Торможение небесных тел частицами т¼мной материи

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Òîðìîæåíèå íåáåñíûõ òåë ÷àñòèöàìè ò¼ìíîé ìàòåðèè
Ãîðÿ÷óê È
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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].
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