SISSA 13/2024/FISI

Neutrino Tomography of the Earth:
the Earth Total Mass, Moment of Inertia and
Hydrostatic Equilibrium Constraints

S. T. Petcov$\mbox{}^{a,b)}$ 111Also at: Institute of Nuclear Research and
Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria.

a INFN/SISSA, Via Bonomea 265, 34136 Trieste, Italy

b Kavli IPMU (WPI), University of Tokyo, 5-1-5 Kashiwanoha, 277-8583 Kashiwa,
Japan

We investigate the implications of the constraints following from the precise
knowledge of the total Earth mass, $M_{\oplus}$, and moment of inertia,
$I_{\oplus}$, and from the requirement that Earth be in hydrostatic
equilibrium (EHE), in the neutrino tomography studies of the Earth density
structure. In order to estimate the sensitivity of a given neutrino detector
to possible deviations of the inner core (IC), outer core (OC), core (IC + OC)
and mantle Earth densities from those obtained using geophysical and
seismological data and described by the preliminary reference Earth model
(PREM), in the statistical analyses performed within the neutrino tomography
studies one typically varies the density of each of these structures. These
variations, however, must respect the $M_{\oplus}$, $I_{\oplus}$ and EHE
constraints. Working with PREM average densities we derive the $M_{\oplus}$,
$I_{\oplus}$ and EHE constraints on the possible density variations when one
approximates the Earth density structure with i) three layers - mantle, outer
core and inner core, and ii) four layers - upper mantle, lower mantle, outer
core and inner core. We get drastically different results in the two cases.

##  1 Introduction

At present our knowledge about the interior composition of the Earth and its
density structure is based primarily on seismological and geophysical data
(see, e.g., [1, 2, 3, 4, 5, 6, 7, 8, 9]). These data were used to construct
the Preliminary Reference Earth Model (PREM) [10] of the density distribution
of the Earth. In the PREM model, the Earth density distribution $\rho_{\rm E}$
is assumed to be spherically symmetric, $\rho_{\rm E}=\rho_{\rm E}(r)$, $r$
being the distance from the Earth center, and there are three major density
structures (or layers)- the inner core, outer core and the mantle, and a
certain number of substructures (shells). In addition there are three shells
below, but very close to, the Earth surface - the ocean and the two crust
“surface” layers. The mantle has five shells in the model. The mean Earth
radius is $R_{\oplus}=6371$ km; the Earth core has a radius of $R_{c}=3480$
km, with the IC and OC extending respectively from $r=0$ to $r=1221.5$ km, and
from $r=1221.5$ km to $r=3480$ km. The mantle layer is located between the
outer core and the lower crust layer, i.e., for $r$ in the interval $R_{\rm
C}\leq r\leq R_{\rm man}$, with $R_{\rm man}=6346.6$ km. The mean densities of
the mantle, the inner and out core are respectively $\bar{\rho}_{\rm
man}=4.66$ g/cm3, $\bar{\rho}_{\rm IC}=12.89$ g/cm3 and $\bar{\rho}_{\rm
OC}=10.90$ g/cm3. The change of density between the mantle and the outer core
is described by a step function.

The determination of the radial density distributions in the mantle, outer
core and inner core (or core) $\rho_{\rm man}(r)$, $\rho_{\rm OC}(r)$ and
$\rho_{\rm IC}(r)$ (or $\rho_{\rm C}(r)$) from seismological and geophysical
data is not direct and suffers from uncertainties [2, 3, 4, 5]. An approximate
and perhaps rather conservative estimate of this uncertainty for
$\rho_{man}(r)$ is $\sim 5\%$; for the core density $\rho_{c}(r)$ it is larger
and can be significantly larger [2, 3, 4, 5]. It was concluded in [4] (see
also [5]), in particular, that the density increase across the inner core -
outer core boundary is known with an uncertainty of about 20%.

A precise knowledge of the Earth’s density distribution and of the average
densities of the Earth’s three different major structures - the mantle, outer
core and inner core - is essential for understanding the physical conditions
and fundamental aspects of the structure and properties of the Earth’s
interior (including the dynamics of mantle and core, the bulk composition of
the Earth’s three structures, the generation, properties and evolution of the
Earth’s magnetic field and the gravity field of the Earth) [2, 11, 1, 12]. The
thermal evolution of the Earth’s core, in particular, depends critically on
the density change across the inner core - outer core boundary (see, e.g.,
[13]).

A unique alternative method of obtaining information about the density profile
of the Earth is the neutrino tomography of the Earth [14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27]. The propagation of the active flavour
neutrinos and antineutrinos $\nu_{\alpha}$ and $\bar{\nu}_{\alpha}$,
$\alpha=e,\mu,\tau$, in the Earth is affected by the Earth matter. The
original idea of neutrino Earth tomography is based on the observation that
the cross section of the neutrino-nucleon interaction rises with energy. For
neutrinos with energies $E_{\nu}\gtrsim$ a few TeV, the inelastic scattering
off protons and neutrons leads to absorption of neutrinos and thus to
attenuation of the initial neutrino flux. The magnitude of the attenuation
depends on the Earth matter density along the neutrino path. Attenuation data
for neutrinos with different path-lengths in the Earth carry information about
the matter density distribution in the Earth interior. The absorption method
of Earth tomography with accelerator neutrino beams, which is difficult (if
not impossible) to realise in practice was discussed first in [14, 15] and
later in grater detail in [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27].

The oscillations between the active flavour neutrinos and antineutrinos,
$\nu_{\alpha}\leftrightarrow\nu_{\beta}$ and
$\bar{\nu}_{\alpha}\leftrightarrow\bar{\nu}_{\beta}$, $\alpha,\beta=e,\mu$
having energies in the range $E\sim(0.1-15.0)$ GeV and traversing the Earth
can be strongly modified by the Earth matter effects (see, e.g., [28]). These
modifications depend on the Earth matter density along the path of the
neutrinos 111More precisely, they depend on the electron number density
$N_{e}(r)$, $N_{e}(r)=\rho_{rmE}(r)Y_{e}(r)/m_{N}$, $Y_{e}(r)$ and $m_{N}$
being the electron fractions number and nucleon mass. It follows from the
studies of the Earth matter composition that $Y_{e}\cong(0.490-0.0496)$ in the
mantle and $Y_{e}\cong(0.466-0.471)$ in the core (see, e.g., [29, 30, 31,
32]). Taking into account these uncertainties has no effect on the results on
the densities of the mantle and the core obtained in neutrino tomography
studies (see, e.g.,[33]). The relation between $N_{e}(r)$ and $\rho_{rmE}(r)$
can be used to obtain information about $Y_{e}(r)$, i.e., the composition of
the core and mantle layers of the Earth (see, e.g., [34, 35]). . Thus, by
studying the effects of Earth matter on the oscillations of, e.g., $\nu_{\mu}$
and $\nu_{e}$ ($\bar{\nu}_{\mu}$ and $\bar{\nu}_{e}$) neutrinos traversing the
Earth along different trajectories it is possible to obtain information about
the Earth density distribution.

Atmospheric neutrinos (see, e.g., [36, 37]) are a perfect tool for performing
Earth tomography. Consisting of significant fluxes of muon and electron
neutrinos and antineutrinos, $\nu_{\mu}$, $\nu_{e}$, $\bar{\nu}_{\mu}$ and
$\bar{\nu}_{e}$, produced in the interactions of cosmic rays with the Earth
atmosphere, they have a wide range of energies spanning the interval from a
few MeV to multi-GeV to multi-TeV. Being produced isotropically in the upper
part of the Earth atmosphere at a height of $\sim 15$ km, they travel
distances from $\sim 15$ km to 12742 km before reaching detectors located on
the Earth surface, crossing the Earth along all possible directions and thus
“scanning” the Earth interior. The interaction rates that allow to get
information about the Earth density distribution can be obtained in the
currently taking data IceCube experiment [38, 39] and its planned upgrade
[39], in the ORCA [40], Hyper Kamiokande [41] and DUNE [42] experiments which
are under cosntruction, and in the planned INO [43] experiment.

The idea of using the absorption method of Earth tomography with atmospheric
neutrinos was discussed first, to our knowledge, in [27]. In 2018 in [33] the
authors used the data of the IceCube experiment on multi-TeV atmospheric
$\nu_{\mu}$ and $\bar{\nu}_{\mu}$ with sufficiently long paths in the Earth
and obtained information about the Earth density distribution, which, although
not very precise, broadly agrees with the PREM model. More specifically, in
[33] it is assumed that the Earth density distribution is spherically
symmetric. The analysis is performed with a five layer Earth model: the inner
core, two equal width layers of the outer core and two equal width layers of
the mantle. The densities in each of the five layers are varied independently.
The external constraints on the Earth total mass $M_{\oplus}$ and moment of
inertia $I_{\oplus}$, which are known with a remarkable high precision [44,
45, 46], as well as the Earth hydrostatic equilibrium (EHE) constraint, were
not applied. The results are obtained with the IceCube data on the zenith
angle dependence of the fluxes of up-going atmospheric $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ with energies in the interval $E=(1.5-20.0)$ TeV [47]. Four
different models of the initial fluxes of atmospheric $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ were used in the analysis. The value of the Earth mass found
in [33], $M^{\nu}_{\oplus}=(6.0^{+1.6}_{-1.3})\times 10^{24}$ kg, is in good
agreement with gravitationally determined value [44, 45],
$M_{\oplus}=(5.9722\pm 0.0006)\times 10^{24}~{}{\rm kg}$. Thus, the Earth was
“weighted” with neutrinos. The results obtained in [33] contain evidence at
$2\sigma$ C.L. that the core is denser than the mantle:
$\bar{\rho}^{\nu}_{c}({\rm 3layer})-\bar{\rho}^{\nu}_{man}({\rm
2layer})=(13.1^{+5.8}_{-6.3})$ g/cm3, where $\bar{\rho}^{\nu}_{c}({\rm
3layer})$ and $\bar{\rho}^{\nu}_{man}({\rm 2layer})$ are the values of the
average core and mantle densities determined in [33]. This was the first time
the study of neutrinos traversing the Earth provided information of the Earth
interior and marked the beginning of real experimental data driven neutrino
tomography of the Earth.

The neutrino oscillation tomography of the Earth with IceCube, ORCA, DUNE and
INO detectors is discussed in [48, 34, 49, 50, 35, 51, 52, 53, 54, 55, 56, 57,
58] (the early studies include [59, 60] briefly described in [52]). In order
to estimate the sensitivity of a given neutrino detector to possible
deviations of the inner core (IC), outer core (OC), core (IC + OC) and mantle
Earth densities from those obtained using geophysical and seismological data
and described by PREM, in the statistical analyses performed within the
neutrino tomography studies one typically varies the density of each of these
structures. These variations, however, must respect the $M_{\oplus}$,
$I_{\oplus}$ and EHE constraints. In the studies performed so far only in [53]
all three $M_{\oplus}$, $I_{\oplus}$ and EHE constraints were taken into
account. The $M_{\oplus}$ and EHE constraints, but not the $I_{\oplus}$ one,
were implemented in the analyses performed in [52, 57].

The existing studies provide very limited information about the implications,
and thus on the relevance, if any, of the three external constraints -
$M_{\oplus}$, $I_{\oplus}$ and EHE - in the neutrino tomography studies of the
Earth density distribution. The aim of the present letter is to assess the
importance of the three constraints in the neutrino tomography studies of the
Earth interior. Working with PREM average densities we derive the
$M_{\oplus}$, $I_{\oplus}$ and EHE constraints on the possible density
variations in two illustrative and widely considered cases, namely, when one
approximates the Earth density structure with i) three layers - mantle, outer
core and inner core, and ii) four layers - upper mantle, lower mantle, outer
core and inner core.

The article is organised as follows. In Section 2 we derive the contributions
of the “surface” layers, the mantle, outer core and inner core to the Earth
total mass and moment of inertia in terms of average densities of these layers
calculated using the corresponding PREM density distributions. In Section 3 we
implement the $M_{\oplus}$, $I_{\oplus}$ and EHE constraints considering
uniform ($r-$independent) deviations of the average densities of the Earth
layers from their respective PREM values. In sub-sections 3.1 and 3.2 We
consider the cases when one approximates the Earth density structure with i)
three layers - mantle, outer core and inner core, and ii) four layers - upper
mantle, lower mantle, outer core and inner core. We obtain the equations which
the deviation parameters should satisfy and provide the analytic solutions of
these equations. We further comment on the implications of the results
obtained. Section 4 contains a brief summary.

##  2 Earth Total Mass and Moment of Inertia: the Mantle, Outer Core, Inner
Core and “Surface” Layers Contributions

In order to estimate the sensitivity of any detector to the IC, OC, core (IC +
OC) and mantle densities, one typically varies the density in one of these
structures - the structure of interest. These variations, however, must
respect the constraints following form the precise knowledge of the total
Earth mass, $M_{\oplus}$, and momentum of inertia, $I_{\oplus}$ [44, 45, 46]:

$\displaystyle M_{\oplus}=(5.9722\pm 0.0006)\times 10^{24}~{}{\rm kg}\,,$ (1)
$\displaystyle I_{\oplus}=(8.01736\pm 0.00097)\times 10^{37}~{}{\rm
kg\,m^{2}}\,.$ (2)

The mean value of $I_{\oplus}$ can be related to the Earth mean total mass and
radius $R_{\oplus}$:

$I_{\oplus}=C_{\rm G(PREM)}\,M_{\oplus}\,R^{2}_{\oplus}\,.$ (3)

where $C_{\rm G}$ ($C_{\rm PREM}$) is a constant determined in gravity
experiments (the PREM model):

$C_{\rm G}=0.330745\,,~{}~{}~{}C_{\rm PREM}=0.33080\,.$ (4)

One can implement the total Earth mass and moment of ineria contraints by
compensating the variation of the density in the structure of interest by
corresponding change of the densities in at least two of the other structures.

Following PREM, we will assume that the Earth density distribution $\rho_{\rm
E}$ is spherically symmetric, $\rho_{\rm E}=\rho_{\rm E}(r)$, where $r$ is the
distance from the Earth center. The Earth density structure consists of inner
core, outer core, mantle and three relatively thin layers close to the Earth
surface - the ocean and two crust “surface” layers, having masses $M_{\rm
IC}$, $M_{\rm OC}$, $M_{\rm man}$ and $M_{\rm o+c1+c2}$, and moments of
inertia $I_{\rm IC}$, $I_{\rm OC}$, $I_{\rm man}$ and $I_{\rm o+c1+c2}$,
respectively. Correspondingly, the total Earth mass in PREM is given by:

$\displaystyle M_{\oplus}=M_{\rm IC}+M_{\rm OC}+M_{\rm man}+M_{\rm
o+c1+c2}=\int^{R_{\oplus}}_{0}4\pi\rho_{\rm E}(r)r^{2}dr$
$\displaystyle=4\pi\left[\int^{R_{\rm IC}}_{0}\rho_{\rm
IC}(r)r^{2}dr+\int^{R_{\rm C}}_{R_{\rm IC}}\rho_{\rm
OC}(r)r^{2}dr+\int^{R_{\rm man}}_{R_{\rm C}}\rho_{\rm
man}(r)r^{2}dr+\int^{R_{\oplus}}_{R_{\rm man}}\rho_{\rm
o+c1+c2}(r)r^{2}dr\right]\,$ (5)

where $\rho_{\rm IC}(r)$, $\rho_{\rm OC}(r)$, $\rho_{\rm mant}(r)$ and
$\rho_{\rm o+c1+c2}(r)$ are the inner core (IC), outer core (OC), the mantle
and the “surface” layers densities as a function of $r$ and, according to
PREM,

$R_{\rm IC}=1221.5~{}{\rm km}\,,~{}R_{\rm OC}=R_{\rm C}=3480~{}{\rm
km}\,,~{}R_{\rm man}=6346.6~{}{\rm km}\,,~{}R_{\oplus}=6371~{}{\rm km}\,.$ (6)

Note that $R_{\rm man}$ is smaller than the mean radius of the Earth
$R_{\oplus}$ due to the presence in PREM of an “ocean” and two crust “surface”
layers.

Similarly, the Earth total moment of inertia can be written as:

$\displaystyle I_{\oplus}=I_{\rm IC}+I_{\rm OC}+I_{\rm man}+I_{\rm
o+c1+c2}=\int^{R_{\oplus}}_{0}\dfrac{8\pi}{3}\rho_{\rm E}(r)r^{4}dr$
$\displaystyle=\dfrac{8\pi}{3}\left[\int^{R_{\rm IC}}_{0}\rho_{\rm
IC}(r)r^{4}dr+\int^{R_{\rm C}}_{R_{\rm IC}}\rho_{\rm
OC}(r)r^{4}dr+\int^{R_{\rm man}}_{R_{\rm C}}\rho_{\rm
man}(r)r^{4}dr+\int^{R_{\oplus}}_{R_{\rm man}}\rho_{\rm
o+c1+c2}(r)r^{4}dr\right]\,.$ (7)

It follows from the seismological and geophysical data that in order for Earth
to be in hydrostatic equilibrium the following inequalities should always
hold:

$\rho_{man}<\rho_{OC}<\rho_{IC}\,.$ (8)

We will refer to these inequalities as Earth hydrostatic equilibrium (EHE)
constraint. The EHE constraint should also be implemented in the studies of
the Earth density structure using the method of neutrino tomography.

In our further analysis we will use the the IC, OC, mantle and the “surface”
layers density distributions given by PREM. In PREM, the mantle is represented
by five layers with different distributions in each layer. In addition, as we
have already indicated, there are three thin layers with constant density
close to the Earth’s surface: one “ocean” and two crust “surface” layers.

In PREM the IC and OC density distributions have the form:

$\displaystyle\rho_{\rm
IC}(r)=13.0885-8.8381\dfrac{r^{2}}{R^{2}_{\oplus}}\,,~{}~{}r=(0-1221.5)~{}{\rm
km}\,;$ (9) $\displaystyle\rho_{\rm
OC}(r)=12.5815-1.2638\dfrac{r}{R_{\oplus}}-3.6426\dfrac{r^{2}}{R^{2}_{\oplus}}-5.5281\dfrac{r^{3}}{R^{3}_{\oplus}}\,,~{}r=(1221.5-3480.0)~{}{\rm
km}\,,$ (10)

where $\rho_{\rm IC}(r)$ and $\rho_{\rm OC}(r)$ are in units of g/cm3 and we
have given also for convenience the “widths” of the IC and OC layers.
Similarly, the densities (in units of g/cm3) and the widths of the ocean, the
two crust and the five mantle layers read [10]:

$\displaystyle\rho_{\rm ocean}(r)=1.020\,,~{}r=(6368-6371)~{}{\rm km}\,;$ (11)
$\displaystyle\rho_{\rm crust1}(r)=2.600\,,~{}r=(6356-6368)~{}{\rm km}\,;$
$\displaystyle\rho_{\rm crust2}(r)=2.900\,,~{}r=(6346.6-6356.0)~{}{\rm km}\,;$
(12) $\displaystyle\rho_{\rm
m1}(r)=2.6910+0.6924\dfrac{r}{R_{\oplus}}\,,~{}r=(6151-6346.6)~{}{\rm km}\,;$
$\displaystyle\rho_{\rm
m2}(r)=7.1089-3.8045\dfrac{r}{R_{\oplus}}\,,~{}r=(5971-6151)~{}{\rm km}\,;$
$\displaystyle\rho_{\rm
m3}(r)=11.2494-8.0298\dfrac{r}{R_{\oplus}}\,,~{}r=(5771-5971)~{}{\rm km}\,;$
$\displaystyle\rho_{\rm
m4}(r)=5.3197-1.4836\dfrac{r}{R_{\oplus}}\,,~{}r=(5701-5771)~{}{\rm km}\,;$
$\displaystyle\rho_{\rm
m5}(r)=7.9565-6.4761\dfrac{r}{R_{\oplus}}+5.5283\dfrac{r^{2}}{R^{2}_{\oplus}}-3.0807\dfrac{r^{3}}{R^{3}_{\oplus}}\,,~{}r=(3480-5701)~{}{\rm
km}\,.$ (13)

Using Eqs. (5) and (7) - (13), it is not difficult to obtain the contributions
of the IC, OC, the mantle and “surface” layers in $M_{\oplus}$ and
$I_{\oplus}$. For the contributions of the IC and OC layers, $M_{\rm IC}$,
$I_{\rm IC}$ and $M_{\rm OC}$, $I_{\rm OC}$, we get:

$\displaystyle M_{\rm IC}=\dfrac{4\pi}{3}\,\bar{\rho}_{\rm IC}\,R^{3}_{\rm
IC}\,,$ (14) $\displaystyle I_{\rm
IC}=\dfrac{2}{5}\,\dfrac{4\pi}{3}\,\bar{\rho}^{\rm I}_{\rm IC}\,R^{5}_{\rm
IC}=\dfrac{2}{5}\,\dfrac{\bar{\rho}^{\rm I}_{\rm IC}}{\bar{\rho}_{\rm
IC}}\,M_{\rm IC}\,R^{2}_{\rm IC}\,,$ (15) $\displaystyle M_{\rm
OC}=\dfrac{4\pi}{3}\,\bar{\rho}_{\rm OC}\,\left(R^{3}_{\rm C}-R^{3}_{\rm
IC}\right)\,,~{}~{}R_{\rm C}=R_{\rm OC}\,,$ (16) $\displaystyle I_{\rm
OC}=\dfrac{2}{5}\,\dfrac{4\pi}{3}\,\bar{\rho}^{\rm I}_{\rm OC}\left(R^{5}_{\rm
C}-R^{5}_{\rm IC}\right)=\dfrac{2}{5}\,\dfrac{\bar{\rho}^{\rm I}_{\rm
OC}}{\bar{\rho}_{\rm OC}}\,M_{\rm OC}\,\dfrac{R^{5}_{\rm C}-R^{5}_{\rm
IC}}{R^{3}_{\rm C}-R^{3}_{\rm IC}}\,,$ (17)

where $\bar{\rho}_{\rm IC}$, $\bar{\rho}^{\rm I}_{\rm IC}$, $\bar{\rho}_{\rm
OC}$ and $\bar{\rho}^{\rm I}_{\rm OC}$ are average densities,

$\displaystyle\bar{\rho}_{\rm IC}=\dfrac{4\pi\int^{R_{\rm IC}}_{0}\rho_{\rm
IC}(r)r^{2}dr}{4\pi\int^{R_{\rm
IC}}_{0}r^{2}dr}=13.0885-8.8381\,\dfrac{3R^{2}_{\rm IO}}{5R^{2}_{\oplus}}\,,$
(18) $\displaystyle\bar{\rho}^{\rm I}_{\rm
IC}=\dfrac{\frac{8\pi}{3}\int^{R_{\rm IC}}_{0}\rho_{\rm
IC}(r)r^{4}dr}{\frac{8\pi}{3}\int^{R_{\rm
IC}}_{0}r^{4}dr}=13.0885-8.8381\,\dfrac{5R^{2}_{\rm IC}}{7R^{2}_{\oplus}}\,,$
(19) $\displaystyle\bar{\rho}_{\rm OC}=\dfrac{4\pi\int^{R_{\rm C}}_{R_{\rm
IC}}\rho_{\rm OC}(r)r^{2}dr}{4\pi\int^{R_{\rm C}}_{R_{\rm IC}}r^{2}dr}$
$\displaystyle=12.5815-1.2638\,\dfrac{3(R^{4}_{\rm OC}-R^{4}_{\rm
IO})}{4R_{\oplus}(R^{3}_{\rm OC}-R^{3}_{\rm IO})}-3.6426\,\dfrac{3(R^{5}_{\rm
OC}-R^{5}_{\rm IO})}{5R^{2}_{\oplus}(R^{3}_{\rm OC}-R^{3}_{\rm
IO})}-5.5281\,\dfrac{3(R^{6}_{\rm OC}-R^{6}_{\rm
IO})}{6R^{3}_{\oplus}(R^{3}_{\rm OC}-R^{3}_{\rm IO})}\,,$ (20)
$\displaystyle\bar{\rho}^{\rm I}_{\rm OC}=\dfrac{\frac{8\pi}{3}\int^{R_{\rm
C}}_{R_{\rm IC}}\rho_{\rm OC}(r)r^{4}dr}{\frac{8\pi}{3}\int^{R_{\rm
C}}_{R_{\rm IC}}r^{4}dr}$ $\displaystyle=12.5815-1.2638\,\dfrac{5(R^{6}_{\rm
OC}-R^{6}_{\rm IO})}{6R_{\oplus}(R^{5}_{\rm OC}-R^{5}_{\rm
IO})}-3.6426\,\dfrac{5(R^{7}_{\rm OC}-R^{7}_{\rm
IO})}{7R^{2}_{\oplus}(R^{5}_{\rm OC}-R^{5}_{\rm
IO})}-5.5281\,\dfrac{5(R^{8}_{\rm OC}-R^{8}_{\rm
IO})}{8R^{3}_{\oplus}(R^{5}_{\rm OC}-R^{5}_{\rm IO})}\,.$ (21)

Using the values of $R_{\rm IC}$, $R_{\rm OC}$ and $R_{\oplus}$ we find:

$\displaystyle\bar{\rho}_{\rm
IC}=12.8936\,\dfrac{g}{cm^{3}}\,,~{}~{}\bar{\rho}^{\rm I}_{\rm
IC}=12.8564\,\dfrac{g}{cm^{3}}=0.9971\,\bar{\rho}_{\rm IC}\,,$ (22)
$\displaystyle\bar{\rho}_{\rm
OC}=10.90070\,\dfrac{g}{cm^{3}}\,,~{}~{}\bar{\rho}^{\rm I}_{\rm
OC}=10.6583\,\dfrac{g}{cm^{3}}=0.9778\,\bar{\rho}_{\rm OC}\,.$ (23)

Note that $\bar{\rho}^{\rm I}_{\rm IC}$ and $\bar{\rho}^{\rm I}_{\rm OC}$
differ respectively from $\bar{\rho}_{\rm IC}$ and $\bar{\rho}_{\rm OC}$,
being somewhat smaller.

We can consider also the the total mass and moment of inertia of the Earth
core:

$\displaystyle M_{\rm C}=M_{\rm IC}+M_{\rm
OC}=\dfrac{4\pi}{3}\,\bar{\rho}_{\rm C}\,R^{3}_{\rm C}\,,$ (24) $\displaystyle
I_{\rm C}=I_{\rm IC}+I_{\rm OC}=\dfrac{2}{5}\,\dfrac{4\pi}{3}\,\bar{\rho}^{\rm
I}_{\rm C}\,R^{5}_{\rm C}=\dfrac{2}{5}\,\dfrac{\bar{\rho}^{\rm I}_{\rm
C}}{\bar{\rho}_{\rm C}}\,M_{\rm C}\,R^{2}_{\rm C}\,,$ (25)

where

$\displaystyle\bar{\rho}_{\rm C}=\bar{\rho}_{\rm IC}\,\dfrac{R^{3}_{\rm
IC}}{R^{3}_{\rm C}}+\bar{\rho}_{\rm OC}\,\left(1-\dfrac{R^{3}_{\rm
IC}}{R^{3}_{\rm C}}\right)=10.9869\,\dfrac{g}{cm^{3}}\,,$ (26)
$\displaystyle\bar{\rho}^{\rm I}_{\rm C}=\bar{\rho}^{\rm I}_{\rm
IC}\,\dfrac{R^{5}_{\rm IC}}{R^{5}_{\rm C}}+\bar{\rho}^{\rm I}_{\rm
OC}\,\left(1-\dfrac{R^{5}_{\rm IC}}{R^{5}_{\rm
C}}\right)=10.6700\,\dfrac{g}{cm^{3}}=0.9712\,\bar{\rho}_{\rm C}\,.$ (27)

It is not difficult to calculate in a similar way the contributions of the
five mantle layers to $M_{\oplus}$ and $I_{\oplus}$. They can be written in
the form:

$\displaystyle M_{\rm man}=\dfrac{4\pi}{3}\,\bar{\rho}_{\rm
man}\,\left(R^{3}_{\rm man}-R^{3}_{\rm C}\right)\,,~{}\bar{\rho}_{\rm
man}=\dfrac{4\pi\int^{R_{\rm man}}_{R_{\rm C}}\rho_{\rm
man}(r)r^{2}dr}{4\pi\int^{R_{\rm man}}_{R_{\rm C}}r^{2}dr}$ (28)
$\displaystyle I_{\rm man}=\dfrac{8\pi}{15}\,\bar{\rho}^{\rm I}_{\rm
man}\,\left(R^{5}_{\rm man}-R^{5}_{\rm
C}\right)=\dfrac{2}{5}\,\dfrac{\bar{\rho}^{\rm I}_{\rm man}}{\bar{\rho}_{\rm
man}}\,M_{\rm man}\,\dfrac{R^{5}_{\rm man}-R^{5}_{\rm C}}{R^{3}_{\rm
man}-R^{3}_{\rm C}}\,,~{}\bar{\rho}^{\rm I}_{\rm
man}=\dfrac{\dfrac{8\pi}{3}\int^{R_{\rm man}}_{R_{\rm C}}\rho_{\rm
man}(r)r^{4}dr}{\dfrac{8\pi}{3}\int^{R_{\rm man}}_{R_{\rm C}}r^{4}dr}\,.$ (29)

For the average densities $\bar{\rho}_{\rm man}$ and $\bar{\rho}^{\rm I}_{\rm
man}$ we get:

$\bar{\rho}_{\rm man}=4.6651\,\dfrac{g}{cm^{3}}\,,~{}~{}\bar{\rho}^{\rm
I}_{\rm man}=4.2833\,\dfrac{g}{cm^{3}}=0.9182\,\bar{\rho}_{\rm man}\,.$ (30)

We remark that if we include the “surface” layers in the mantle, integrating
from $R_{C}=3480$ km to $R_{\oplus}=6371$ km we obtain: $\bar{\rho}_{\rm
man}=4.4551\,{\rm g/cm^{3}}$, $\bar{\rho}^{\rm I}_{\rm man}=4.2481\,{\rm
g/cm^{3}}=0.9535\,\bar{\rho}_{\rm man}$.

All results derived above are within the PREM framework. For the Earth total
mass and moment on inertia in this framework we have:
$M_{\oplus}=M_{o+c1+c2}+M_{\rm man}+M_{\rm C}$ and
$I_{\oplus}=I_{o+c1+c2}+I_{\rm man}+I_{\rm C}$, where $M_{\rm C}=M_{\rm
IC}+M_{\rm OC}$, $I_{\rm C}=I_{\rm IC}+I_{\rm OC}$ and $M_{o+c1+c2}$ and
$I_{o+c1+c2}$ are the contributions from the ocean and the two crust “surface”
layers. Using the PREM results shown in Eqs. (9) - (13), we find
$M_{o+c1+c2}\ll M_{\rm man}+M_{\rm C}$ and $I_{o+c1+c2}\ll I_{\rm man}+I_{\rm
C}$: $M_{o+c1+c2}/(M_{\rm man}+M_{\rm C})\cong 5\times 10^{-3}$ and
$I_{o+c1+c2}/(I_{\rm man}+I_{\rm C})\cong 0.01060$.

We note that the “surface” layers play subleading (if not negligible) role in
neutrino tomography of the Earth due to their relatively small widths and
contributions to $M_{\oplus}$ and $I_{\oplus}$. As far as $M_{o+c1+c2}$ and
$I_{o+c1+c2}$ are kept constant, they do not play any role in our analysis and
the results we will obtain are independent of the values of $M_{o+c1+c2}$ and
$I_{o+c1+c2}$, as it will become clear further. We will keep them for
consistency. For this purpose we introduce their average densities
$\bar{\rho}_{\rm oc1c2}$ and $\bar{\rho}^{\rm I}_{\rm oc1c2}$:

$\displaystyle M_{\rm o+c1+c2}=\dfrac{4\pi}{3}\,\bar{\rho}_{\rm
oc1c2}\,\left(R^{3}_{\oplus}-R^{3}_{\rm man}\right)\,,$
$\displaystyle\bar{\rho}_{\rm
oc1c2}=\dfrac{4\pi\left[\int^{R_{\oplus}}_{R_{\rm ocean}}\rho_{\rm
ocean}(r)r^{2}dr+\int^{R_{\rm ocean}}_{R_{\rm crust1}}\rho_{\rm
crust1}(r)r^{2}dr+\int^{R_{\rm crust1}}_{R_{\rm crust2}}\rho_{\rm
crust2}(r)r^{2}dr\right]}{4\pi\int^{R_{\oplus}}_{R_{\rm man}}r^{2}dr}\,,$ (31)
$\displaystyle I_{\rm o+c1+c2}=\dfrac{8\pi}{15}\,\bar{\rho}^{\rm I}_{\rm
oc1c2}\,\left(R^{5}_{\oplus}-R^{5}_{\rm man}\right)\,,$
$\displaystyle\bar{\rho}^{\rm I}_{\rm
oc1c2}=\dfrac{\dfrac{8\pi}{3}\left[\int^{R_{\oplus}}_{R_{\rm ocean}}\rho_{\rm
ocean}(r)r^{4}dr+\int^{R_{\rm ocean}}_{R_{\rm crust1}}\rho_{\rm
crust1}(r)r^{4}dr+\int^{R_{\rm crust1}}_{R_{\rm crust2}}\rho_{\rm
crust2}(r)r^{4}dr\right]}{\dfrac{8\pi}{3}\int^{R_{\oplus}}_{R_{\rm
man}}r^{4}dr}\,,$ (32)

where $R_{\rm ocean}=6368$ km, $R_{\rm crust1}=6356$ km and $R_{\rm
crust2}\equiv R_{\rm man}=6346.6$ km. For $\bar{\rho}_{\rm oc1c2}$ and
$\bar{\rho}^{\rm I}_{\rm oc1c2}$ we get:

$\bar{\rho}_{\rm oc1c2}=2.5204\,\dfrac{g}{cm^{3}}\,,\quad\bar{\rho}^{\rm
I}_{\rm oc1c2}=2.5194\,\dfrac{g}{cm^{3}}=0.9996\,\bar{\rho}_{\rm oc1c2}\,.$
(33)

Summarising the results obtained so far we have:

$\displaystyle M_{\oplus}=M_{\rm man}+M_{\rm OC}+M_{\rm IC}+M_{o+c1+c2}$
$\displaystyle=\dfrac{4\pi}{3}\,R^{3}_{\oplus}\left[\bar{\rho}_{\rm
man}\,\dfrac{R^{3}_{\rm man}}{R^{3}_{\oplus}}\left(1-\dfrac{R^{3}_{\rm
C}}{R^{3}_{\rm man}}\right)+\bar{\rho}_{\rm OC}\,\dfrac{R^{3}_{\rm
C}}{R^{3}_{\oplus}}\,\left(1-\dfrac{R^{3}_{\rm IC}}{R^{3}_{\rm
OC}}\right)+\bar{\rho}_{\rm IC}\,\dfrac{R^{3}_{\rm
IC}}{R^{3}_{\oplus}}+\bar{\rho}_{\rm oc1c2}\,\left(1-\dfrac{R^{3}_{\rm
man}}{R^{3}_{\oplus}}\right)\right]\,,$ (34) $\displaystyle I_{\oplus}=I_{\rm
man}+I_{\rm OC}+I_{\rm IC}+I_{o+c1+c2}$
$\displaystyle=\dfrac{2}{5}\,\dfrac{4\pi}{3}\,R^{5}_{\oplus}\left[\bar{\rho}^{\rm
I}_{\rm man}\,\dfrac{R^{5}_{\rm man}}{R^{5}_{\oplus}}\left(1-\dfrac{R^{5}_{\rm
C}}{R^{5}_{\rm man}}\right)+\bar{\rho}^{\rm I}_{\rm OC}\dfrac{R^{5}_{\rm
C}}{R^{5}_{\oplus}}\left(1-\dfrac{R^{5}_{\rm IC}}{R^{5}_{\rm
OC}}\right)+\bar{\rho}^{\rm I}_{\rm IC}\dfrac{R^{5}_{\rm
IC}}{R^{5}_{\oplus}}+\bar{\rho}^{\rm I}_{\rm oc1c2}\left(1-\dfrac{R^{5}_{\rm
man}}{R^{5}_{\oplus}}\right)\,\right]\,.$ (35)

In terms of mantle and total core densities the expressions for $M_{\oplus}$
and $I_{\oplus}$ read:

$\displaystyle M_{\oplus}=M_{\rm man}+M_{\rm C}+M_{o+c1+c2}$
$\displaystyle=\dfrac{4\pi}{3}\,R^{3}_{\oplus}\left[\bar{\rho}_{\rm
man}\,\dfrac{R^{3}_{\rm man}}{R^{3}_{\oplus}}\left(1-\dfrac{R^{3}_{\rm
C}}{R^{3}_{\rm man}}\right)+\bar{\rho}_{\rm C}\,\dfrac{R^{3}_{\rm
C}}{R^{3}_{\oplus}}+\bar{\rho}_{\rm oc1c2}\left(1-\dfrac{R^{3}_{\rm
man}}{R^{3}_{\oplus}}\right)\right]\,,$ (36) $\displaystyle I_{\oplus}=I_{\rm
man}+I_{\rm C}+I_{o+c1+c2}$
$\displaystyle=\dfrac{2}{5}\,\dfrac{4\pi}{3}\,R^{5}_{\oplus}\left[\bar{\rho}^{\rm
I}_{\rm man}\,\dfrac{R^{5}_{\rm man}}{R^{5}_{\oplus}}\left(1-\dfrac{R^{5}_{\rm
C}}{R^{5}_{\rm man}}\right)+\bar{\rho}^{\rm I}_{\rm C}\dfrac{R^{5}_{\rm
C}}{R^{5}_{\oplus}}+\bar{\rho}^{\rm I}_{\rm oc1c2}\left(1-\dfrac{R^{5}_{\rm
man}}{R^{5}_{\oplus}}\right)\right]\,,$ (37)

where we have used Eqs. (24) - (27) and (31) - (32).

It proves convenient to express the constants $\bar{\rho}^{({\rm I})}_{i}$ in
Eqs. (35) and (37), $i={\rm IC,OC,C,man,oc1c2}$, as $\bar{\rho}^{({\rm
I})}_{i}=r_{i}\bar{\rho}_{i}$. For the ratios $r_{i}$ we have:

$\displaystyle r_{\rm man}=\dfrac{\bar{\rho}^{\rm I}_{\rm
man}}{\bar{\rho}_{\rm man}}=0.9182\,,r_{\rm OC}=\dfrac{\bar{\rho}^{\rm I}_{\rm
OC}}{\bar{\rho}_{\rm OC}}=0.9778\,,~{}r_{\rm IC}=\dfrac{\bar{\rho}^{\rm
I}_{\rm IC}}{\bar{\rho}_{\rm IC}}=0.9971\,,~{}$ $\displaystyle r_{\rm
C}=\dfrac{\bar{\rho}^{\rm I}_{\rm C}}{\bar{\rho}_{\rm C}}=0.9712\,,~{}r_{\rm
oc1c2}=\dfrac{\bar{\rho}^{\rm I}_{\rm oc1c2}}{\bar{\rho}_{\rm
oc1c2}}=0.9996\,.$ (38)

The values of these ratios are given in Eqs. (22), (23), (27), (30) and (33).

It also proves convenient to introduce the average Earth density:

$\displaystyle\bar{\rho}_{\oplus}\equiv\dfrac{3}{4\pi}\dfrac{M_{\oplus}}{R^{3}_{\oplus}}=a_{\rm
man}\,\bar{\rho}_{\rm man}+a_{\rm OC}\,\bar{\rho}_{\rm OC}+a_{\rm
IC}\,\bar{\rho}_{\rm IC}+a_{\rm oc1c2}\,\bar{\rho}_{\rm oc1c2}$ (39)
$\displaystyle=a_{\rm man}\,\bar{\rho}_{\rm man}+a_{\rm C}\,\bar{\rho}_{\rm
C}+a_{\rm oc1c2}\,\bar{\rho}_{\rm oc1c2}\,.$ (40)

where

$\displaystyle a_{\rm man}=\dfrac{R^{3}_{\rm
man}}{R^{3}_{\oplus}}\left(1-\dfrac{R^{3}_{\rm C}}{R^{3}_{\rm
man}}\right)\cong 0.8256\,,~{}a_{\rm OC}=\dfrac{R^{3}_{\rm
C}}{R^{3}_{\oplus}}\left(1-\dfrac{R^{3}_{\rm IC}}{R^{3}_{\rm C}}\right)\cong
0.15592\,,~{}a_{\rm C}=\dfrac{R^{3}_{\rm C}}{R^{3}_{\oplus}}\cong 0.1639\,,$
$\displaystyle a_{\rm IC}=\dfrac{R^{3}_{\rm IC}}{R^{3}_{\oplus}}\cong
7.0479\times 10^{-3}\,.~{}a_{\rm oc1c2}=\left(1-\dfrac{R^{3}_{\rm
man}}{R^{3}_{\oplus}}\right)\cong 0.0114\,.$ (41)

Replacing $\bar{\rho}^{({\rm I})}_{i}$ with $r_{i}\bar{\rho}_{i}$ in Eqs. (35)
and (37), $i={\rm IC,OC,C,man,oc1c2}$, and using Eq. (39) we get
$I_{\oplus}=C\,M_{\oplus}\,R^{2}_{\oplus}$, which implies:

$\displaystyle\dfrac{I_{\oplus}}{R^{5}_{\oplus}}\dfrac{5}{2}\,\dfrac{3}{4\pi}=C^{\prime}\,\bar{\rho}_{\oplus}=b_{\rm
man}\,\bar{\rho}_{\rm man}+b_{\rm OC}\,\bar{\rho}_{\rm OC}+b_{\rm
IC}\,\bar{\rho}_{\rm IC}+b_{\rm oc1c2}\,\bar{\rho}_{\rm oc1c2}$ (42)
$\displaystyle=b_{\rm man}\,\bar{\rho}_{\rm man}+b_{\rm C}\,\bar{\rho}_{\rm
C}+b_{\rm oc1c2}\,\bar{\rho}_{\rm oc1c2}\,$ (43)

where $C^{\prime}$ is a constant, $C^{\prime}=5C/2\cong 0.8270$ and

$\displaystyle b_{\rm man}=r_{\rm man}\dfrac{R^{5}_{\rm
man}}{R^{5}_{\oplus}}\left(1-\dfrac{R^{5}_{\rm C}}{R^{3}_{\rm
man}}\right)\,,~{}b_{\rm OC}=r_{\rm OC}\,\dfrac{R^{5}_{\rm
C}}{R^{5}_{\oplus}}\left(1-\dfrac{R^{5}_{\rm IC}}{R^{5}_{\rm
C}}\right)\,,~{}b_{\rm C}=r_{\rm C}\,\dfrac{R^{5}_{\rm C}}{R^{5}_{\oplus}}\,,$
$\displaystyle b_{\rm IC}=r_{\rm IC}\dfrac{R^{5}_{\rm
IC}}{R^{5}_{\oplus}}\,,~{}~{}b_{\rm oc1c2}=r_{\rm
oc1c2}\,\left(1-\dfrac{R^{5}_{\rm man}}{R^{5}_{\oplus}}\right)\,.$ (44)

The constant parameters $r_{\rm man}$, $r_{\rm OC}$, $r_{\rm IC}$ and $r_{\rm
oc1c2}$ are defined and their values are given in Eq. (38). They are all close
to, but different from, 1. The numerical values of $b_{\rm man}$, $b_{\rm
OC}$, $b_{\rm C}$, $b_{\rm IC}$ and $b_{\rm oc1c2}$ read:

$b_{\rm man}\cong 0.8561\,,~{}b_{\rm OC}\cong 0.0473\,,~{}b_{\rm C}\cong
0.0472\,,~{}b_{\rm IC}\cong 2.58\times 10^{-4}\,,~{}b_{\rm oc1c2}\cong
0.0190\,.$ (45)

The left-hand sides of Eqs. (39), (40), (42) and (43) are fixed since
$M_{\oplus}$, $I_{\oplus}$ and $R_{\oplus}$ are known with relatively high
precision. Thus, within the aspects of neutrino tomography of the Earth we are
considering, Eqs. (39), (40) and (42), (43) represent the constraints on the
variations of $\bar{\rho}_{\rm man}$, and $\bar{\rho}_{\rm OC}$,
$\bar{\rho}_{\rm IC}$ or $\bar{\rho}_{\rm C}$, from the knowledge of
$M_{\oplus}$ and $I_{\oplus}$.

We will assume that $R_{\rm man}$, $R_{\rm OC}$ and $R_{\rm IC}$ and,
correspondingly, the values of the constants $r_{\rm man}$, $r_{\rm OC}$ and
$r_{\rm IC}$ are correctly given by PREM. These are standard assumptions made
in the neutrino tomography of the Earth studies of possible deviations of
$\bar{\rho}_{\rm man}$, $\bar{\rho}_{\rm OC}$ and $\bar{\rho}_{\rm IC}$ from
their PREM values. In our further analyses we will consider the “surface”
terms in Eqs. (39) and (42), $a_{\rm oc1c2}\,\bar{\rho}_{\rm oc1c2}$ and
$b_{\rm oc1c2}\,\bar{\rho}_{\rm oc1c2}$, as fixed. Under these conditions,
Eqs. (39) and (42) represent two equations for the three densities of
interest: $\bar{\rho}_{\rm man}$, $\bar{\rho}_{\rm OC}$ and $\bar{\rho}_{\rm
IC}$.

##  3 Implementing the $M_{\oplus}$, $I_{\oplus}$ and EHE Constraints

###  3.1  The Mantle, Outer Core and Inner Core Case

Suppose one is interested in obtaining information about possible deviations
of the densities of the mantle $\rho_{\rm man}$, the outer core, $\rho_{\rm
OC}$, and the inner core, $\rho_{\rm IC}$ from their PREM values. In the
corresponding statistical analysis one varies the density in the chosen layer
of interest, say $\rho_{\rm OC}$. In order to satisfy the $M_{\oplus}$ and
$I_{\oplus}$ constraints one has to compensate the variation of $\rho_{\rm
OC}$ by an appropriate change of the densities in the other two layers,
$\rho_{\rm man}$ and $\rho_{\rm IC}$ In the analyses performed so far it is
assumed that the indicated variations and changes of interest are by constant
$r$-independent factors $\kappa_{i}$, which can be positive, negative or zero:

$\rho^{\prime}_{i}=(1+\kappa_{i})\,\rho_{i}\,,~{}i=man,OC,IC\,.$ (46)

We will analyse this possibility below.

The total Earth mass and momentum of inertia constraints in the simple case we
are considering of average densities imply that $\bar{\rho}^{\prime}_{i}$
should satisfy Eqs. (39) and (42) and the following two equations:

$\displaystyle a_{\rm man}\,(1+\kappa_{\rm man})\bar{\rho}_{\rm man}+a_{\rm
OC}\,(1+\kappa_{\rm OC})\bar{\rho}_{\rm OC}+a_{\rm IC}\,(1+\kappa_{\rm
IC})\bar{\rho}_{\rm IC}+a_{\rm oc1c2}\,\bar{\rho}_{\rm
oc1c2}=\bar{\rho}_{\oplus}\,,$ $\displaystyle b_{\rm man}\,(1+\kappa_{\rm
man})\bar{\rho}_{\rm man}+b_{\rm OC}\,(1+\kappa_{\rm OC})\bar{\rho}_{\rm
OC}+b_{\rm IC}\,(1+\kappa_{\rm IC})\bar{\rho}_{\rm IC}+b_{\rm
oc1c2}\,\bar{\rho}_{\rm oc1c2}=C^{\prime}\,\bar{\rho}_{\oplus}\,.$ (47)

Combining these equations with Eqs. (39) and (42) and taking into account that
the terms $a_{\rm oc1c2}\,\bar{\rho}_{\rm oc1c2}$ and $b_{\rm
oc1c2}\,\bar{\rho}_{\rm oc1c2}$ are fixed, we get neglecting the uncertainties
in the determination of $M_{\oplus}$ and $I_{\oplus}$:

$\displaystyle a_{\rm man}\,\kappa_{\rm man}\bar{\rho}_{\rm man}+a_{\rm
OC}\,\kappa_{\rm OC}\bar{\rho}_{\rm OC}+a_{\rm IC}\,\kappa_{\rm
IC}\bar{\rho}_{\rm IC}=0\,,$ $\displaystyle b_{\rm man}\,\kappa_{\rm
man}\bar{\rho}_{\rm man}+b_{\rm OC}\,\kappa_{\rm OC}\bar{\rho}_{\rm OC}+b_{\rm
IC}\,\kappa_{\rm IC}\bar{\rho}_{\rm IC}=0\,.$ (48)

Suppose we vary $\rho_{\rm OC}$ and are interested in the sensitivity of a
given detector to possible deviations of $\rho^{\prime}_{\rm OC}$ from the
PREM prediction, i.e., to values of $\kappa_{\rm OC}\neq 0$. In the simple
case of average densities we are considering we can express $\kappa_{\rm man}$
and $\kappa_{\rm IC}$ in terms of $\kappa_{\rm OC}$ using Eq. (48):

$\displaystyle\kappa_{\rm man}=\dfrac{\kappa_{\rm
OC}}{D_{OC}}\,\left(a^{\prime}_{\rm IC}\,b_{\rm OC}-a_{\rm
OC}\,b^{\prime}_{\rm IC}\right)\,,$ $\displaystyle\kappa_{\rm
IC}=\dfrac{\kappa_{\rm OC}}{D_{OC}}\,\left(a_{\rm OC}\,b^{\prime}_{\rm
man}-a^{\prime}_{\rm man}\,b_{\rm OC}\right)\,.$ (49)

Here

$D_{OC}\equiv a^{\prime}_{\rm man}\,b^{\prime}_{\rm IC}-\,a^{\prime}_{\rm
IC}\,b^{\prime}_{\rm man}\,,$ (50)

and

$a^{\prime}_{j}\equiv\dfrac{\bar{\rho}_{j}}{\bar{\rho}_{\rm
OC}}\,a_{j}\,,~{}~{}b^{\prime}_{j}\equiv\dfrac{\bar{\rho}_{j}}{\bar{\rho}_{\rm
OC}}\,b_{j}\,,~{}~{}j=man,IC\,.$ (51)

It follows, in particular, from Eq. (49) that in order to satisfy the Earth
total mass and moment of inertia constraints, the variations of
$\bar{\rho}_{\rm OC}$ have to be compensated by changes of the densities of at
least two other Earth layers. Using the PREM values of $\bar{\rho}_{\rm IC}$,
$\bar{\rho}_{\rm OC}$ and $\bar{\rho}_{\rm man}$ given in Eqs. (22), (23) and
(30), as well as the values of $a_{i}$ (in Eq. (41)) and $b_{i}$ (in Eq. (44))
we find:

$\kappa_{\rm man}\cong-\,0.1104\,\kappa_{\rm OC}\,,~{}~{}\kappa_{\rm
IC}\cong-\,13.9558\,\kappa_{\rm OC}\,.$ (52)

In the case of $\kappa_{\rm OC}>0$ the relevant Earth hydrostatic equilibrium
(EHE) constraint is $\bar{\rho}_{\rm OC}(1+\kappa_{\rm OC})<\bar{\rho}_{\rm
IC}(1+\kappa_{\rm IC})$, with $\kappa_{\rm IC}<0$. Using the relation between
$\kappa_{\rm IC}$ and $\kappa_{\rm OC}$ given in Eq. (52), we find:

$\kappa_{\rm OC}<\dfrac{\bar{\rho}_{\rm IC}-\bar{\rho}_{\rm
OC}}{\bar{\rho}_{\rm OC}+13.9558\bar{\rho}_{\rm IC}}\cong 0.0104\,,$ (53)

where we have used the PREM values of $\bar{\rho}_{\rm IC}$ and
$\bar{\rho}_{\rm OC}$ (see Eq. (22)).

For $\kappa_{\rm OC}<0$ the relevant EHE constraint is instead
$\bar{\rho}_{\rm man}(1+\kappa_{\rm man})<\bar{\rho}_{\rm OC}(1-|\kappa_{\rm
OC}|)$. This inequality together with the relation between $\kappa_{\rm man}$
and $\kappa_{\rm OC}$ we have derived and the PREM values of $\bar{\rho}_{\rm
man}$ and $\bar{\rho}_{\rm OC}$ leads to the following lower limit:

$\kappa_{\rm OC}>-\,\dfrac{\bar{\rho}_{\rm OC}-\bar{\rho}_{\rm
man}}{\bar{\rho}_{\rm OC}+0.1104\bar{\rho}_{\rm
man}}\cong-\,0.546\,,~{}~{}\kappa_{\rm OC}<0\,.$ (54)

However, the obtained lower limit on $\kappa_{\rm OC}<0$ allows excessively
large values of $\kappa_{\rm IC}>0$ and thus of $\bar{\rho}_{\rm
IC}(1+\kappa_{\rm IC})$, as large as, e.g., 7.6 times bigger than the PREM
value of $\bar{\rho}_{\rm IC}$. The existing seismological data on
$\bar{\rho}_{\rm IC}$ is claimed to have an uncertainty not exceeding $\sim
10\%$. Even if we allow a rather large maximum deviation of $\bar{\rho}_{\rm
IC}$ from its PREM value of 25% (50%), which corresponds to $\kappa_{\rm
IC}\leq 0.25~{}(0.50)$, this leads to the following lower limit on
$\kappa_{\rm OC}<0$: $\kappa_{\rm OC}\ \raisebox{-1.72218pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.72218pt}{$>$}\ -\,0.018~{}(0.036)$.

Thus, we see that in the considered case of three layer Earth density
distribution, the Earth mass and moment of inertia constraints together with
the requirement of the Earth being in hydrostatic equilibrium and the value of
the inner core average density lead to severe constraints on the possible
uniform deviations of $\bar{\rho}_{\rm OC}$ from its PREM value:

$-\,0.018~{}(0.036)\ \raisebox{-1.72218pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.72218pt}{$<$}\ \kappa_{\rm OC}<0.010\,.$ (55)

These very restrictive constraints can be traced to the relatively small
contributions of the inner core to $M_{\oplus}$ and especially to
$I_{\oplus}$.

We note that we get practically the same results if we consider the “surface”
layers as part of the mantle. In this case the mantle extends from $R_{\rm
C}=3480$ km to $R_{\oplus}=6371$ km (instead of $R_{\rm man}=6346.6$ km).

In this analysis we did not take into account the uncertainties in the
determination of $M_{\oplus}$ and $I_{\oplus}$. As it follows from Eqs. (1)
and (2), the relative $1\sigma$ uncertainties in the values of $M_{\oplus}$
and $I_{\oplus}$ are $\sim 10^{-4}$. These uncertainties are too small to have
any significant effect, if taken into account, on the results we have
obtained.

###  3.2 The case of Two Mantle Layers, Outer Core and Inner Core

We consider next the case of two different mantle layers and the already
discussed outer core, inner core and “surface” layers. The mantle is divided
into i) lower mantle, extending from $R_{c}=3480$ km to $R_{\rm lman}=5701$
km, with densities $\bar{\rho}_{\rm lman}$ and $\bar{\rho}^{\rm I}_{\rm
lman}$, and ii) upper mantle, extending from $R_{\rm lman}=5701$ km to $R_{\rm
man}=6346.6$ km, with densities $\bar{\rho}_{\rm uman}$ and $\bar{\rho}^{\rm
I}_{\rm uman}$. It is not difficult to calculate the PREM values of
$\bar{\rho}_{\rm lman}$, $\bar{\rho}^{\rm I}_{\rm lman}$, $\bar{\rho}_{\rm
uman}$ and $\bar{\rho}^{\rm I}_{\rm uman}$:

$\displaystyle\bar{\rho}_{\rm
lman}=4.9035~{}\dfrac{g}{cm^{3}}\,,~{}~{}\bar{\rho}^{\rm I}_{\rm lman}=r_{\rm
lman}\bar{\rho}_{\rm lman}\,,~{}~{}r_{\rm lman}=0.9771\,,$ (56)
$\displaystyle\bar{\rho}_{\rm
uman}=3.6046~{}\dfrac{g}{cm^{3}}\,,~{}~{}\bar{\rho}^{\rm I}_{\rm man}=r_{\rm
lman}\bar{\rho}_{\rm lman}\,,~{}~{}r_{\rm uman}=0.9425\,.$ (57)

As in the preceding subsection we will assume that that the variations and
changes of $\bar{\rho}_{\rm IC}$, $\bar{\rho}_{\rm OC}$, $\bar{\rho}_{\rm
lman}$ and $\bar{\rho}_{\rm uman}$ are by constant $r$-independent factors
$\kappa_{\rm IC}$, $\kappa_{\rm OC}$, $\kappa_{\rm lman}$ and $\kappa_{\rm
uman}$:

$\rho^{\prime}_{i}=(1+\kappa_{i})\,\rho_{i}\,,~{}i={\rm IC,OC,lman,uman}\,.$
(58)

The analogs of the equations in Eq. (48) now read:

$\displaystyle a_{\rm lman}\,\kappa_{\rm lman}\bar{\rho}_{\rm lman}+a_{\rm
uman}\,\kappa_{\rm uman}\bar{\rho}_{\rm uman}+a_{\rm OC}\,\kappa_{\rm
OC}\bar{\rho}_{\rm OC}+a_{\rm IC}\,\kappa_{\rm IC}\bar{\rho}_{\rm IC}=0\,,$
$\displaystyle b_{\rm lman}\,\kappa_{\rm lman}\bar{\rho}_{\rm lman}+b_{\rm
uman}\,\kappa_{\rm uman}\bar{\rho}_{\rm uman}+b_{\rm OC}\,\kappa_{\rm
OC}\bar{\rho}_{\rm OC}+b_{\rm IC}\,\kappa_{\rm IC}\bar{\rho}_{\rm IC}=0\,.$
(59)

For the new parameters in these equations we find:

$\displaystyle a_{\rm lman}=\dfrac{R^{3}_{\rm
lman}}{R^{3}_{\oplus}}(1-\dfrac{R^{3}_{\rm C}}{R^{3}_{\rm lman}})\cong
0.5535\,,~{}a_{\rm uman}=\dfrac{R^{3}_{\rm
man}}{R^{3}_{\oplus}}\left(1-\dfrac{R^{3}_{\rm lman}}{R^{3}_{\rm
man}}\right)\cong 0.2720\,,$ (60) $\displaystyle b_{\rm lman}=r_{\rm
lman}\dfrac{R^{5}_{\rm lman}}{R^{5}_{\oplus}}\left(1-\dfrac{R^{5}_{\rm
c}}{R^{5}_{\rm lman}}\right)\cong 0.5131\,,~{}b_{\rm uman}=r_{\rm
uman}\dfrac{R^{5}_{\rm man}}{R^{5}_{\oplus}}\left(1-\dfrac{R^{5}_{\rm
lman}}{R^{5}_{\rm man}}\right)\cong 0.3838\,.$ (61)

where we have used the values of $R_{\rm C}$, $R_{\rm lman}$, $R_{\rm man}$,
$R_{\oplus}$, $r_{\rm lman}$ and $r_{\rm uman}$.

Equations (59) represent the constraints following from the knowledge of
$M_{\oplus}$ and $I_{\oplus}$ on the the possible deviations of
$\bar{\rho}_{\rm IC}$, $\bar{\rho}_{\rm OC}$, $\bar{\rho}_{\rm lman}$ and
$\bar{\rho}_{\rm uman}$ from their PREM values.

Suppose we vary $\rho_{\rm OC}$ and are interested in the sensitivity of a
given detector to possible deviations $\kappa_{\rm OC}\neq 0$ of the PREM
prediction for $\rho_{\rm OC}$. Consider the simple case of $\kappa_{\rm
IC}=0$. Working with the average densities and using Eq. (59) we can express
$\kappa_{\rm lman}$ and $\kappa_{\rm uman}$ in terms of $\kappa_{\rm OC}$:

$\displaystyle\kappa_{\rm lman}=\dfrac{\kappa_{\rm
OC}}{D^{(m)}_{OC}}\,\left(a^{\prime}_{\rm uman}\,b_{\rm OC}-a_{\rm
OC}\,b^{\prime}_{\rm uman}\right)\,,$ $\displaystyle\kappa_{\rm
uman}=\dfrac{\kappa_{\rm OC}}{D^{(m)}_{OC}}\,\left(a_{\rm OC}\,b^{\prime}_{\rm
lman}-a^{\prime}_{\rm lman}\,b_{\rm OC}\right)\,.$ (62)

Here

$D^{(m)}_{OC}\equiv a^{\prime}_{\rm lman}\,b^{\prime}_{\rm
uman}-\,a^{\prime}_{\rm uman}\,b^{\prime}_{\rm lman}\,,$ (63)

and

$a^{\prime}_{j}\equiv\dfrac{\bar{\rho}_{j}}{\bar{\rho}_{\rm
OC}}\,a_{j}\,,~{}~{}b^{\prime}_{j}\equiv\dfrac{\bar{\rho}_{j}}{\bar{\rho}_{\rm
OC}}\,b_{j}\,,~{}~{}j=lman,uman\,.$ (64)

Using the PREM values of $\bar{\rho}_{\rm OC}$, $\bar{\rho}_{\rm lman}$ and
$\bar{\rho}_{\rm uman}$ given in Eqs. (23) and (56) and (57), as well as the
values of $a_{i}$ and $b_{i}$, $i=OC,lman,uman$, from Eqs. (41), (45), (60)
and (61), we get:

$\kappa_{\rm lman}=f_{\rm lman}\,\kappa_{\rm OC}\,,~{}f_{\rm
lman}\cong-\,1.4327;~{}~{}~{}\kappa_{\rm uman}=f_{\rm uman}\,\kappa_{\rm
OC}\,,~{}f_{\rm uman}\cong 2.2329.$ (65)

The EHE constraint for $\kappa_{\rm OC}>0$ is $\bar{\rho}_{\rm
OC}(1+\kappa_{\rm OC})<\bar{\rho}_{\rm IC}$ (recall that we set $\kappa_{\rm
IC}=0$). Using the PREM values of $\bar{\rho}_{\rm IC}$ and $\bar{\rho}_{\rm
OC}$ (see Eq. (22)) we find:

$\kappa_{\rm OC}<\dfrac{\bar{\rho}_{\rm IC}-\bar{\rho}_{\rm
OC}}{\bar{\rho}_{\rm OC}}\cong 0.18\,.$ (66)

In the case of $\kappa_{\rm OC}<0$ the EHE constraint reads:
$\bar{\rho}^{\prime}_{\rm man}<\bar{\rho}_{\rm OC}(1-|\kappa_{\rm OC}|)$,
where $\bar{\rho}^{\prime}_{\rm man}$ is the modified mantle density due to
the changes of the lower and upper mantle densities $\bar{\rho}_{\rm
lman}(1+\kappa_{\rm lman})$ and $\bar{\rho}_{\rm uman}(1+\kappa_{\rm uman})$.
It is not difficult to show that

$\bar{\rho}^{\prime}_{\rm man}=\bar{\rho}_{\rm lman}(1+\kappa_{\rm
lman})\,C_{\rm lman}+\bar{\rho}_{\rm uman}(1+\kappa_{\rm uman})\,C_{\rm
uman}\,,$ (67)

where

$C_{\rm lman}=\dfrac{R^{3}_{\rm lman}}{R^{3}_{\rm
man}}\left(1-\dfrac{R^{3}_{\rm C}}{R^{3}_{\rm man}}\right)\cong
0.6705\,,~{}~{}C_{\rm uman}=\dfrac{1-\dfrac{R^{3}_{\rm lman}}{R^{3}_{\rm
man}}}{1-\dfrac{R^{3}_{\rm C}}{R^{3}_{\rm man}}}\cong 0.3295\,,$ (68)

where we have used the values of $R_{\rm lman}$, $R_{\rm man}$ and $R_{\rm
C}$. The EHE constraint in the considered case has the form:

$\kappa_{\rm OC}>-\,\dfrac{\bar{\rho}_{\rm OC}-\bar{\rho}_{\rm lman}\,C_{\rm
lman}-\bar{\rho}_{\rm uman}\,C_{\rm uman}}{\bar{\rho}_{\rm OC}+\bar{\rho}_{\rm
lman}\,C_{\rm lman}\,f_{\rm lman}+\bar{\rho}_{\rm uman}\,C_{\rm uman}\,f_{\rm
uman}}\,,~{}~{}\kappa_{\rm OC}<0\,.$ (69)

Using the values of $\bar{\rho}_{\rm OC}$, $\bar{\rho}_{\rm lman}$,
$\bar{\rho}_{\rm uman}$, $f_{\rm lman}$, $f_{\rm uman}$, $C_{\rm lman}$ and
$C_{\rm uman}$ given in Eqs. (23), (56), (57), (65) and (68) we get:

$\kappa_{\rm OC}>-\,0.73\,.$ (70)

Combining the two limits we have:

$-\,0.73<\kappa_{\rm OC}<0.18\,.$ (71)

Practically the same results are obtained if we include the “surface” layers
in the upper mantle. In this case the “surface” layers and the upper mantle
form one “upper mantle” layer, which extends from $R_{\rm lman}=3701$ km to
$R_{\oplus}=6371$ km (instead of $R_{\rm man}=6346.6$ km). Similarly, the
results change insignificantly if one joins the inner core and the outer core
in one core layer.

Thus, we see that in the case of two mantle, outer and inner core (or one
core) layers, the $M_{\oplus}$, $I_{\oplus}$ and EHE constraints on the
possible variation of $\bar{\rho}_{\rm OC}$ are much less restrictive than in
the case of one mantle, outer and inner core layers, considered in the
preceding subsection. The main reason for this difference is that the
contributions of the inner core to $M_{\oplus}$ and $I_{\oplus}$ are not
crucial for satisfying the $M_{\oplus}$ and $I_{\oplus}$ constraints when
there are two mantle layers, while, being relatively small, they are critical
in the one mantle layer case.

One may consider other layer structures of the Earth density distribution, for
example, inner core, two equal width outer core layers and mantle layer. We
did not performed the numerical analysis in this case but expect the
$M_{\oplus}$, $I_{\oplus}$ and EHE constraints to be relevant but much less
restrictive than in the case of the three layer case we have considered.

##  4 Summary and Conclusions

Summarising, we have shown that the impact of the Earth total mass,
$M_{\oplus}$, moment of inertia, $I_{\oplus}$, and the Earth hydrostatic
equilibrium (EHE) constraints in the neutrino tomography studies of the Earth
density structure depends crucially on the Earth layer structure assumed in
the studies. More specifically, we find that if one approximates the Earth
density distribution with three layers - inner core, outer core and mantle,
the $M_{\oplus}$, $I_{\oplus}$ and EHE constraints lead to severe limits on
the possible uniform deviations of the outer core and mantle densities from
their PREM values even if one assumes that the inner core density, which plays
a crucial role in satisfying the constraints, can be by 50% larger than its
PREM density. We have shown also that, in contrast, the $M_{\oplus}$,
$I_{\oplus}$ and EHE constraints on the possible deviations of the outer core
and mantle densities from their PREM value are much less restrictive if one
assumes a four layer Earth density structure consisting of inner core, outer
core, and two - upper and lower - mantle layers. In this case the inner core
density plays an insignificant (or no) role in satisfying the constraints.

It follows from our results that the implementation of $M_{\oplus}$,
$I_{\oplus}$ and EHE constraints in the neutrino tomography studies of the
Earth density structure can have a very strong impact on the conclusions
reached in the studies.

## Acknowledgements

This work was supported in part by the European Union’s Horizon 2020 research
and innovation programme under the Marie Sklodowska-Curie grant agreement No.
860881-HIDDeN, by the INFN program on Theoretical Astroparticle Physics and by
the World Premier International Research Center Initiative (WPI Initiative,
MEXT), Japan.

## References

  * [1] W.F. McDonough, “Treatise on Geochemistry: The Mantle and Core”, vol. 2 (ed. R. W. Carlson, Elsevier-Pergamon, Oxford, 2003), p. 547.
  * [2] B. A. Bolt, Q. J. R. Astron. Soc. 32, 367 (1991).
  * [3] B. L. N. Kennett, Geophys. J. Int. 132 (1998) 374.
  * [4] G. Masters and D. Gubbins, Phys. Earth Planet. Inter. 140 (2003) 159.
  * [5] F. Nimmo, talk given at the International Workshop “Multi-Messenger Tomography of the Earth”, Snowbird, Utah, U.S.A., July 30-31, 2022.
  * [6] K.D. Koper, talk given at the International Workshop “Multi-Messenger Tomography of the Earth”, Snowbird, Utah, U.S.A., July 30-31, 2022.
  * [7] L. Waszek, talk given at the International Workshop “Multi-Messenger Tomography of the Earth”, APC, Paris, July 4-7, 2023.
  * [8] W.F. McDonough, talk given at the International Workshop “Multi-Messenger Tomography of the Earth”, APC, Paris, July 4-7, 2023.
  * [9] S. Noe, talk given at the International Workshop “Multi-Messenger Tomography of the Earth”, APC, Paris, July 4-7, 2023.
  * [10] A. M. Dziewonski and D. L. Anderson, Phys. Earth Planet. Interiors 25 (1981) 297.
  * [11] C. F. Yoder, “Global Earth Physics”, vol. 1, (ed. T. J. Ahrens, American Geophysical Union, Washington DC, 1995), p. 1.
  * [12] W. F. McDonough and R. Arevalo, J. Phys. Conf. Ser. 136 (2008), 022006 (doi:10.1088/1742-6596/136/2/022006).
  * [13] B.A. Baffet, H.E. Huppert, J.R. Lister and A.W. Woods, Submitted to Nature 1991.
  * [14] A. Placci and E. Zavattini, “On the possibility of using high-energy neutrinos to study the Earth’s interior”, https://cds.cern.ch/record/2258764 (1973), CERN Report.
  * [15] L. V. Volkova and G. T. Zatsepin, Izv. Akad. Nauk Ser. Fiz. 38N5 (1974) 1060.
  * [16] I. P. Nedyalkov, preprint JINR (Dubna), JINR-P2-81-645, 1981.
  * [17] I. P. Nedyalkov, Bolgarskaia Akademiia Nauk 34 (1981) 177.
  * [18] A. De Rujula, S. L. Glashow, R. R. Wilson, G. Charpak, Phys. Rept. 99 (1983) 341.
  * [19] T. L. Wilson, Nature 309 (1984) 38.
  * [20] G. A. Askar’yan, Usp. Fiz. Nauk 144 (1984) 523 [Sov. Phys. Usp. 27 (1984) 896].
  * [21] A. B. Borisov, B. A. Dolgoshein, A. N. Kalinovsky, Yad. Fiz. 44 (1986) 681.
  * [22] A. B. Borisov, B. A. Dolgoshein, Phys. Atom. Nucl. 56 (1993) 755.
  * [23] W. Winter, Earth Moon Planets 99 (2006) 285.
  * [24] C. Kuo, et al., Earth Plan. Sci. Lett. 133 (1995) 95.
  * [25] P. Jain, J. P. Ralston, G. M. Frichter, Astropart. Phys. 12 (1999) 193.
  * [26] M. M. Reynoso, O. A. Sampayo, Astropart. Phys. 21 (2004) 315.
  * [27] M. C. Gonzalez-Garcia, F. Halzen, M. Maltoni and H. K. M. Tanaka, Phys. Rev. Lett. 100 (2008) 061802 [arXiv:0711.0745 [hep-ph]].
  * [28] M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98 (2018) 030001 (doi:10.1103/PhysRevD.98.030001). See therein the review “Neutrino Masses, Mixing and Oscillations” by K. Nakamura and S.T. Petcov.
  * [29] J. Badro et al., Proc. Natl. Acad. Sci. U. S. A. 112(40) (2015) 12310-12314.
  * [30] E. Kaminski and M. Javoy, Earth Plan. Sci. Lett. 365 (2013) 97-107.
  * [31] T. Sakamaki et al., Earth and Planetary Science Letters 287 (2009) 293-297.
  * [32] https://earthref.org/GERMRD/datamodel/ (cit. on pp. 114, 115).
  * [33] A. Donini, S. Palomares-Ruiz and J. Salvado, Nature Phys. 15 (2019) 37 [arXiv:1803.05901 [hep-ph]].
  * [34] C. Rott, A. Taketa and D. Bose, Sci. Rep. 5 (2015) 15225 [arXiv:1502.04930 [physics.geo-ph]].
  * [35] S. Bourret, J. Coelho, E. Kaminski and V. Van Elewyck, PoS ICRC2019 (2020), 1024 doi:10.22323/1.358.1024
  * [36] T. K. Gaisser and M. Honda, Ann. Rev. Nucl. Part. Sci. 52 (2002) 153 [arXiv:hep-ph/0203272].
  * [37] M. Honda, M. Sajjad Athar, T. Kajita, K. Kasahara and S. Midorikawa, Phys. Rev. D 92 (2015) 023004 [arXiv:1502.03916].
  * [38] R. Abbasi et al. [IceCube Collab.], Astropart. Phys. 35 (2012) 615;
M.G. Aartsen et al. [IceCube Collab.], Phys. Rev. Lett. 120 (2018) 071801; P.
Eller et al. [IceCube], PoS ICRC2023 (2023) 1036 [arXiv:2307.15295].

  * [39] J.A. Sanchez [IceCube Collab], talk given at the International Conference on Neutrino Physics and Astrophysics, June 17-22, 2024, Milano, Italy.
  * [40] S. Adrian-Martinez et al. [KM3Net], “Letter of intent for KM3NeT 2.0,” J. Phys. G 43 (2016) 084001 [arXiv:1601.07459].
  * [41] K. Abe et al. [Hyper-Kamiokande], “Hyper-Kamiokande Design Report,” arXiv:1805.04163;
M. Ishitsuka [on behalf of the Hyper-Kamiokande Proto-Collab.], talk given at
the XXIX Int. Conference on Neutrino Physics and Astrophysics, Chicago, June
22 - July 2, 2020 (virtual conference), DOI:10.5281/zenodo.3959585.

  * [42] B. Abi et al. [DUNE], [arXiv:1807.10334 [physics.ins-det]].
  * [43] S. Ahmed et al. [ICAL], Pramana 88 (2017) 79 [arXiv:1505.07380].
  * [44] B. Luzum et al., Celest. Mech. Phys. 110 (2011) 110.
  * [45] H.USAO, USNO, UKHO, “The Astronomical Almanac”, http://asa.usno.navy.mil/.
  * [46] W. Chen, J. Ray, W.B. Shen and C.L. Huang, J. Geod. 89 (2015) 179.
  * [47] M. G. Aartsen et al. [IceCube], Phys. Rev. Lett. 117 (2016) 071801 [arXiv:1605.01990].
  * [48] S. K. Agarwalla, T. Li, O. Mena and S. Palomares-Ruiz, [arXiv:1212.2238].
  * [49] W. Winter, Nucl. Phys. B 908 (2016), 250-267 doi:10.1016/j.nuclphysb.2016.03.033 [arXiv:1511.05154 [hep-ph]].
  * [50] S. Bourret et al. [KM3NeT], J. Phys. Conf. Ser. 888 (2017) no.1, 012114 doi:10.1088/1742-6596/888/1/012114 [arXiv:1702.03723 [physics.ins-det]].
  * [51] A. Kumar and S.Kumar Agarwalla, [arXiv:2104.11740].
  * [52] F. Capozzi and S.T. Petcov, Eur. Phys. J. C 82 (2022) 461 [arXiv:2111.13048].
  * [53] K.J. Kelly, P.A.N. Machado, I. Martinez-Soler and Y.F. Perez-Gonzalez, “DUNE atmospheric neutrinos: Earth tomography,” JHEP 05 (2022) 187 [arXiv:2110.00003].
  * [54] P. B. Denton and R. Pestes, Phys. Rev. D 104 (2021) 113007 [arXiv:2110.01148].
  * [55] L. Maderer, E. Kaminski, J.A.B. Coelho, S. Bourret and V. Van Elewyck, Front. Earth Sci. 11 (2023) 1008396 [arXiv:2208.00532].
  * [56] A.K. Upadhyay, A. Kumar, S.K. Agarwalla and A. Dighe, JHEP 04 (2023) 068 [arXiv:2211.08688].
  * [57] D. Raikwal and S. Choubey, Phys. Rev. D 109 (2024) 073011 [arXiv:2309.12573].
  * [58] A.K. Upadhyay, A. Kumar, S.K. Agarwalla and A. Dighe, [arXiv:2405.04986].
  * [59] S. Choubey, P. Ghoshal and S.T. Petcov, studies performed in the period 2008 - 2011 for the 100 kt liquid Argon and 1 Mt water Cerenkov (Hyper-Kamiokande) detectors, unpublished.
  * [60] S. Choubey and S.T. Petcov, studies performed in 2014 for the PINGU detector, unpublished.