{ "page": 1, "width": 1448, "height": 2048, "coordinates": "x1,y1,x2,y2 normalised to 0-1000", "blocks": [ { "label": "chart", "bbox": [ 152, 94, 497, 272 ], "text": "\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
r (km)Gνₑ|Gνₑ⊥|
03.52.5
502.01.5
1000.80.7
1500.30.3
2000.10.1
2500.050.05
3000.00.0
", "continues": false }, { "label": "chart", "bbox": [ 500, 94, 842, 272 ], "text": "\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
r (km)Gν̄ₑ|Gν̄ₑ⊥|
03.02.5
501.51.0
1000.50.3
1500.20.1
2000.10.05
2500.050.02
3000.020.01
", "continues": false }, { "label": "caption", "bbox": [ 113, 287, 880, 325 ], "text": "Figure 4.2: Geometrical factors for neutrinos (left) and antineutrinos (right figure), as a function of distance, in our schematic model based on binary star mergers. The two curves correspond to $G_{\\nu}$ (4.38) (solid line) and $G_{\\nu}^{\\perp}$ (4.41) (dashed line) in the self-interaction Hamiltonian Eq. (4.37) and Eq. (4.40) respectively. Results correspond to Model C (Table 4.3).", "continues": false }, { "label": "table", "bbox": [ 111, 338, 934, 577 ], "text": "\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
SymbolNameExpression
$h_0$ Vacuum contribution to the Hamiltonian $h_0 = \\omega \\begin{pmatrix} -c_{2\\theta} & s_{2\\theta} \\\\ s_{2\\theta} & c_{2\\theta} \\end{pmatrix}$
$h_{\\text{mat}}$ Diagonal matter Hamiltonian $h_{\\text{mat},\\alpha\\beta} = \\frac{\\sqrt{2}}{2} G_F n_B \\delta_{\\alpha\\beta} \\left[ 2\\delta_{\\alpha e} Y_e - (1 - Y_e) \\right]$
$\\hat{\\epsilon}_q^* \\cdot \\vec{V}_{\\text{mat}}$ Off-diagonal matter Hamiltonian $\\hat{\\epsilon}_q^* \\cdot \\vec{V}_{\\text{mat},\\alpha\\beta} = \\frac{\\sqrt{2}}{2} G_F n_B \\beta \\delta_{\\alpha\\beta} \\left[ 2\\delta_{\\alpha e} Y_e - (1 - Y_e) \\right]$
$G_{\\nu_\\alpha}$ Geometrical factor $G_{\\nu_\\alpha} = \\int_{\\Omega_{\\nu_\\alpha}} \\mathrm{d}\\Omega (1 - \\hat{q} \\cdot \\hat{p})$
$G_{\\nu_\\alpha}^{\\perp}$ Perpendicular geometrical factor $G_{\\nu_\\alpha}^{\\perp} = \\int_{\\Omega_{\\nu_\\alpha}} \\mathrm{d}\\Omega (\\hat{\\epsilon}^*(\\hat{q}) \\cdot \\hat{p})$
$h_{\\nu\\nu}$ Diagonal self-interaction Hamiltonian $h_{\\nu\\nu} = \\sqrt{2} G_F \\sum_\\alpha \\int \\mathrm{d}p \\left[ G_{\\nu_\\alpha} \\rho_{\\nu_\\alpha} \\frac{L_{\\nu_\\alpha} f_{\\nu_\\alpha}}{\\pi^2 R_{\\nu_\\alpha}^2 \\langle E_{\\nu_\\alpha} \\rangle} - \\bar{\\rho}_{\\bar{\\nu}_\\alpha} G_{\\bar{\\nu}_\\alpha} \\frac{L_{\\bar{\\nu}_\\alpha} f_{\\bar{\\nu}_\\alpha}}{\\pi^2 R_{\\bar{\\nu}_\\alpha}^2 \\langle E_{\\bar{\\nu}_\\alpha} \\rangle} \\right]$
$h_{\\nu\\nu}^{\\perp}$ Off-diagonal self-interaction Hamiltonian $h_{\\nu\\nu}^{\\perp} = \\sqrt{2} G_F \\sum_\\alpha \\int \\mathrm{d}p \\left[ G_{\\nu_\\alpha}^{\\perp} \\rho_{\\nu_\\alpha} \\frac{L_{\\nu_\\alpha} f_{\\nu_\\alpha}}{\\pi^2 R_{\\nu_\\alpha}^2 \\langle E_{\\nu_\\alpha} \\rangle} - \\bar{\\rho}_{\\bar{\\nu}_\\alpha} G_{\\bar{\\nu}_\\alpha}^{\\perp} \\frac{L_{\\bar{\\nu}_\\alpha} f_{\\bar{\\nu}_\\alpha}}{\\pi^2 R_{\\bar{\\nu}_\\alpha}^2 \\langle E_{\\bar{\\nu}_\\alpha} \\rangle} \\right]$
", "continues": false }, { "label": "caption", "bbox": [ 113, 601, 803, 612 ], "text": "Table 4.1: Symbols, names and expressions of the relevant quantities involved in the neutrino evolution equations (see text).", "continues": false }, { "label": "text", "bbox": [ 112, 638, 888, 875 ], "text": "where the dependence on the emission variables is shown. The explicit expression for $\\Gamma^{\\perp}$ is given by Eqs. (C.9)-(C.10) C. Figures 4.2 and 4.3 show the geometrical factor (4.41) as a function of the distance travelled by the neutrinos from the neutrinospheres. The results correspond to the cases A and C (Table 4.3) which can be considered as representative of the the typical behaviors of $G_{\\nu}^{\\perp}$ , as we have been observing in our runs. One can see that $G_{\\nu}^{\\perp}$ have a similar $r$ dependence as $G_{\\nu}$ (4.38), as expected. Their absolute values turn out to be suppressed by a few percents up to several factors, with respect to the $G_{\\nu}$ value. As we will discuss, the $r$ dependence of $G_{\\nu}$ plays a crucial role on the possibility to have multiple crossings and a nonlinear feedback mechanism in presence of helicity coherence (see Section 4.4). By including Eq.(4.41) into (4.40) one gets the same expression Eq. (4.37) for $h_{\\nu\\nu}^{\\perp}$ with $G_{\\nu_\\alpha}^{\\perp}$ Eq.(4.41) replacing $G_{\\nu_\\alpha}$ Eq.(4.36). Table 4.1 sums up the relevant quantities involved in the generalized Hamiltonian.", "continues": false }, { "label": "text", "bbox": [ 133, 884, 887, 900 ], "text": "The neutrino total luminosities and spectra at the neutrinospheres are an essential ingredient of the self-", "continues": false }, { "label": "page_number", "bbox": [ 488, 937, 511, 953 ], "text": "83", "continues": false } ] }