---
abstract: |
    We perform a detailed study of the Earth matter effects on supernova neutrinos with neutrino oscillation parameter LMA and small $\theta_{13}$. The Earth effects show significant dependences on the neutrino path length inside the Earth and the value of $\Delta m^{2}_{12}$. We investigate rather optimistically a possibility that we can probe the value of $\Delta m^{2}_{12}$ by the Earth effects. We assume that $\theta_{12}$ and the direction of the supernova are known with enough accuracy and that the resonance that occurs at higher density in supernova envelope is completely nonadiabatic. Further the neutrino spectra before neutrinos go through the Earth are assumed to be known. Then we show that making use of these dependences, we can obtain implication for the value of $\Delta m^{2}_{12}$ by comparing the observed energy spectrum to the predicted one. When SK detects neutrinos from supernova at 10kpc which traveled through the Earth (nadir angle $<$ 80 degree), $\Delta m^{2}_{12}$ can be determined with an accuracy of $\sim 10\%$. In much of the neutrino-detection-time-$\Delta m^{2}_{12}$ plane, $\Delta m^{2}_{12}$ might be determined with an accuracy equal to or better than $\pm 0.5 \times 10^{-5} {\rm eV}^{2}$.

    $PACS:$ 14.60.Pq; 14.60.Lm; 96.40.Tv; 97.60.Bw;

    $Keywords:$ Neutrino oscillation; Supernovae; Earth effects;
author:
- |
    Keitaro Takahashi$^{\rm a}$ and Katsuhiko Sato$^{\rm a,b}$\
    [*$^{\rm a}$Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo,*]{}\
    [*Tokyo 113-0033, Japan*]{}\
    [*$^{\rm b}$Research Center for the Early Universe, University of Tokyo,*]{}\
    [*7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan*]{}
title: Earth effects on supernova neutrinos and their implications for neutrino parameters
---

Introduction
============

A type II supernova is a prodigious source of neutrinos. Almost all of the binding energy, $E_{b} \sim 10^{53}$erg, is radiated away as neutrinos. These neutrinos carry information about both the core collapse process and intrinsic properties of the neutrinos. In fact, the detection of the neutrino burst from SN1987A [@K2_SN1987A; @IMB_SN1987A] induced a number of studies on these areas [@Arafune; @Smirnov1987A; @Sato; @Goldman].

Neutrino oscillation, which is confirmed by the observations of solar and atmospheric neutrinos, can affect the energy spectrum of supernova neutrino drastically. Neutrinos of all flavors are produced in the high dense region of the iron core [@Totani] and interact with matter before emerging from the supernova. The presence of non-zero masses and mixing in vacuum among various neutrino flavors results in strong matter dependent effects, including conversion from one flavor to another. Hence, the observed neutrino flux in the detectors may be dramatically different for certain neutrino flavors and for certain values of mixing parameters, due to neutrino oscillation. Similar matter effect occurs in the Earth, too.

These oscillation effects on neutrino spectra depend strongly on neutrino oscillation parameters: mixing angles and mass spectrum. Some of them are firmly established but others are not. All the existing experimental results on the atmospheric neutrinos can be well described in terms of the $\nu_{\mu} \leftrightarrow \nu_{\tau}$ vacuum oscillation with mass squared difference and the mixing angle given by [@Fukuda], $$\Delta m^{2}_{\rm atm} \approx 7 \times 10^{-3} {\rm eV}^2 \; , \;   
\sin^{2} 2 \theta \approx 1 .$$ In contrast, for the observed $\nu_{e}$ suppression of solar neutrinos four solutions are still possible [@Bahcall; @Krastev; @Garcia]: large mixing angle (LMA), small mixing angle (SMA), low $\Delta m^{2}$ (LOW), and vacuum oscillation (VO). $$\begin{aligned}
({\rm LMA}) \;\Delta m^{2}_{\odot} & \approx & (1 \sim 10) \times 10^{-5}  {\rm eV}^2 \; , \; 
 \sin^{2} 2 \theta_{\odot} \approx 0.7 \sim 0.95 \\
({\rm SMA}) \;\Delta m^{2}_{\odot} & \approx & (4 \sim 10) \times 10^{-6}  {\rm eV}^2 \; , \;
 \sin^{2} 2 \theta_{\odot} \approx (2 \sim 10) \times 10^{-3} \\
({\rm LOW}) \;\Delta m^{2}_{\odot} & \approx & (0.5 \sim 2) \times 10^{-7}  {\rm eV}^2 \; , \;
 \sin^{2} 2 \theta_{\odot} \approx 0.9 \sim 1.0 \\
({\rm VO}) \;\Delta m^{2}_{\odot} & \approx & (0.6 \sim 6) \times 10^{-10}  {\rm eV}^2 \; , \; 
 \sin^{2} 2 \theta_{\odot} \approx 0.8 \sim 1.0 \end{aligned}$$ For $\theta_{13}$, the mixing angle between mass eigenstate $\nu_{1}, \nu_{3}$, only upper bound has been known from reactor experiment [@CHOOZ] and combined three generation analysis [@Fogli; @Yasuda; @MinakataYasuda]. Also the nature of neutrino mass hierarchy (normal or inverted) is still a matter of controversy[@Minakata; @Barger].

Supernova is a completely different system from solar, atmospheric, accelerator, and reactor neutrinos in regard to neutrino energy and flavor of produced neutrinos, propagation length and so forth. Then neutrino emission from a supernova is expected to give valuable information that can not be obtained from neutrinos from other sources.

There have been some studies on the supernova neutrino as an oscillation parameter prober. Dighe and Smirnov [@Dighe] have studied the role the supernova neutrinos can play in the reconstruction of the neutrino mass spectrum. Dutta et al. [@Dutta] showed numerically that the events involving oxygen targets increase dramatically when there is neutrino mixing. We have shown in our previous work [@KT2] that the degeneracy of the solutions of the solar neutrino problem can be broken by the combination of the SK and SNO detections of a future Galactic supernova.

The Earth effects have also been studied by several authors. In our previous work [@KT1], we analyzed numerically the time-integrated energy spectra in a simple case and found the possibility to probe the mixing angle $\theta_{13}$. Lunardini and Smirnov [@SmirnovEarth] performed a study of the Earth effects taking the arrival directions of neutrinos into account and showed that studies of the Earth effects will select the solution of the solar neutrino problem, probe the mixing $U_{e3}$ and identify the hierarchy of the neutrino mass spectrum.

In this paper, we perform a detailed study of the Earth matter effects on supernova neutrinos and show that the detection of Earth matter effects allows us to probe $\Delta m^{2}_{12}$ more accurately than by solar neutrino observations.

Three-flavor formulation
========================

In the framework of three-flavor neutrino oscillation, the time evolution equation of the neutrino wave functions can be written as follows:

$$i\frac{d}{dt}\left(
    \begin{array}{ccc}\nu_e\\ \nu_{\mu}\\ \nu_{\tau}
    \end{array}\right)
= H(t)\left(
    \begin{array}{ccc}\nu_e\\ \nu_{\mu}\\ \nu_{\tau}
    \end{array}\right) \label{Schrodinger}$$

$$\begin{aligned}
H(t)  \equiv 
U\left(
    \begin{array}{ccc}
        0 & 0 & 0\\
        0 & \Delta m^2_{21} /2E & 0\\
        0 & 0 & \Delta m^2_{31} /2E
    \end{array}\right)U^{-1} 
    +\left(
    \begin{array}{ccc}
        A(t) & 0 & 0\\
        0 & 0 & 0\\
        0 & 0 & 0
    \end{array}\right),\end{aligned}$$

where $A(t)=\sqrt{2}G_{F}n_{e}(t)$, $G_{F}$ is Fermi constant, $n_{e}(t)$ is the electron number density, $\Delta m^2_{ij}$ is the mass squared differences, and $E$ is the neutrino energy. In case of antineutrino, the sign of $A(t)$ changes. Here U is a unitary 3 $\times$ 3 mixing matrix in vacuum: $$U  =  \left(\begin{array}{ccc}
c_{12}c_{13} & s_{12}c_{13} & s_{13}\\
-s_{12}c_{23}-c_{12}s_{23}s_{13} & c_{12}c_{23}-s_{12}s_{23}s_{13} 
& s_{23}c_{13}\\
s_{12}s_{23}-c_{12}c_{23}s_{13} & -c_{12}s_{23}-s_{12}c_{23}s_{13} 
& c_{23}c_{13}
\end{array}\right)\label{mixing_matrix},$$ where $s_{ij} = \sin{\theta_{ij}}, c_{ij} = \cos{\theta_{ij}}$ for $i,j=1,2,3 (i<j)$. We have here put the CP phase equal to zero in the CKM matrix.

In $H(t)$, the first term is the origin of vacuum oscillation, and the second term A(t), which is the only time-dependent term in H(t), is the origin of MSW effect.

Determination of neutrino oscillation parameters
================================================

In this section we summarize our previous studies on the effects of neutrino oscillation on supernova neutrino and show that future detection of neutrino from Galactic supernova allows us to break the degeneracy of the solutions of the solar neutrino problem and to probe $\theta_{13}$.

In [@KT2], three-flavor neutrino oscillation in the star is studied. We calculated the expected event rate and energy spectra, and their time evolution at the SuperKamiokande(SK) and the Sudbury Neutrino Observatory(SNO). For the calculations of them, we used a realistic neutrino burst model based on numerical simulations of supernova explosions [@Totani; @Wilson] and employed a realistic density profile based on a presupernova model [@Woosley]. The distance between the supernova and the Earth was set to 10kpc.

By solving numerically the differential equations (\[Schrodinger\]) from the center of supernova to the outside of supernova, we obtained conversion probabilities $P(\nu_{\alpha \rightarrow \beta})$, i.e., probabilities that a neutrino of flavor $\alpha$ produced at the center of supernova is observed as a neutrino of flavor $\beta$.

We assumed normal mass hierarchy and used four sets of mixing parameters shown in table \[table:parameter\]. Here $\theta_{12}$ and $\Delta m_{12}^{2}$ correspond to the solutions of solar neutrino problem and $\theta_{23}$ and $\Delta m_{13}^{2}$ correspond to the solution of atmospheric neutrino problem. The value of $\theta_{13}$ is taken to be consistent with current upper bound from reactor experiment [@CHOOZ]. These models are named after their values of mixing angle: LMA-L means that $\theta_{12}$ is set to be LMA of solar neutrino problem and $\theta_{13}$ is large.

We then found that when there is neutrino oscillation, neutrino spectra are harder than those in absence of neutrino oscillation. This is because average energies of $\nu_{e}$ and $\bar{\nu}_{e}$ are smaller than those of $\nu_{x}$(either of $\nu_{\mu}$, $\nu_{\tau}$, and their antineutrinos), and neutrino oscillation produces high energy $\nu_{e}$ and $\bar{\nu}_{e}$ which was originally $\nu_{x}$. This feature was used as a criterion of magnitude of neutrino oscillation. We calculated the following ratio of events at both detectors: $$R_{\rm SK} \equiv \frac{\makebox{number of events at } 30<E<70\rm{MeV}}
        {\makebox{number of events at } 5<E<20\rm{MeV}}$$ $$R_{\rm SNO} \equiv \frac{\makebox{number of events at } 25<E<70\rm{MeV}}
        {\makebox{number of events at } 5<E<20\rm{MeV}}$$ The plots of $R_{\rm SK}$ vs. $R_{\rm SNO}$ are shown in Fig.\[figure:ratio\]. The errorbars include only statistical errors. The difference among the following three groups is clear: (1)LMA-L and LMA-S, (2)SMA-L, and (3)SMA-S and no oscillation.

In [@KT1], the Earth effects on the energy spectra are studied using the result of [@KT2]. For mass squared differences in table \[table:parameter\], the supernova neutrinos arrive at Earth in mass eigenstates. This is because neutrino eigenstates with different masses lose coherence on the way from the supernova to the Earth. Since the eigenstates of Hamiltonian in matter differ from the mass eigenstates in vacuum, supernova neutrinos begin to oscillate again in the Earth. The numbers of neutrinos of each mass eigenstates at the surface of the Earth are determined by neutrino oscillation in the supernova [@KT1].

We analyzed numerically the time-integrated energy spectra of neutrino in a mantle-core-mantle step function model of the Earth’s matter density profile. We assumed that neutrino arrived at the detectors after traveling through the Earth along its diameter. We then found that the Earth matter affects the $\nu_{e}$ spectrum significantly in model LMA-S and $\bar{\nu}_{e}$ spectrum slightly in model LMA-S and LMA-L. In other models, there are no significant Earth effects. We concluded that we can differentiate LMA-L from LMA-S, by observing the Earth matter effect in $\nu_{e}$.

Thus we can distinguish all the four models in table \[table:parameter\]. But there are some ambiguities besides statistical errors. One is the mass of the progenitor star. Supernovae with different progenitor masses may result in different original neutrino spectra and neutrino oscillation effects. Studies on this point are now in progress. But dependence of shape of neutrino spectra on progenitor mass is not so large [@mass] and we would still be able to distinguish the models.

Supernova model which we use does not consider rotation of the progenitor star. In general, however, stars rotate through their lives and recent numerical simulation indicates that the rotation facilitate supernova explosion [@Shimizu]. Rotation of the progenitor star can affect its density profile and the dynamics of the neutrino oscillation in supernova.

Another ambiguity is the direction of supernova. The trajectories of neutrinos change according to the location of the supernova, the position of the detector and the time $t$ of the day at which the burst arrives at the Earth [@SmirnovEarth]. Fig \[figure:distance\] shows the dependence of the length $d$ of neutrino trajectory on the time $t$ for the three detectors, SK, SNO and LVD. $d$ is in unit of the diameter of the Earth. Neutrino trajectory can also be described by the nadir angle $\theta_{n}$ of the supernova with respect to the detector. In the next section, we perform a detailed study of the Earth matter effect.

Detailed study of the Earth matter effect
=========================================

In this section, we concentrate on model LMA-S and discuss the dependence of the Earth matter effect on the distance which neutrinos travel in the Earth and $\Delta m^{2}_{12}$. We consider three detectors: SK, SNO and LVD. But energy spectra at SK and LVD in model LMA-L are the same as in model LMA-S, since most of the events at SK and LVD are induced by $\bar{\nu}_{e}$s which are influenced by the Earth matter in both LMA-L and LMA-S. The dependence on $\Delta m^{2}_{12}$ is expected because the oscillation length in the Earth between mass eigenstates $\nu_{1}$ and $\nu_{2}$ is comparable $(\sim 1000 {\rm km})$ to the diameter of the Earth in LMA-S model [@KT1].

Here we use the realistic Earth density profile [@profile], while we used step-like model in our previous work [@KT1]. By solving (\[Schrodinger\]) along the Earth density profile numerically, we obtain conversion probability $P_{\nu_{i} \rightarrow
\nu_{\alpha}}$, i.e. probability that $i$th mass eigenstate at the surface of the Earth is detected as neutrino of flavor $\alpha$. Then combining these probabilities with the result of neutrino oscillation in supernova [@KT2], we obtain neutrino flux and at each detector.

neutrino events at SuperKamiokande
----------------------------------

SuperKamiokande (SK) is a water Cherenkov detector with 32,000 ton pure water based at Kamioka in Japan. The relevant interactions of neutrinos with water are as follows: $$\begin{aligned}
\bar{\nu_e} +p & \rightarrow & n + e^+ \quad(\rm{CC}) 
\label{interaction:anuep}\\
\nu_e + e^- & \rightarrow & \nu_e + e^- \quad(\rm{CC \quad and \quad NC})\\
\bar{\nu_e} + e^- & \rightarrow & \bar{\nu_e} + e^- 
\quad(\rm{CC \quad and \quad NC})\\
\nu_x + e^- & \rightarrow & \nu_x + e^- \quad(\rm{NC})\\
\nu_e + {\rm O} & \rightarrow & {\rm F} + e^- \quad(\rm{CC})\\
\bar{\nu_e} + {\rm O} & \rightarrow & {\rm N} + e^+ 
\quad(\rm{CC})\label{interaction:nuebarO}\end{aligned}$$ where CC and NC stand for charged current and neutral current interactions, respectively. For the cross sections of these interactions, we refer to [@SKcross].

SK is not available now due to the unfortunate accident [@SKaccident] but is expected to be restored within the year. Since the number of PMTs will decrease to half the original, energy threshold and energy resolution will become worse. The detection efficiency is expected to be 90% at 8MeV[@SKefficiency]. In the above interactions, the $\bar{\nu_e}p$ CC interaction \[Eq.(\[interaction:anuep\])\] has the largest contribution to the detected events at SK. Hence the energy spectrum detected at SK (including all the reactions) is almost the same as the spectrum derived from the interaction Eq.(\[interaction:anuep\]) only.

neutrino events at SNO
----------------------

Sudbury Neutrino Observatory(SNO) is a water $\check{\rm{C}}$herenkov detector based at Sudbury, Ontario. SNO is unique in its use of 1000 tons of heavy water, by which both the charged-current and neutral-current interactions can be detected. The interactions of neutrinos with heavy water are as follows, $$\begin{aligned}
\nu_e + d & \rightarrow & p + p + e^-\quad(\rm{CC})\label{eq:SNO_CC_nue}\\
\bar{\nu_e} + d & \rightarrow & n + n + e^+\quad(\rm{CC})
    \label{eq:SNO_CC_anue}\\
\nu_x + d & \rightarrow & n + p + \nu_x\quad(\rm{NC})\\
\bar{\nu_x} + d & \rightarrow & n + p + \bar{\nu_x}\quad(\rm{NC})\end{aligned}$$ The two interactions written in Eqs.(\[eq:SNO\_CC\_nue\]) and (\[eq:SNO\_CC\_anue\]) are detected when electrons emit $\check{\rm{C}}$herenkov light. These reactions produce electrons and positrons whose energies sensitive to the neutrino energy, and hence the energy spectra of electrons and positrons give us the information on the original neutrino flux. In this work, we mainly take into account these two charged current interactions, since the number of neutral current events does not change by neutrino oscillation. For the cross sections, we refer to [@SNOcross]. The efficiency of detection is set to be one, because we have no information about it.

The SNO detector has also 7,000 tons of light water which can be used to detect neutrinos. This can be considered to be a miniature of SuperKamiokande (32,000 tons of light water). Then the number of events detected by light water at SNO is 7/32 of that at SuperKamiokande.

neutrino events at LVD
----------------------

The Large Volume Detector (LVD) in the Gran Sasso Underground Laboratory is a $\nu$ observatory mainly designed to study low energy neutrinos from gravitational stellar collapse. LVD consists of 2 kinds of detectors, namely: liquid scintillator, for a total mass of 1840 tons, and streamer tubes for a total surface of about $7000 {\rm m}^{2}$.

The bulk of events is due to the capture reaction: $$\bar{\nu}_{e} + p \rightarrow n + e^{+} \; \; \; \; \; \; 
n + p \rightarrow d + \gamma_{2.2 {\rm MeV}}$$ Further, about $5\%$ of the events are due to neutral current interactions with ${}^{12}$C which deexcites emitting a 15.1 MeV $\gamma$. Moreover, $3\%$ of events are due to elastic scattering of all neutrino flavours on electrons, and less than $1\%$ to charged current interactions of $\nu_{e}$ and $\bar{\nu}_{e}$ with ${}^{12}$C nuclei. For the cross sections, we refer to [@LVDcross]. The appropriate detection efficiency curve is also taken into account [@SmirnovEarth].

nadir angle dependence of energy spectra
----------------------------------------

Fig.\[figure:SK\_angle\], \[figure:SNO\_angle\] and \[figure:LVD\_angle\] shows the nadir angle dependence of neutrino spectrum at SK, SNO and LVD, respectively. Neutrino oscillation parameters are set to the values in model LMA-S except $\Delta m^{2}_{12} = 2 \times 10^{-5} {\rm eV}^{2}$. Since the oscillation lengths are comparable to the radius of the Earth, the Earth effects are strongly dependent on the nadir angle.

$\Delta m^{2}_{12}$ dependence of energy spectra and determination of $\Delta m^{2}_{12}$
=========================================================================================

method
------

Fig.\[figure:SK\_mass\], \[figure:SNO\_mass\] and \[figure:LVD\_mass\] shows the $\Delta m^{2}_{12}$ dependence of energy spectrum at SK, SNO and LVD. Neutrino oscillation parameters are set to the values in model LMA-S except $\Delta m^{2}_{12}$. Nadir angle is set to 0 degree. The larger $\Delta m^{2}_{12}$ results in higher-frequency oscillation in the energy spectra with respect to the energy. This is because the neutrino oscillation length is proportional to the inverse of $\Delta m^{2}$.

As can be seen, there is significant dependence of the Earth effect on $\Delta m^{2}_{12}$. Making use of this dependence, we can probe $\Delta m^{2}_{12}$ by the observations of the supernova neutrino spectra more accurately than by the observations of solar neutrino, if neutrino oscillation parameters are as in model LMA-S.

We assume that the direction of the supernova is given by direct optical observations or by the experimental study of the neutrino scattering on electrons [@direction1; @direction2]. Other parameters of neutrino oscillation except $\Delta m^{2}_{12}$ are also assumed to be known with enough accuracy. But in fact, $\theta_{23}$ does not affect the Earth matter effects and $\theta_{13}$ affects only the magnitude of the Earth matter effects through the adiabaticity of the H-resonance in supernova envelope[@Dighe]. If $\sin^{2}{\theta_{13}} {\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}10^{-5}$, the H-resonance is completely nonadiabatic for broad range of $\Delta m_{13}^{2}$ and in this case the Earth matter effects do not depend on the value of $\theta_{13}$. So what we need with enough accuracy is $\theta_{12}$ only if we assume the H-resonance is completely nonadiabatic.

Comparing the observed neutrino spectra to the predicted spectra of various values of $\Delta m^{2}_{12}$, we can determine the value of $\Delta m^{2}_{12}$. We perform Monte Carlo simulation to obtain the expected observed spectra at each detector. Here we take statistical fluctuation in number of neutrino events and energy resolution of each detector into account. Energy spectra are then binned according to the energy resolution. For the value of the energy resolution, we refer to [@SmirnovEarth] but that of SK after restoration is expected to be $\sqrt{2}$ times as bad as before the accident[@SKefficiency]. The energy resolution of each detector is shown in Fig.\[figure:resolution\].

The simulated spectrum is compared to the predicted spectrum with a $\chi^{2}$ method. The definition of the reduced $\chi^{2}$, which is a function with respect to $\Delta m^{2}_{12}$, is, $$\chi^{2}(\Delta m^{2}_{12}) = \frac{1}{d} \mathop{\sum}^{d}_{i=1} 
\frac{(N^{\rm sim}_{i}-N^{\rm pre}_{i}(\Delta m^{2}_{12}))^{2}}
{N^{\rm pre}_{i}(\Delta m^{2}_{12})},$$ where $d$ is the number of bins and $N^{\rm sim}_{i}$ and $N^{\rm pre}_{i}(\Delta m^{2}_{12})$ are the simulated and predicted number of events in $i$th bin, respectively. The $\Delta m^{2}_{12}$ with the least $\chi^{2}(\Delta m^{2}_{12})$ is considered to be the value determined by this method.

representative results
----------------------

We make 1000 “supernovae” and “observe” the neutrino spectra assuming that the true value of $\Delta m^{2}_{12}$ is $4 \times 10^{-5} {\rm eV}^{2}$. Fig.\[figure:representative\] show the representative results of $\Delta m^{2}_{12}$ determination by the method described above. These are relative frequencies that the observed spectrum is identified with the theoretical spectrum with the various values of $\Delta m^{2}_{12}$. The nadir angle is set to be 0 degree at each detector and the $\chi^{2}$ method is performed with the data of each detector only.

As can be seen, if the supernova occur at the direction which the nadir angle is 0 degree at SK or SNO, $\Delta m^{2}_{12}$ can be determined with a high accuracy $(4.00 \pm 0.07) \times 10^{-5} 
{\rm eV}^{2}$ or $(4.00 \pm 0.20) \times 10^{-5} {\rm eV}^{2}$ in case of SK and SNO, respectively. However, in case of LVD, it is difficult to determine $\Delta m^{2}_{12}$. The difference in the accuracy of determination of $\Delta m^{2}_{12}$ is due to the difference in the properties of the detectors. Here the properties of the detectors mean the energy resolution, the detectability of $\nu_{e}$ and the number of events.

Since the Earth effects are large in $\nu_{e}$ channel and relatively small in $\bar{\nu}_{e}$ channel, SK and LVD, at which most of events are $\bar{\nu}_{e}$, are disadvantageous. But the statistical error is very small at SK due to the large event number, $\sim 10000$ (10 kpc), and SK is the most efficient to identify the Earth effects. SNO is also efficient due to its high energy-resolution and detectability of $\nu_{e}$ although the number of events is small. Then LVD is not favorable for detecting the Earth effects. The properties of the detectors are summarized in the Table \[table:detector\].

It should be noted that the effectiveness of determining $\Delta m^{2}_{12}$ depend on the value of $\Delta m^{2}_{12}$ itself and nadir angle. These dependences are discussed in the following subsection.

determination of $\Delta m^{2}_{12}$
------------------------------------

Fig.\[figure:SNO\_best\_mass\] shows the relative frequencies at SNO that the simulated spectrum based on the theoretical spectrum with $\Delta m^{2}_{12} = 2 {\rm (solid)}, 4 {\rm (dashed)}, 6 {\rm (long-dashed)} 
\times 10^{-5} {\rm eV}^{2}$ is identified with the theoretical spectrum with the various values of $\Delta m^{2}_{12}$. The nadir angle is set to be 0 degree. When $\Delta m^{2}_{12}$ is large, the oscillation in energy spectrum becomes rapid and then statistical errors make it difficult to identify the Earth effect and determine $\Delta m^{2}_{12}$. However at SK, its large number of events allows us to determine $\Delta m^{2}_{12}$ rather accurately irrespective of the true value of $\Delta m^{2}_{12}$ (Fig.\[figure:SK\_best\_mass\]).

Fig.\[figure:SNO\_best\_angle\] shows the relative frequencies at SNO at various nadir angles: 15 (solid), 80 (dashed) and 85 (long-dashed). The value of $\Delta m^{2}_{12}$ is set to be $4 \times 10^{-5} {\rm eV}^{2}$. Since the oscillation length of 60 MeV $\nu$ in Earth matter is $\sim 6000$km [@KT1], the Earth effects do not appear unless nadir angle is smaller than $\sim 80$ degree.

Now we consider the problem, ’How accurately can we determine $\Delta m^{2}_{12}$ when the true value of $\Delta m^{2}_{12}$ and the arrival time of neutrinos are given?’

As is discussed above, when the nadir angle at SK is smaller than 80 degree (time $= 0 \sim 8$ and $19 \sim 24$ hour), the value of $\Delta m^{2}_{12}$ can be determined accurately using only the SK data irrespective of the value of $\Delta m^{2}_{12}$ in the allowed region of the solar neutrino problem. When the nadir angle is smaller than 80 degree at SNO and greater than 80 degree at SK (time $= 10 \sim 19$ hour), the value of $\Delta m^{2}_{12}$ can be determined accurately using only the SNO data if the true value of $\Delta m^{2}_{12}$ is smaller than $\sim 4 \times 10^{-5} {\rm eV}^{2}$. However, if $\Delta m^{2}_{12} > 4 \times 10^{-5} {\rm eV}^{2}$, the single data of SNO is not sufficient. We performed then a $\chi^{2}$ method combining data of various detectors for time $= 8 \sim 19$ hour.

Fig.\[figure:contour\] is the contour map of the probability that the value $\Delta m^{2}_{12}$ can be determined with an accuracy equal to or better than $\pm 0.5 \times 10^{-5} {\rm eV}^{2}$. The probability is more than 0.7 for three fifths of the time-$\Delta m^{2}_{12}$ plane.

Discussion and conclusion
=========================

In this paper, we performed a detailed study of the Earth matter effects on supernova neutrinos with neutrino oscillation parameter LMA and small $\theta_{13}$. We showed that we can probe $\Delta m^{2}_{12}$ accurately by comparing the observed energy spectra to the predicted one. In much of the time-$\Delta m^{2}_{12}$ plane, $\Delta m^{2}_{12}$ can be determined with an accuracy equal to or better than $\pm 0.5 \times 10^{-5} {\rm eV}^{2}$. When SK detect neutrinos from supernova at 10kpc which traveled through the Earth (nadir angle $<$ 80 degree), $\Delta m^{2}_{12}$ can be determined with an accuracy of $\sim 10\%$. This accuracy is amazing compared to that from current solar neutrino experiments, which determine only the order of $\Delta m^{2}_{12}$. But as we stated before, $\theta_{12}$ and the direction of the supernova need to be known with enough accuracy. How much our results are affected by the uncertainties of the other parameters will be our future work.

$\Delta m^{2}_{12}$ can also be probed by KamLAND experiment[@KamLAND]. If the solution of solar neutrino problem is LMA, $\Delta m^{2}_{12}$ is expected to be known by three-year data-taking as precisely as by supernova neutrino discussed above. There is some possibility that we can know it from a Galactic supernova before the completion of the KamLAND experiment. Even if the results of KamLAND experiment is earlier than future galactic supernova, it is important to cross-check the results of ground-based experiments by astrophysical observations. Furthermore, it is meaningful itself to notice that we can obtain implication for $\Delta m^{2}$ from supernova neutrino.

We assumed in this paper that the neutrino spectra at the surface of the Earth were given. But since the original fluxes based on the numerical supernova model have some ambiguities, the sensitivity to $\Delta m_{12}^{2}$ might be worse than our estimation. Estimation of uncertainties of the numerical supernova model is hard because there is currently no successful simulation other than that by the Lawrence Livermore group [@Wilson]. There are, however, some studies about the temperatures of the produced neutrinos. According to them, they are typically[@Janka], $$\langle E_{\nu_{e}} \rangle = 10 - 12 {\rm MeV}, \;\;
\langle E_{\bar{\nu}_{e}} \rangle = 14 - 17 {\rm MeV}, \;\;
\langle E_{\nu_{x}} \rangle = 24 - 27 {\rm MeV}.$$ On the other hand, in the model that we used, they are $$\langle E_{\nu_{e}} \rangle \sim 13 {\rm MeV}, \;\;
\langle E_{\bar{\nu}_{e}} \rangle \sim 16 {\rm MeV}, \;\;
\langle E_{\nu_{x}} \rangle \sim 23 {\rm MeV}, \;\;$$ although the spectra are not exactly the Fermi-Dirac distribution[@Totani]. These values seem close to the typical values and the analysis taking these uncertainties into account will be performed in our future work.

Furthermore, supernova neutrino spectra depend on the mass of the progenitor star. However, we will be able to reconstruct the neutrino fluxes at the surface of the Earth from the data from the detector which detect the neutrinos directly from the supernova. Although the statistical errors then become larger and the accuracy of the determination of $\Delta m^{2}_{12}$ becomes worse, the Earth effects on the supernova neutrinos will still give valuable information which cannot be obtained from the other neutrino sources.

Acknowledgments
===============

We would like to thank A.Suzuki for advice on cross sections of neutrino interaction in scintillator experiment. We also thank Super Kamiokande people including Y. Totsuka, Y. Suzuki, T.Kajita, M. Nakahata and Y. Fukuda for showing the most recent SK data, and fruitful discussion. This work was supported in part by Grants-in-Aid for Scientific Research provided by the Ministry of Education, Science and Culture of Japan through Research Grant No.07CE2002.

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   model   $\sin^{2} 2 \theta_{12}$   $\sin^{2} 2 \theta_{23}$   $\sin^{2} 2 \theta_{13}$   $\Delta m_{12}^{2}({\rm eV}^{2})$   $\Delta m_{13}^{2}({\rm eV}^{2})$   $\nu_{\odot}$ problem
  ------- -------------------------- -------------------------- -------------------------- ----------------------------------- ----------------------------------- -----------------------
   LMA-L             0.87                       1.0                       0.043                   $7.0 \times 10^{-5}$                $3.2 \times 10^{-3}$                   LMA
   LMA-S             0.87                       1.0                $1.0 \times 10^{-6}$           $7.0 \times 10^{-5}$                $3.2 \times 10^{-3}$                   LMA
   SMA-L     $5.0 \times 10^{-3}$               1.0                       0.043                   $6.0 \times 10^{-6}$                $3.2 \times 10^{-3}$                   SMA
   SMA-S     $5.0 \times 10^{-3}$               1.0                $1.0 \times 10^{-6}$           $6.0 \times 10^{-6}$                $3.2 \times 10^{-3}$                   SMA

             detector                        SNO                    SK                LVD
  ------------------------------ --------------------------- ----------------- -----------------
            main event            ${\nu}_{e},\bar{\nu}_{e}$   $\bar{\nu}_{e}$   $\bar{\nu}_{e}$
   number of events (10[kpc]{})              300                   10000              800
        energy resolution                 $<$ 3 MeV              $<$ 6 MeV         $<$ 7 MeV
