclar ful th star cools down and its intrnal prssur cannot withstand gravitational comprssion anymor. As a rsult, th star is contracting until its substanc undrgos transition to a nw phas in which all lctrons ar «shard», all nucli «float» in th lctron «gas», and only th prssur of this gas halts furthr collaps. Th star can xist in this stat for a long tim; howvr, if th mass of its dns cor gradually incrass and xcds M 1, 5M th cor bcoms unstabl and collapss. At th onst of cr C collaps such a cor usually has a radius about km with approximatly qual numbrs of protons and nutrons. Soon aftr th contraction starts th rat of lctron-nuclar collisions bcoms high, which rsults in nutronization of th star substanc du to raction of «invrs - dcay» ( p n ). Elctron disapparanc rducs th prssur of lctron gas acclrating th nutronization vn mor. Th whol procss is ssntially th trmndous xplosion laving in th aftrmath a nutron star and an «outr nvlop» flying outwards. Astronomrs call such an xplosion «Suprnova xplosion» or simply «Suprnova». A nutron star is indd composd mostly of tightly packd nutrons, so its dnsity is approximatly qual to th dnsity of atomic nucli. 7) Estimat an ordr of magnitud of th nrgy rlasd du to comprssion of th stllar cor from th initial radius to th nutron star. Calculat th numrical answr in Jouls. Th rlasd nrgy convrts to kintic nrgy of th star rmnants (th flying outr layrs and rotation of th nutron star) and to th nrgy of lctromagntic radiation and nutrinos. Suprnova xplosion is on of th most powrful sourc of nutrinos (somtims it is calld «nutrino bombs»), calculations show that mor than half of th rlasd nrgy convrts to th nrgy of radiatd nutrinos. Nutrinos ar radiatd both at th nutronization stag and aftr formation of th nutron star which is initially xtrmly hot and subsquntly cools down mostly by radiating nutrinos. Th nutronization and cooling tak just svral sconds. Notic that only lctron nutrinos ar cratd during th nutronization and nutrino-antinutrino pairs of various flavors ar cratd during th cooling. 10

11 8) Estimat th numbr of nutrinos cratd by a Suprnova xplosion assuming that th mass of th initial stllar cor is approximatly 1,5M C (th stllar substanc dos not «go away» with outr layrs), th nrgy of radiatd nutrinos is 80% of th rlasd nrgy, and th man nrgy of radiatd nutrinos and antinutrinos is approximatly 10 MV. You could assum that nutron mass and radius ar approximatly th sam as thos of proton. Calculat th numrical valus. 9) Suprnova SN1987А xplodd at th distanc of R 50 kpc from th Earth (in th Larg Magllanic Cloud). What is th total numbr of nutrinos and antinutrinos passd through an Earth basd dtctor of th cross-sctional ara S 100 m? Estimat th xpctd numbr of dtctd nutrinos and antinutrinos assuming that th dtctor on avrag 14 rgistrs 310 % of nutrinos of any flavor in th corrsponding nrgy rang. Calculat th numrical valus. It is important that nutrino radiation is asymmtric with rspct to th star magntic axis du to «pculiar» nutrino bhavior undr mirror rflction: th powr di radiatd in th infinitsimal di I solid angl d sin dd is [1 cos ], whr 10, is th angl of nutrino d 4 mission to th axis, and is th angl of rotation around th axis. 10) Estimat th spd gaind by th star du to nutronization and cooling. Calculat th numrical answr (in km/s). 11