Presupernova Neutrinos as Probes of New Physics Yong-Zhong Qian - - PowerPoint PPT Presentation
Presupernova Neutrinos as Probes of New Physics Yong-Zhong Qian - - PowerPoint PPT Presentation
Presupernova Neutrinos as Probes of New Physics Yong-Zhong Qian University of Minnesota Workshop on Neutrino-Electron Scattering at Low Energies Amherst Center for Fundamental Interactions April 25-27, 2019 Summary Neutrino cooling is the
Summary Neutrino cooling is the standard energy loss mechanism governing evolution of massive stars Presupernova neutrinos provide advance warning for nearby supernovae within ~1 day JUNO might be able to distinguish the neutrino mass ordering through detection of presupernova neutrinos Dark photons complicate stellar evolution but might be constrained by detection of presupernova neutrinos
How to Become a Star Virial theorem for a contracting gas cloud
Tc + ~2 2med2 ∼ GMmp R ✓ M mp ◆ d3 ∼ R3 ⇒ Tc ∼ GMmp R − ~2 2me ✓ M mp ◆2/3 1 R2 ⇒ Tc,max ∝ M 4/3
Massive stars are radiation dominated !
T 4
c
ρc/mp ∼ GMmp R , ρc ∼ M R3
<latexit sha1_base64="3lG5nCaEJQ7Rc2Qv58r7S05Mw=">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</latexit>⇒ Tc ∝ M 1/2 R
<latexit sha1_base64="CfyU2asf/dn98lOw8Hhq6Sfsz9M=">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</latexit>⇒ T 3
c
ρc ∝ M 1/2
<latexit sha1_base64="gO5SvmwOY4cXmYhtz3PNM0OLkI=">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</latexit>Janka 2012
Fe Si, S, Ar, Ca
O, Mg, Ne He, C
H, He He
900 R 1 R 0.1 R 0.01 R
25 M Presupernova Star
Tc ∝ ρ1/3
c
Woosley, Heger, Weaver 2002
Fuel Main Product
Secondary Product
T (109 K) Time (yr)
Main Reaction
H He
14N
0.02 107
CNO
4 H 4He
He O, C
18O, 22Ne
s-process
0.2 106
3 He4 12C
12C
)16O
C
Ne, Mg
Na 0.8 103
12C + 12C
Ne O, Mg Al, P 1.5 3
20Ne
)16O
20Ne
)24Mg
O Si, S
Cl, Ar, K, Ca
2.0 0.8
16O + 16O
Si,S Fe
Ti, V, Cr, Mn, Co, Ni
3.5 0.02
28Si
)…
Nuclear burning stages
(20 MꙨ stars)
来自大质量恒星和伽玛射线暴的中微子 上海交通大学博士学位论文 图 大质量恒星中能量产生和损失速率。
表中的数值在数量级上吻合。 中微子只参与弱相互作用,因此它的产生过程必定与弱相互作用直接相关;在实 际物理系统中,大多数中微子的生成反应也涉及到电磁相互作用和强相互作用(核力) 。 由于恒星温度比较高,各种基本粒子和核素的大量存在,中微子可以通过很多反应通道 生成。表中列举了恒星不同演化阶段中微子的亮度、产生方式以及味道组分。在下 文中,我们将对此作简要的阐述。 中微子 太阳处于主星序阶段时, 主要通过将氢燃烧成氦从而释放能量, 并产生大量 。 通 过对太阳中微子的探测,人们可以更好的检验太阳模型以及相关的恒星结构和演 化理论。由于人们对太阳中微子更加了解,且基于太阳中微子的重要性,我们将 它们归为独立的一类来作介绍。与太阳一样,大质量恒星处于主星序阶段时,也 会以相同的方式产生 ,不过由于温度与密度的不同,各过程中 的流量将与太 阳中微子有所区别。 在主序星中,氢的燃烧过程主要通过两种方式进行, 链式反应( )和
Energy Generation vs. Loss
Processes of Thermal Neutrino Emission Pair annihilation Plasmon decay Photo-neutrino emission Bremsstrahlung
(Z, A) + e− → (Z, A) + e− + ν + ¯ ν e− + e+ → ν + ¯ ν γ + e− → e− + ν + ¯ ν γpl → ν + ¯ ν
上海交通大学博士学位论文 第二章 来自大质量恒星的中微子
- 结果分析与讨论
- 中微子能损速率和各热过程的主导区域
前文已提到, 由于中微子冷却对恒星 包括白矮星和中子星等 的演化十分重要, 因 此人们主要侧重于计算中微子能损速率。对前文中得到的中微子能谱作简单的积分,我 们即可得到各热过程中微子的能损速率。在本小节中,我们先来比较我们得到的中微子 能量损失速率与前人的计算结果,从而检验我们计算结果的正确性。 及其合作者曾 细致计算过上述这些热过程的中微子能损速率, 并得到了各过程的拟合公式。 这些拟 合公式在天体物理学和恒星演化中被广泛应用。为了讨论的方便,下文将比较我们的数 值结果与 拟合公式之间的差别。
)]
- 3
)/(g cm
e
µ / ρ Log[(
1 2 3 4 5 6 7 8 9 10 11
Log[T/K]
7 7.5 8 8.5 9 9.5 10 10.5
Pair Plasma Photo Brem Recom C O Si Fe 20M
F
T = . 3 T = 1 Γ = 1 Γ
图 在 上中各不同热过程中微子能损速率的主导区域。在各区域中,该热过程对总中 微子能损的贡献超过 。图中,我们直接通过 拟合公式得到重组过程的能损速率。
图给出了不同温度和密度下中微子能损速率的计算结果与 拟合公式的比 较。需要特别强调的是, 等人关于各过程的拟合公式只在该过程占中重要地位时才 准确,不难发现,当考虑所有热过程的贡献时,我们中微子能损总速率与 等人的拟
Guo & Qian 2016
What Can Pre-Supernova Neutrinos Tell Us ? Advance warning of supernovae Test of stellar models: progenitor mass Probe of neutrino mass ordering: NH/IH ~ 3.4 Last-day events of 1 to 4 MeV for 1 kpc & 20 kton (JUNO)
12 Msun 15 Msun 20 Msun 25 Msun 6.1 11.9 20.2 24.2 1.8 3.5 5.9 7.1 ¯ νe + p → n + e+
<latexit sha1_base64="0prUyzNXOt9r7COKklwoaInUxY=">AB/XicdVDLSgNBEJz1GeMrPm5eBoMgBMLuJiTxFvTiMYJ5QDaG2UknGTI7u8zMCjEf8WLB0W8+h/e/BtnkwgqWtBQVHXT3eVHnClt2x/W0vLK6tp6aiO9ubW9s5vZ2+oMJYU6jTkoWz5RAFnAuqaQ6tSAIJfA5Nf3SR+M1bkIqF4lqPI+gEZCBYn1GijdTNHo+kZ6Iu5CLPB1ikYObXDeTtfNnlZJbLGE7b9tlx3US4paLhSJ2jJIgixaodTPvXi+kcQBCU06Uajt2pDsTIjWjHKZpL1YQEToiA2gbKkgAqjOZXT/FJ0bp4X4oTQmNZ+r3iQkJlBoHvukMiB6q314i/uW1Y92vdCZMRLEGQeL+jH5s0kCtxjEqjmY0MIlczciumQSEK1CSxtQvj6FP9PGm7eKeTdq2K2er6I4WO0DE6RQ4qoyq6RDVURxTdoQf0hJ6te+vRerFe561L1mLmAP2A9fYJzKCUzg=</latexit>Three Neutrino Mixing
|ναi = X
i
U ∗
αi|νii, |νii =
X
α
Uαi|ναi
U = c12c13 s12c13 s13e−iδ −s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ s23c13 s12s23 − c12c23s13eiδ −c12s23 − s12c23s13eiδ c23c13
Neutrino Mixing in Vacuum Uαi = hνα|νii, ¯ Uαi = h¯ να|¯ νii Neutrino Flavor Evolution in Matter (MSW only) normal mass hierarchy inverted mass hierarchy N¯
νe/N 0 ¯ νe ≈ 0.76
N¯
νe/N 0 ¯ νe ≈ 0.21
(MeV)
ν
E
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Events/MeV
2 4 6 8 10 12 14 16 18 12, NH 15, NH 20, NH 25, NH ν Reactor- ν Geo-
Last-day Pre-SN neutrinos at JUNO for stars at 1 kpc Guo, Qian, & Heger 2019
time to core collapse [days]
3 −
10
2 −
10
1 −
10 1 10
Rates/day
1 −
10 1 10
2
10
3
10 12, NH 15, NH 20, NH 25, NH ν Reactor- ν Geo-
Pre-SN neutrinos of 1 to 4 MeV at JUNO for stars at 1 kpc Guo, Qian, & Heger 2019
µNH
s
<latexit sha1_base64="m/MiVRS8/XfiH7Ef2/CnHIrVdZY=">ACdHicdVFNTxsxFPRuaQuhH4EeOMDBNIrUQxXWCYJwQ3CASxBFBJCycreBMLe21sb6totb+Af8eNn8GFM142iLZqn2RpNDN+9psXK86MDYJ7z3+z8Pbd+8Wl2vKHj58+1dWL4zMNKF9IrnUVzE2lLOU9i2znF4pTbGIOb2Mrw9L/fIn1YbJ9NzOFB0KPElZwgi2jorqt80w0Zjk5xEZbRd5qKcyIlsiUkVomICVeNQrifys+B7eZHgMK1dpqPSe0adotYMz9hkarHW8hd87dt56VuESktlJeyNcrTVLmqhyCIzcqAJ8dFVG8ErSAIEKwBGh3J3Bgb6/bRl2ISslVA8zrNKrfhWNJMkFTSzg2ZoACZYc51pYRTl37zFCFyTWe0IGDKRbUDPn0ArYdMwYJlK7k1r4zP5+I8fCmJmInVNgOzV/ayX5L2Q2aQ7zFmqMktTUj2UZBy6ucsNwDHTlFg+cwATzdxfIZliF5Z1e6q5EF4mhf8HF+0W6rTaP7Yb+wfzOBbBOvgKvgEdsE+OAanoA8IePDWPOhteo/+ht/wm5XV9+Z3voA/ym89Acn6v6c=</latexit>µIH
s
<latexit sha1_base64="bH4G16U7UHD1yPZkCnPI/NH1g=">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</latexit>p1 = 95%
<latexit sha1_base64="sFHpTZ7uwfPNi/rLEasJw9SMCPU=">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</latexit>p2 = 95%
<latexit sha1_base64="tCl+8dtIDMXNzFtW0WJx9cmOs=">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</latexit>Frequency of occurrences Number of events
[days]
s
T
2 −
10
1 −
10 1
L [kpc]
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
= 95%
2
= p
1
p
25 20 15 12
Guo, Qian, & Heger 2019
α
σ
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
NH s
µ
2
10
= 1 day
s
T
12, L = 222 pc 15, L = 222 pc 20, L = 222 pc 25, L = 222 pc = 95%
2
= p
1
p 20 20
Guo, Qian, & Heger 2019
Chang, Essig, & McDermott 2017
log T log electron density dark photon emission competes with neutrino emission too few pairs too heavy electron mass in medium e− + e+ → A∗ vs. e− + e+ → ν + ¯ ν
<latexit sha1_base64="lZJyOI8J+AtN1WZVvk+JFi1hPDc=">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</latexit>˙ QA∗ vs. ˙ Qν¯
ν
<latexit sha1_base64="eUvOERSJbf3HkPm7rIv0tCEqVbg=">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</latexit>mA∗ = 1.21 MeV
<latexit sha1_base64="0uhs+ySe30FMKSXviSOJmygQMcA=">ACBXicdVDLSgMxFM34rPVdamLYLGIizKpYtuFUnHjRqhgH9CpJZNm2tBkZkgyQhm6ceOvuHGhiFv/wZ1/Y6atoKIHLhzOuZd73FDzpS27Q9rZnZufmExtZReXldW89sbNZVElCayTgWy6WFHOfFrTHPaDCXFwuW04Q7OE79xS6VigX+thyFtC9zmcI1kbqZHZyohOf3RyMThwnUP5AnJg7EgBL2l91Mlk7bxt2wghmBUPLYNKZdLBVSCKLEMsmCKaifz7nQDEgnqa8KxUi1kh7odY6kZ4XSUdiJFQ0wGuEdbhvpYUNWOx1+M4J5RutALpClfw7H6fSLGQqmhcE2nwLqvfnuJ+JfXirRXasfMDyNfTJZ5EUc6gAmkcAuk5RoPjQE8nMrZD0scREm+DSJoSvT+H/pF7Io8N84eoWzmdxpEC2AX7AMEiqACLkAV1ABd+ABPIFn6956tF6s10nrjDWd2QI/YL19Ahilmo=</latexit>mA∗ = 2 MeV
<latexit sha1_base64="PEwfH74Aj7odqlpgR+7zg6f2xH0=">ACAnicdVDLSgMxFM34rPVdSVugsUiLsqkim0XSsWNG6GCfUCnlkyaUOTmSHJCGUobvwVNy4UcetXuPNvzLQVPTAhcM593LvPW7ImdK2/WHNzM7NLymltLK6tr65mNzboKIklojQ8kE0XK8qZT2uaU6boaRYuJw23MF54jduqVQs8K/1MKRtgXs+8xjB2kidzHZOdOKzm4PRieOkcwUHxo4U8JLWR51M1s7bto0QglBxWPbkHK5VEAliBLIAumqHYy7043IJGgviYcK9VCdqjbMZaEU5HaSdSNMRkgHu0ZaiPBVXtePzCO4ZpQu9QJryNRyr3ydiLJQaCtd0Cqz76reXiH95rUh7pXbM/DS1CeTRV7EoQ5gkgfsMkmJ5kNDMJHM3ApJH0tMtEktbUL4+hT+T+qFPDrMF6OspXTaRwpsAN2wT5AoAgq4AJUQ0QcAcewBN4tu6tR+vFep20zljTmS3wA9bJ7vPlbw=</latexit>mA∗ = 3 MeV
<latexit sha1_base64="yFvS2o3GNc+NobOFCsz0ERF+gs=">ACAnicdVDLSgMxFM3UV62vqitxEywWcVEmrdh2oVTcuBEq2Ad0asmkaRuazAxJRihDceOvuHGhiFu/wp1/Y6atoKIHLhzOuZd73EDzpS27Q8rMTe/sLiUXE6trK6tb6Q3t+rKDyWhNeJzXzZdrChnHq1pjltBpJi4XLacIfnsd+4pVIx37vWo4C2Be57rMcI1kbqpHeyohOd3RyOTxwnlS04MHKkgJe0Pu6kM3bOtm2EIwJKh7bhpTLpTwqQRbBhkwQ7WTfne6PgkF9ThWKkWsgPdjrDUjHA6TjmhogEmQ9ynLUM9LKhqR5MXxnDfKF3Y86UpT8OJ+n0iwkKpkXBNp8B6oH57sfiX1wp1r9SOmBeEmnpkuqgXcqh9GOcBu0xSovnIEwkM7dCMsASE21S5kQvj6F/5N6PocKufzVUaZyOosjCXbBHjgACBRBVyAKqgBAu7A3gCz9a9Wi9WK/T1oQ1m9kGP2C9fQK9XpW9</latexit>mA∗ = 5 MeV
<latexit sha1_base64="3U2JxpNO04hB4l9uLtO5IUm1P/c=">AB/nicdVDLSgMxFM34rOrKq7cBIsgLsqkPtoulIobN0IF+4BOLZk0bUOTmSHJCGUY8FfcuFDErd/hzr8x01ZQ0QMXDufcy73eCFnSjvOhzUzOze/sJhZspdXVtfWsxubdRVEktAaCXgmx5WlDOf1jTnDZDSbHwOG14w4vUb9xRqVjg3+hRSNsC93WYwRrI3Wy26ITn98eJKf2sQtjVwp4RetJ5tz8o7jIRgSlDxDGkXC4VUAmi1DLIgSmqney72w1IJKivCcdKtZAT6naMpWaE08R2I0VDTIa4T1uG+lhQ1Y7H5ydwzyhd2AukKV/Dsfp9IsZCqZHwTKfAeqB+e6n4l9eKdK/UjpkfRpr6ZLKoF3GoA5hmAbtMUqL5yBMJDO3QjLAEhNtErNCF+fwv9JvZBHh/nC9VGucjaNIwN2wC7YBwgUQVcgiqoAQJi8ACewLN1bz1aL9brpHXGms5sgR+w3j4BiWqUkw=</latexit>