Торможение небесных тел частицами т¼мной материи
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Òîðìîæåíèå íåáåñíûõ òåë ÷àñòèöàìè ò¼ìíîé ìàòåðèè Ãîðÿ÷óê È 217 ãðóïïà Ñ íåäàâíèõ ïîð ïðîáëåìå èññëåäîâàíèÿ ñâîéñòâ ò¼ìíîé ìàòåðèè ñòàëè óäåëÿòü âñ¼ áîëüøå âíèìàíèÿ. Ó÷¼íûõ èíòåðåñóþò âîïðîñû îá ýâîëþöèè Âñåëåííîé â ïðîøëîì è î òîì, ÷òî îæèäàåò å¼ â áóäóùåì. Äåëî â òîì, ÷òî äëÿ îáúÿñíåíèÿ íåêîòîðûõ àñòðîíîìè÷åñêèõ ÿâëåíèé ýêñïåðèìåíòàëüíàÿ îöåíêà âçàèìîäåéñòâóþùåé ñî ñâåòîì ìàññû Âñåëåííîé îêàçûâàåòñÿ íåäîñòàòî÷íîé. Ïðèõîäèòñÿ ââîäèòü ìàññèâíóþ ò¼ìíóþ ìàòåðèþ, íå èçëó÷àþùóþ è íå ïîãëîùàþùóþ ôîòîíû. Ýòî ñëåäóåò, íàïðèìåð èç èçìåðåíèé êðèâûõ âðàùåíèÿ íåêîòîðûõ ñïèðàëüíûõ ãàëàêòèê. Îêàçûâàåòñÿ, ÷òî ñêîðîñòü âðàùåíèÿ îñòà¼òñÿ ïîñòîÿííîé, íà÷èíàÿ ñ íåêîòîðîãî ðàññòîÿíèÿ äî öåíòðà. Òàêîé âèä çàâèñèìîñòè ìîæíî îáúÿñíèòü, òîëüêî åñëè ïðåäïîëîæèòü ñóùåñòâîâàíèå äîïîëíèòåëüíîé íåâèäèìîé ìàññû â ïðåäåëàõ ãàëàêòèêè - íåáàðèîííîãî ñôåðè÷åñêîãî ò¼ìíîãî ãàëî. Äëÿ íåêîòîðûõ ãàëàêòèê îêàçûâàåòñÿ, ÷òî åãî ìàññà íàìíîãî ïðåâûøàåò ñâåòÿùóþñÿ. Ñóùåñòâîâàíèå íåáàðèîííîé ìàòåðèè òàêæå ïîäòâåðæäàåòñÿ òåì, ÷òî îöåíêà áàðèîííîé êîñìîëîãè÷åñêîé ïëîòíîñòè Âñåëåííîé, ñäåëàííàÿ ïî ñâåòèìîñòè ãàëàêòèê (Ωb < 0.02) äàæå ñ ó÷¼òîì ïîïðàâîê íà ìåæçâ¼çäíóþ ïûëü, êîðè÷íåâûå è áåëûå êàðëèêè, íåéòðîííûå çâ¼çäû, ÷¼ðíûå äûðû, à òàêæå ñîâðåìåííûõ äàííûõ î ìàññèâíûõ êîìïàêòíûõ ãàëàêòè÷åñêèõ îáúåêòàõ (MACHO) íå ñîîòâåòñòâóåò äèíàìèêè ãàëàêòè÷åñêèõ êëàñòåðîâ (äëÿ ãðàâèòàöèîííî çàìåäëÿþùåãîñÿ ðàñøèðåíèÿ Âñåëåííîé òðåáóåòñÿ Ωb ≥ 1 , ÷òî ñëåäóåò èç èçìåðåíèé ðåëèêòîâîãî èçëó÷åíèÿ â ýêñïåðèìåíòå Áó- ìåðàíã). [1] Íåðåëÿòèâèñòñêèå ìàññèâíûå ñëàáîâçàèìîäåéñòâóþùèå ÷àñòèö ò¼ìíîé ìàòåðèè (WIMP) ìîæíî ðàññìàòðèâàòü êàê èäåàëüíûé ãàç. Âåðîÿòíîñòü èõ âçàèâîäåéñòâèÿ ñ âåùåñòâîì îïèñûâàåòñÿ ïîëíûì ýôôåêòèâíûì ñå÷åíèåì σ. Îöåíèì ñèëó òîðìîæåíèÿ íåáåñíûõ òåë - Çåìëè è Ñîëíöà, ñâÿçàííóþ ñ èõ äâèæåíèåì â ýòîì ãàçå. Äëÿ ýòîãî ðàññìîòðèì äâèæåíèå øàðà ðàäèóñà ns (ìàññà êîòîðûõ ms ) ñî ñêîðîñòüþ Vrel R ñ êîíöåíòðàöèåé ÷àñòèö â ñèñòåìå îòñ÷¼òà ãàçà. Ñ÷èòàåì, ÷òî ãàç èìååò èçîòðîïíîå ðàñïðåäåëåíèå ïî ñêîðîñòÿì â ñâîåé ñèñòåìå. Åãî êîíöåíòðàöèÿ ng , à ìàññà ÷àñòèö mg . Ïåðåéä¼ì â ñèñòåìó øàðà, ãäå íà íåãî íàëåòàåò ïîòîê ÷àñòèö. Ñîãëàñíî îáû÷íîìó îïðåäåëåíèþ ñå÷åíèÿ [2], d(dN ) = σVrel ns ng dV dt åñòü ÷èñëî âçàèìîäåéñòâèé â îáú¼ìå dV dt çà âðåìÿ dN dt , èçìåíÿåòñÿ èõ ïîòîê ÷åðåç ïëîùàäêó ×àñòèöû âçàèìîäåéñòâóþò ñî ñêîðîñòüþ èç-çà âçàèìîäåéñòâèÿ) dI = − ñàì ýòîò ïîòîê d(dN ) dtS dN dxrel dN dN = = Vrel = Vrel ng dtS dt dxrel S dV σVrel ns ng dV dI = − = −σVrel ns ng dx = −σIns dx S I= Íàéä¼ì îñëàáëåíèå ïîòîêà íà ïóòè x (âäîëü îòíîñèòåëüíîé ñêîðîñòè) ZI dI = I (−σns )dx 0 Io ln Zx I Io = −σns x 1 S (óìåíüøàåòñÿ I = Io e−σns x Íàéä¼ì îñëàáëåíèå ïîòîêà â øàðå ðàäèóñà R (ãäå dNr ÷àñòèö ïðîøëè áåç âçàèìîäåéñòâèÿ); ïðîâåä¼ì èíòåãðèðîâàíèå ïî öåíòðàëüíîìó ïîïåðå÷íîìó ñå÷åíèþ dNr Ir = = dt Z2π ZR dϕ 0 = Io π ZR ρdρI = 0 √ −e Z2π −σns 2 R2 −ρ2 ZR dϕ 0 √ ρdρIo e −σns 2 R2 −ρ2 ZR = 2πIo 0 2 d R −ρ 2 1 −σns 2√R2 −ρ2 2 e dρ = 2 0 = Io π ZR √ 2 2 p p R2 − ρ2 ≡ −2 R2 − ρ2 e−σns 2 R −ρ d 0 0 Z0 ≡ Io 2π −te−σns 2t dt = Io 2π ZR R ZR −t −1 te−σns 2t − e−σns 2t dt = d e−σns 2t = I0 2π 2σns 2σns 0 0 R = Io 2π −1 2σns 0 e−σns 2t R −1 e−σns 2R 1 Re−σns 2R − = Io 2π Re−σns 2R + − = −2σns 2σns 2σns 2σns 0 ! Re−σns 2R 1 − e−σns 2R − = Io 2π ≡ Sk Io 2 2σns (2σns ) Èñõîäíîå ÷èñëî ÷àñòèö dNro îáðàçîâûâàëè ïîòîê Iro = Èòàê, â åäèíèöó âðåìåíè ïðîèñõîäèò âûðàæåíèåì dNro = πR2 Io ≡ SIo dt (S − Sk ) Io âçàèìîäåéñòâèé, ãäå ïëîòíîñòü ïîòîêà îïðåäåëÿåòñÿ Io = Vrel ng Ðàññìîòðèì ïîäðîáíåå ñòîëêíîâåíèå äâóõ ÷àñòèö.  ñèñòåìå ãàçà (ãäå 0z ïðîòèâîïîëîæíà ñêîðîñòè ÷àñòèöû øàðà) èìååì ~ = (Vgx , Vgy , Vgz ) = (V sin θ cos ϕ, V sin θ sin ϕ, V ~rel = (0, 0, −Vrel ) - ñêîðîñòü ÷àñòèöû øàðà V ~cm = V~ mg +V~rel ms - ñêîðîñòü öåíòðà ìàññ ÷àñòèö V V cos θ) - ñêîðîñòü ÷àñòèöû ãàçà mg +ms Ïåðåõîäÿ â ñèñòåìó öåíòðà ìàññ (o ), èñïîëüçóåì ïðåîáðàçîâàíèÿ Ãàëèëåÿ (çäåñü m̄ = mg ms mg +ms - ïðè- âåä¼ííàÿ ìàññà) ~ ~ ~ ~ ~og = V ~ −V ~cm = V ~ − V mg + Vrel ms = V ms − Vrel ms = V ~ −V ~rel m̄ V mg + ms mg + ms mg + ms mg ~ ~ ~ ~ ~o = V ~rel − V ~cm = V ~rel − V mg + Vrel ms = Vrel mg − V mg = − V ~ −V ~rel m̄ V mg + ms mg + ms mg + ms ms Ïðè ñòîëêíîâåíèè ïî ÇÑÈ 0 0 ~og + ms V ~o = mg V ~og ~os mg V + ms V 0 0 ~ −V ~rel m̄ + − V ~ −V ~rel m̄ = mg V ~og ~os V + ms V mg ~ 0 ms ~ 0 0 0 ~ ~ V èëè Vog = − V Vos = − ms og mg os Ñ÷èòàåì ñòîëêíîâåíèå óïðóãèì, òîãäà ïî ÇÑÝ 2 02 02 ~og ~og ~o2 ~os mg V mg V ms V ms V + = + 2 2 2 2 2 2 ~ −V ~rel m̄2 mg V 2 ~ −V ~rel m̄2 ms V + 02 02 ~og ~os mg V ms V + 2 2 = 2m2g 2m2s 2 1 1 02 02 ~ −V ~rel ~og ~os m̄2 V + = mg V + ms V mg ms 2 2 ms ~ 02 mg m2s ~ 02 ms ~ 02 ms ms + mg ~ 02 (ms + mg ) ~ 02 02 ~ −V ~rel = mg V ~og V Vos + V + Vos = V = V = os m̄ m̄ m̄ m2g os m̄ os m̄ mg m2g ~ −V ~rel | mg |V ; ms + mg 0 ~os |V |= Ïóñòü çäåñü ðàñïðåäåëåíèå ïî Ñ÷èòàÿ, ÷òî ÷àñòèö ñ äàííìè Z 0 hms Vosz i = 0 ~og |V |= 0 ~og V èçîòðîïíî (à V, ϕ, θ ìíîãî, 0 dΩ ms Vosz 0 0 Z = Ω â ñèëó 0 ~0 ~og V ||Vos = è ïî 0 Vosz = 0 Vosx = 0 ~os |V | cos θ0 0 ~os |V | sin θ0 cos ϕ0 0 Vosy = 0 ~os |V | sin θ0 sin ϕ0 0 dΩ 0 ~os ms |V | cos θ0 0 = Ω Z2π 0 Zπ Zπ dϕ 0 0 ~os | ms|V 2π 4π 0 ~os V ) óñðåäíèì èìïóëüñ ÷àñòèö øàðà ïî Ω0 Ω0 ~ −V ~rel | ms |V ms + mg Ω0 1 0 ~os sin θ0 dθ0 ms |V | cos θ0 = 4π 0 1 sin 2θ0 dθ0 = 0 2 0 0 hms Vosx i Z = 0 dΩ ms Vosx 0 Ω 0 Z = Ω0 0 dΩ 0 ~os ms |V | sin θ0 cos ϕ0 0 = Ω Ω0 Z2π 0 dϕ 0 2π Zπ 1 0 ~os sin θ0 dθ0 ms |V | sin θ0 cos ϕ0 = 4π 0 π Z 0 Z ~os | ms |V 0 0 cos ϕ dϕ sin2 θ0 dθ0 = 0 = 4π 0 hms Vosy i= Z 0 ms Vosy Ω0 dΩ0 = Ω0 Z 0 0 0 ~os ms |V | sin θ0 sin ϕ0 dΩ0 = Ω0 Ω0 0 2π = dϕ0 Zπ 0 ~os sin θ0 dθ0 ms |V | sin θ0 sin ϕ0 0 π Z ~0 |Z ms |V os sin ϕ0 dϕ0 sin2 θ0 dθ0 = 0 4π 0 Òîãäà è Z2π 0 0 ~os hV i=0 Ïåðåõîäèì îáðàòíî â ñèñòåìó ãàçà ~ ~ 0 ~s i = V ~cm + hV ~os ~cm + 0 = V ~cm = V mg + Vrel ms hV i=V mg + ms ~s iz hV = ~s ix hV = ~ s iy hV = V cos θmg − Vrel ms mg + ms V sin θ cos ϕmg mg + ms V sin θ sin ϕmg mg + ms 3 1 = 4π Ñ÷èòàÿ, ÷òî ÷àñòèö ñ äàííîé ïóëüñ ïî V ìíîãî, à ðàñïðåäåëåíèå çäåñü èçîòðîïíî, óñðåäíèì èõ ñðåäíèé èì- Ω Z hms Vsz i = ~s iz dΩ = m s hV Ω Ω Z2π = Zπ dϕ 0 sin θdθ Z ms V cos θmg − Vrel ms dΩ = mg + ms Ω Ω V ms mg Vrel m2s cos θ − mg + ms mg + ms 1 1 = 4π 2 Zπ V ms mg sin 2θdθ− 2 (mg + ms ) 0 0 − 1 Vrel m2s 2 mg + ms Zπ sin θdθ = − π Vrel m2s 1 Vrel m2s (− cos θ) = − 2 mg + ms mg + ms 0 0 Z hms Vsx i = ~s ix dΩ = ms hV Ω Z V sin θ cos ϕmg dΩ = mg + ms Ω ms Ω Ω V m g ms = 4π (mg + ms ) Z2π Zπ cos ϕdϕ 0 Z hms Vsy i = sin2 θdθ = 0 0 ~s iy dΩ = ms hV Ω Z V sin θ sin ϕmg dΩ = mg + ms Ω ms Ω Ω V mg ms = 4π (mg + ms ) Z2π Zπ sin ϕdϕ 0 sin2 θdθ = 0 0 Ïóñòü ðàñïðåäåëåíèå ïî ñêîðîñòÿì ãàçà, íàïðèìåð, Ãàóññîâî, êàê âäàëè îò ãðàâèòàöèîííûõ ïîëåé íåáåñíûõ òåë, (çàìåòèì, ÷òî ýòî ïðåäïîëîæåíèå íå îáÿçàòåëüíî, âåäü èìïóëüñ íîñòè ïîñëå óñðåäíåíèÿ íå çàâèñèò îò ñêîðîñòåé ÷àñòèö ãàçà ~ V ~s i èç-çà èçîòðîïhms V è ðàñïðèäåëåíèå ïî íèì ìîæåò áûòü äðóãèì) − V2 dP = Ae 2σv2 V 2 dV Z∞ 2 −V 1 = Ae 2σv2 V 2 dV 0 Ñðåäíèé èìïóëüñ îäíîé ÷àñòèöû øàðà p0z Z∞ Ahms Vsz ie = − V2 2 2σv V2 Z∞ Z∞ 2 − 2 2 −V Vrel m2s Vrel m2s 2σv V dV = A − V dV = − e Ae 2σv2 V 2 dV = mg + ms mg + ms 2 0 0 0 =− Ïîñëå îäíîãî èç N Vrel m2s mg + ms ñòîëêíîâåíèé èìïóëüñ ÷àñòèöû øàðà èçìåíèòñÿ ñ −Vrel ms íà s −Vrel ms mgm+m s (âäîëü îñè 0z) â åäèíèöó âðåìåíè. Ñèëà òîðìîæåíèÿ ∆pN Fz = = ∆t ms −Vrel ms − (−Vrel ms ) mg + ms Äëÿ ïðîñòîòû ÷èñëåííûõ îöåíîê ðàçëîæèì Sk = 2π 1 − 1 + σns 2R − 1 2 2 (σns 2R) + 1 6 Sk N 2 = Vrel m̄ (S − Sk ) Io ≡ βVrel ∆t â ðÿä äî ëèíåéíûõ ïî 2 (2σns ) 4 − ñëàãàåìûõ 2 (σns 2R) − · · · = 2σns R 1 − σns 2R + 3 (σns 2R) − · · · σns 2R 1 2 = 2π R 1 1 R 1 − R2 + R2 (σns 2R) − + R2 − R2 (σns 2R) + · · · 2σns 2 6 2σns 2 1 2 1 2 = 2π R − R (σns 2R) + · · · 2 3 = Òîãäà 1 S − Sk ≈ 2π R2 − 2π 2 ãäå Ns ρg 1 2 1 2 R − R (σns 2R) 2 3 1 4 = 2π R2 (σns 2R) = πR3 σns = σNs 3 3 - ÷èñëî ÷àñòèö øàðà. Fz = Vrel ãäå ms mg mg ng ρg 2 2 (S − Sk ) Vrel ng ≈ Vrel σ (Ns ms ) = Vrel σMs ms + mg ms + mg ms + mg - ïëîòíîñòü ãàçà, Ms - ïîëíàÿ ìàññà øàðà. Óñêîðåíèå òîðìîæåíèÿ Fz ρg 2 = Vrel σ Ms ms + mg Îöåíèì äëÿ Ñîëíöà, ñ÷èòàÿ, ÷òî âñÿ åãî ìàññà ó÷àñòâóåò âî âçàèìîäåéñòâèÿõ, à åãî ÷àñòèöà - ïðîòîíû (íóêëîíû) ñ ìàññîé â 1GeV. Çäåñü îñíîâíóþ ðîëü èãðàåò òàê íàçûâàåìîå ñïèí-çàâèñèìîå âçàèìîäåéñòâèå. Ýêñïåðèìåíòàëüíûå îãðàíè÷åíèÿ âîçüì¼ì σ = σχp < 10−4 ÷ 10−3 ïèêàáàðí (ýêñïåðèìåíòàëü- íûå ðåçóëüòàòû IceCube [3], SuperKamiokande [4], Áàêñàíñêèé òåëåñêîï [5], ANTARES [6]) äëÿ ìàññ ÷àñòèö ãàçà (WIMP) 1 ÷ 1000GeV ; ïëîòíîñòü ýíåðãèè ñ÷èòàåì ðàâíîé ρg = ρχ = 0.3GeV /cm3 . 0.3 · 106 GeV m Fz m m3 10−4 ÷ 10−3 10−40 m2 = (2 · 10−31 ÷ 9 · 10−28 ) 2 = 250 · 103 Ms s 1GeV + (1 ÷ 1000)GeV s Ñ÷èòàÿ ìàññó Ñîëíöà ðàâíîé 2 · 1030 kg , íàéä¼ì ñèëó F = 2 · 1030 kg(2 · 10−31 ÷ 9 · 10−28 ) m = (0.4 ÷ 2 · 103 )N s2 ×òîáû ïðîâåñòè îöåíêó äëÿ Çåìëè, íåîáõîäèìî ó÷åñòü å¼ äâèæåíèå ïî îðáèòå. Âûøå ïîëó÷èëè ñèëó òîðìîæåíèÿ ~ |V ~| F~ = −β V Ïóñòü γ - óãîë ìåæäó ïëîñêîñòüþ ýêëèïòèêè è íàïðàâëåíèåì ñêîðîñòè ~g V Ñîëíöà â ñèñòåìå íåïî- ϕ - óãîë ìåæäó å¼ ïðîýêöèåé íà ýòó ïëîñêîñòü è íàïðàâëåíèåì èç Ñîëíöà íà Çåìëþ. ~E , òîãäà â ñèñòåìå, ñâÿçàííîé ñ Çåìëÿ äâèæåòñÿ ïî êðóãîâîé îðáèòå ñî ñêîðîñòüþ V äâèæíîãî Ãàëî, Ñ÷èòàåì, ÷òî íåé Vz = Vg sin γ - ïåðïåíäèêóëÿðíî ïëîñêîñòè âðàùåíèÿ ~E Vx = cos γ cos ϕ + VE - âäîëü V ~E Vy = cos γ sin ϕ - ïåðïåíäèêóëÿðíî îñè îðáèòû è V q q ~ | = V = V 2 + V 2 + V 2 = V 2 sin2 γ + (Vg cos γ cos ϕ + VE )2 + V 2 cos2 γ sin2 ϕ = |V x y z g g = q Vg2 + VE2 + 2Vg VE cos γ cos ϕ Ìîäóëü ñèëû F = β Vg2 + VE2 + 2Vg VE cos γ cos ϕ Òàê êàê Vg = 250km/s à VE = 30km/s, òî ìîæíî ñ÷èòàòü, ÷òî ~ V ïðàêòè÷åñêè íå ìåíÿåòñÿ ïî íàïðàâëåíèþ è Z2π hF i = 0 dϕ F = 2π Z2π dϕ β Vg2 + VE2 + 2Vg VE cos γ cos ϕ = β Vg2 + VE2 2π 0 5 Êàê è âûøå, ñ÷èòàåì, ÷òî âñÿ ìàññà Çåìëè ó÷àñòâóåò âî âçàèìîäåéñòâèè, å¼ ÷àñòèöà - ïðîòîí (íóêëîí) ñ ìàññîé â 1GeV . Îäíàêî â äàííîì ñëó÷àå îêàçûâàåòñÿ âàæíåå ñïèí-íåçàâèñèìîå âçàèìîäåé- ñòâèå, è çäåñü îãðàíè÷åíèÿ íà ñå÷åíèå ñèëüíåå. Ïîñëåäíèå ýêñïåðèìåíòàëüíûå äàííûå CDMS [7], [8] è XENON100 [9] óêàçûâàþò íà çíà÷åíèÿ σ = 2 · 10−41 cm2 ïðè ìàññå mg = 9GeV Äëÿ óñêîðåíèÿ Fz ρg = Vg2 + VE2 σ= ME mE + mg 250 · 10 3m 2 s 2 0.3 · 106 GeV m3 + 30 · 10 2 · 10−45 m2 = s 1GeV + 9GeV = 3.8 · 10−30 Òîãäà ñèëà Fz = 3.8 · 10−30 3m m s2 m · 6 · 1024 kg = 2.3 · 10−5 N s2 Óâåëè÷èâàÿ ìàññó ÷àñòèö ò¼ìíîé ìàòåðèè â ðàìêàõ ýêñïåðèìåíòàëüíûõ ðåçóëüòàòîâ, ñîõðàíÿÿ îæèäàåìîå çíà÷åíèå ïëîòíîñòè ýíåðãèè, ìû òîëüêî óìåíüøàåì óñêîðåíèå è ñèëó Fz = ME 250 · 103 m 2 m 2 + 30 · 103 s s Fz = 2.4 · 10−31 0.3 · 106 GeV m m3 2 · 10−46 m2 = 2.4 · 10−31 2 1GeV + 15GeV s m · 6 · 1024 kg = 1.4 · 10−6 N s2 Ìàêñèìóì ïîëó÷èì ïðè íàèìåíüøåé ýêñïåðèìåíòàëüíîé ìàññå ÷àñòèö ò¼ìíîé ìàòåðèè (mg ãäå îãðàíè÷åíèå íà ñå÷åíèå Fz = ME 250 · 10 σ = 6 · 10−41 cm2 ). 3m 2 s = 7GeV , Çäåñü 2 0.3 · 106 GeV m m3 6 · 10−45 m2 = 1.4 · 10−29 2 + 30 · 10 s 1GeV + 7GeV s 3m Fz = 1.4 · 10−29 m · 6 · 1024 kg = 8.4 · 10−5 N s2 Âûøå ïðåäïîëàãàëîñü, ÷òî ÷àñòèöà Çåìëè - ñâîáîäíûé ïðîòîí, îäíàêî ïðîòîíû è íåéòðîíû âåùåñòâà ñâÿçàíû â ÿäðà àòîìîâ. Îöåíèì ñèëó äëÿ êàæäîãî âèäà àòîìîâ îòäåëüíî. Òîãäà è ñå÷åíèå áóäåò ðàçíûì [10], îïðåäåëèì åãî ïî ôîðìóëå 2 σA = σA ãäå mA mg mA mg + mA - ìàññà ÿäðà ñ àòîìíûì âåñîì 2 A, m p - mg + mp mg + mp 2 4 = σA mg + mp mg + mA ìàññà ïðîòîíà. Îöåíèì ñå÷åíèå äëÿ îñíîâíûõ ñîñòàâ- ëÿþùèõ Çåìëè. Æåëåçî (ωF e = 32.1%): σF e = 2 · 10−45 m2 (56)4 Êèñëîðîä (ωO 2 = 4.7 · 10−40 m2 9+1 9 + 16 2 9+1 9 + 28 2 = 2.1 · 10−41 m2 = 15.1%): −45 σSi = 2 · 10 Ìàãíèé (ωM g 9+1 9 + 56 = 30.1%): σO = 2 · 10−45 m2 (16)4 Êðåìíèé (ωSi 2 m (28) 4 = 9.0 · 10−41 m2 = 13.9%): −45 σM g = 2 · 10 2 2 m (24) 4 9+1 9 + 24 6 2 = 6.1 · 10−41 m2 Ñåðà (ωS = 2.9%): σS = 2 · 10 Íèêåëü (ωN i −45 2 4 m (32) 9+1 9 + 32 2 = 1.2 · 10−40 m2 = 1.8%): σN i = 2 · 10−45 m2 (59)4 9+1 9 + 59 2 = 5.2 · 10−40 m2 Òîãäà ñèëó òîðìîæåíèÿ íàéä¼ì, ñóììèðóÿ ωF e + mF e + mg ωO ωSi ωM g ωS ωN i + + + + + ME mO + mg mSi + mg mM g + mg mS + mg mN i + mg F = FF e + FO + FSi + FM g + FS + FN i = Vg2 + VE2 ρg ãäå ω - ìàññîâàÿ äîëÿ êàæäîãî ýëåìåíòà. Ïîäñòàâëÿÿ âûøåóêàçàííûå ÷èñëåííûå çíà÷åíèÿ, íàõîäèì óñêîðåíèå m F = 6.5 · 10−26 2 ME s è ñèëó F = 6.5 · 10−26 m · 6 · 1024 kg = 0.39N s2 Êàê âèäíî, çíà÷åíèÿ çàìåäëÿþùåé ñèëû ÷ðåçâû÷àéíî ìàëû è ñïîñîáíû äàæå çà òàêîå áîëüøîå âðåìÿ, êàê âðåìÿ æèçíè Âñåëåííîé (≈13 ìëðä. ëåò= 4 · 1017 c), óìåíüøèòü ñêîðîñòè íåáåñíûõ òåë (Çåìëè è Ñîëíöà) ëèøü íà äåñÿòêè íàíîìåòðîâ â ñåêóíäó. Îäíàêî ðàíüøå, êîãäà êîíöåíòðàöèè è ñêîðîñòè ÷àñòèö ò¼ìíîé ìàòåðèè áûëè áîëüøå, èìåííî îíà ìîãëà ñûãðàòü îïðåäåëÿþùóþ ðîëü â ýâîëþöèè Âñåëåííîé. Ñïèñîê ëèòåðàòóðû [1] Ïðèðîäà. 2001. 7. ñòð.10-19. [2] Ë. Ä. Ëàíäàó, Å. Ì. Ëèôøèö, Òåîðåòè÷åñêàÿ ôèçèêà., ò.2 Ì. [3] M. G. Aartsen et al. [IceCube Collaboration], Phys. Rev. Lett. 110 "Íàóêà". (2013) 131302 [arXiv:1212.4097 [astro-ph.HE]]. [4] T. Tanaka et al. [Super-Kamiokande Collaboration], Astrophys. J. 742 (2011) 78 [arXiv:1108.3384 [astro-ph.HE]]. [5] M. M. Boliev, S. V. Demidov, S. P. Mikheyev and O. V. Suvorova, arXiv:1301.1138 [astro-ph.HE]. [6] S. Adrian-Martinez et al. [ANTARES Collaboration], arXiv:1302.6516 [astro-ph.HE]. [7] R. Agnese et al. [CDMS Collaboration], [arXiv:1304.3706 [astro-ph.CO]]. [8] R. Agnese et al. [CDMS Collaboration], [arXiv:1304.4279 [hep-ex]]. [9] E. Aprile et al. [XENON100 Collaboration], Phys. Rev. Lett. 109 (2012) 181301 [arXiv:1207.5988 [astro-ph.CO]]. [10] G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. 7 267 (1996) 195 [hep-ph/9506380].
