# High-Energy Neutrino Flavor State Transition Probabilities
## 7 Analysis of Results
When comparing these data to our above probabilities, and adjusting for unit of distance conversions, we find that for solar neutrinos at approximately 50 km we deduce a probability of \(\
u_{e}\)\(\sim\)0.985, \(\
u_{\mu}\)\(\sim\)0.928 and \(\
u_{\tau}\)\(\sim\)0.696 and at 75 km \(\
u_{e}\)\(\sim\)0.473, \(\
u_{\mu}\)\(\sim\)0.419 and \(\
u_{\tau}\)\(\sim\)0.020. In comparison with supernova neutrinos, we find that at the same distances we have probabilities of \(\
u_{e}\)\(\sim\)0.202, \(\
u_{\mu}\)\(\sim\)0.189 and \(\
u_{\tau}\)\(\sim\)0.0.33 and at 75 km \(\
u_{e}\)\(\sim\)0.473, \(\
u_{\mu}\)\(\sim\)0.542 and \(\
u_{\tau}\)\(\sim\)0.763. Based on the above data, and our previous calculations, the probability for cosmic ray neutrinos remain on the order of 10\({}^{-5}\) until they reach an approximate distance of \(\sim\)1\(\times\) 10\({}^{5}\) m or 100 km, which is also shown in the above fraction distribution. These observations were conducted at RENO in South Korea. In 2023, new observations began at JUNO in Guangdong, Southern China. Observations should be concluded in late 2024-early 2025. It will be of great interest to note any similarities and/or differences in the observations. Figure 4: Scatter plot depicting the comparison of probabilities between cosmic ray flavor transitions \(\
u_{e}\to\
u_{\mu}\), \(\
u_{\mu}\to\
u_{\tau}\) and \(\
u_{\tau}\to\
u_{\lambda}\). Where the blue, orange and green dots represent the predicted flavor for second, third and higher-order transitions, respectively. The density chart described in Figure 6 was calculated using a predicted flavor contribution to the solar neutrino flux detected on Earth. With an oscillation distance of \(L_{E}=1.5\times 10^{11}\) m, we find our changes to neutrino flavor abundance (number density) with distance for each observed flavor to be \(\frac{\Delta}{\Delta L}<n_{\
u_{e}},n_{\
u_{\tau}},n_{\
u_{\tau}}>\approx<n_{0}(1 -P_{12}),n_{0}P_{12}(1-P_{23}),n_{0}P_{12}P_{23}(1-P_{3\lambda})>\), where \(n_{0}\) is the original number density of electron neutrinos emitted from proton-proton chains within the Sun, which is approximately \(6.0\times 10^{10}\) cm\({}^{2}\), and \(P_{mn}\) are the corresponding flavor transition probabilities. The neutrino flux can be found by \(\frac{2L_{\
u_{\tau}}}{28 MeV}\frac{1}{4\pi\ell^{2}}/\) cm\({}^{2}\approx 6.0\times 10^{10}\) cm\({}^{2}\)s\({}^{-1}\). With an oscillation distance on the order of one AU, \(10^{11}\) m, we find the number of detected flavors we can expect on Earth from the Sun to be approximately \(1.158\times 10^{11}\) neutrinos per square meter per second. When comparing our probability density with the observed fraction density taken by RENO in South Korea [14], we can see that our predicted flavor detection at much greater distances closely resembles those observed by RENO at a tenth of the distance from the Sun. This tells us that within distances of approximately \(100,000\)-\(1,500,000\) m, our probabilities give us great confidence in our ability to predict flavor transitions over great distances. For comparative purposes, tables for both supernova neutrinos as well as those from cosmic rays were constructed using similar methods. However, for the tables below, a supernova distance of \(1.6\times 10^{21}\) m was used (Figure 7), and for the cosmic rays a shorter distance of \(\sim\)\(10^{11}\) m was used (Figure 8) for these particular densities. We found that for the supernova neutrinos, around 50% should be detected as electron neutrinos while \(\sim\)30% should be detected as muon neutrinos and the final 20% should  
Figure 5: The above image shows the fraction of detected neutrino flavors with their corresponding oscillation distance. Figure used with permission of author [14]. Figure 6: The above image shows the fraction of predicted solar neutrino flavor detection given an oscillation distance of \(1.5\times 10^{11}\) m.  
be detected as tau neutrinos.