#  Exploring the Sun’s birth radius and the distribution of planet building
blocks in the Milky Way galaxy: A multi-zone Galactic chemical evolution
approach

Junichi Baba1,2, Takayuki R. Saitoh3, and Takuji Tsujimoto2
1 Amanogawa Galaxy Astronomy Research Center, Graduate School of Science and
Engineering, Kagoshima University, 1–21–35 Korimoto, Kagoshima 890-0065,
Japan.
2 National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan.
3 Department of planetology, Faculty of Science, Kobe University, 1–1
Rokkodai, Nada-ku, Kobe, Hyogo, 650-0013, Japan.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted xxx. Received xxx; in original form 2023 May 31)

###### Abstract

We explore the influence of the Milky Way galaxy’s chemical evolution on the
formation, structure, and habitability of the Solar system. Using a multi-zone
Galactic Chemical Evolution (GCE) model, we successfully reproduce key
observational constraints, including the age-metallicity ([Fe/H]) relation,
metallicity distribution functions, abundance gradients, and [X/Fe] ratio
trends for critical elements involved in planetary mineralogy, including C, O,
Mg, and Si. Our GCE model suggests that the Sun formed in the inner Galactic
disc, $R_{\rm birth,\odot}\approx 5$ kpc. We also combined a stoichiometric
model with the GCE model to examine the temporal evolution and spatial
distribution of planet building blocks (PBBs) within the Milky Way galaxy,
revealing trends in the condensed mass fraction ($f_{\rm cond}$), iron-to-
silicon mass fraction ($f_{\rm iron}$), and water mass fraction ($f_{\rm
water}$) over time and towards the inner Galactic disc regions. Specifically,
our model predicts a higher $f_{\rm cond}$ in the protoplanetary disc within
the inner regions of the Milky Way galaxy, as well as an increased $f_{\rm
iron}$ and a decreased $f_{\rm water}$ in the inner regions. Based on these
findings, we discuss the potential impact of the Sun’s birth location on the
overall structure and habitability of the Solar System.

###### keywords:

Galaxy: abundances – Galaxy: evolution – stars: abundances – protoplanetary
discs – astrobiology – methods: numerical

##  1 Introduction

Although our Solar system (hereafter referred to as the Sun) is not expected
to be exceptionally unusual compared to exoplanets (Martin & Livio, 2015), it
is interesting to note that the Sun has a higher metallicity (i.e. [Fe/H])
than neighbouring stars of similar age (Edvardsson et al., 1993; Gustafsson et
al., 2010; Casagrande et al., 2011). A potential explanation for this anomaly
is that the Sun formed at a different radius in the Milky Way galaxy than its
current location. This is because the Galactic disc exhibits a negative
gradient in the metallicity of its stars (e.g. Anders et al., 2014; Genovali
et al., 2015), which can be understood within the context of galaxy formation
theories in which the inner disc forms faster and becomes more metal-rich than
the outer disc (Matteucci & Francois, 1989; Chiappini et al., 2001; Grand et
al., 2018). Specifically, given the inferred temporal evolution of the
metallicity gradient of the Galactic disc, it can be deduced that the Sun has
migrated outward 1–4 kpc since its birth111The Sun’s migration is not
exceptional, but rather, radial migration of stars is a common occurrence in
the evolution of the Galactic disc (e.g. Sellwood & Binney, 2002; Roškar et
al., 2008; Schönrich & Binney, 2009; Toyouchi & Chiba, 2018; Hayden et al.,
2020; Lian et al., 2022)., based on [Fe/H] constraints (Wielen et al., 1996;
Nieva & Przybilla, 2012; Minchev et al., 2013, 2018; Feltzing et al., 2020;
Frankel et al., 2020; Tsujimoto & Baba, 2020; Lu et al., 2022). Therefore,
stellar metallicity serves as an effective indicator for determining the birth
radii of stars. This argument was pioneering suggested by Wielen et al.
(1996), which established the foundation for understanding the Sun’s
birthplace within the Milky Way galaxy and the potential implications for its
unique characteristics.

Stellar metallicity, i.e. [Fe/H], also plays a significant role in the context
of planet formation. For instance, observational research has shown that stars
with higher metallicity are more likely to host giant planets (Santos et al.,
2004; Fischer & Valenti, 2005; Johnson et al., 2010; Brewer et al., 2018).
This is likely due to the fact that higher stellar metallicity allows for the
creation of more planet-forming materials (e.g. Dawson et al., 2015; Coleman &
Nelson, 2016). While the relationship between stellar metallicity and the
likelihood of Early-like planets is still being debated (Zhu & Dong, 2021, for
a review), theoretical models suggest the existence of a minimum metallicity
threshold is necessary for the formation of Earth-like planets formation of
[Fe/H]$\gtrsim$-1.5 (Johnson & Li, 2012). In general, both observations and
simulations indicate that metal-enriched environments tend to produce a higher
number of planets compared to metal-poor environments. It is considered that
these correlations between stellar metallicity and planet-hosting probability
are one of the key factors to consider when discussing the Galactic habitable
zone (GHZ; Lineweaver et al., 2004; Prantzos, 2008; Spitoni et al., 2017).

The chemical composition of the host star, including elements such as O, C,
Mg, Fe, and Si, is closely related to the bulk chemical composition of the
planets (Lodders, 2003; Bond et al., 2010; Thiabaud et al., 2015; Unterborn &
Panero, 2017; Bitsch & Battistini, 2020; Adibekyan et al., 2021; Jorge et al.,
2022; Spaargaren et al., 2023). Equilibrium condensation models (Lodders,
2003) predict that the Solar system composition yields approximately 1.5%
total condensate mass ($f_{\rm cond}$), which generally increases with [Fe/H].
These models also suggest that the iron–to–silicate mass fraction ($f_{\rm
iron}$) in the condensate mass is approximately 33%, serving as a proxy for
the metallic iron core–to–silicate mantle ratio in rocky planets (Unterborn et
al., 2014; Unterborn & Panero, 2017). Remarkably, these values align with the
compositions of Earth, Venus, and Mars (32%, 35%, and 24%, respectively;
Breuer et al., 2022, and references therein). Moreover, the water mass
fraction ($f_{\rm water}$) in the condensate mass is estimated to be around
54%, which is essential for known life (Kasting et al., 1993) and plays a
crucial role in planetary plate tectonics, deep water cycles, and carbon
cycles (Korenaga, 2010; Noack et al., 2014; Noack et al., 2017). However, it
is important to note that both $f_{\rm iron}$ and $f_{\rm water}$ are
influenced by the chemical composition of the host star, particularly the
Mg/Si and C/O number ratios (e.g. Gaidos, 2000; Bitsch & Battistini, 2020).

To investigate the chemical composition of planetary building blocks (PBBs)
around thin and thick disc stars, Santos et al. (2017) utilized a simple
stoichiometric model of the equilibrium condensation models (Lodders, 2003)
and found that the PBBs differ in their iron and water mass fractions. Cabral
et al. (2019) also applied this simple stoichiometric model to their stellar
population synthesis simulations of the Milky Way, obtaining similar results.
Bitsch & Battistini (2020) also developed a model to infer the PBBs from the
chemical composition of stars, which was later applied by Cabral et al. (2023)
to the high-resolution spectroscopic survey data such as APOGEE and GALAH to
investigate the PBB composition trends for various stellar populations across
the Milky Way. However, these studies do not examine the temporal evolution of
the PBBs distribution in the Milky Way, which is vital for understanding the
birthplace of the Sun that was formed approximately 4.6 Gyr ago ($\equiv
t_{\rm bk,\odot}$; Bonanno et al., 2002).

In order to comprehend the distribution of the chemical composition of stars
in the past, it is necessary to understand the whole history of chemical
evolution in the Milky Way galaxy. This evolution history is the result of the
interplay of the star formation history, stellar evolution, and
nucleosynthetic yields, as well as the circulation of the gas between the disc
and halo (Tinsley, 1980; Matteucci, 2021). The variation in stellar chemical
compositions is due to the fact that different elements are produced in
different types of stars at various times during the galactic evolution
(Burbidge et al., 1957). For instance, $\alpha$-elements like O, Mg, and Si
are entirely produced by massive stars and subsequently released into the ISM
through core-collapse supernovae (SNe II; e.g. Nomoto et al., 2013). This
enrichment occurs on a relatively short timescale, typically $\lesssim 30$
Myr. On the other hand, a significant amount of Fe is released from SNe Ia
associated with low-mass stars, which occurs over a longer timescale such as
0.1–10 Gyr (Maoz et al., 2014). SNe Ia release only a small fraction of O and
Mg, but a non-negligible fraction of Si (e.g. Iwamoto et al., 1999). In terms
of C, the majority of C originates from low- and intermediate-mass stars
during the asymptotic giant branch (AGB) phase over a longer timescale of
$\gtrsim 1$ Gyr (Karakas & Lattanzio, 2014). However, metal-rich high-mass
stars can also contribute considerable amounts of C through their stellar
winds (e.g. Maeder, 1992; Limongi & Chieffi, 2018).

In this study, we explore the impact of the Milky Way’s chemical evolution on
the birthplace and PBBs that shaped the Solar system. To accomplish this, we
develop a multi-zone Galactic chemical evolution (GCE) model that simulates
the chemical enrichment of O, C, Mg, Si, and Fe in the Milky Way galaxy,
taking into account various nucleosynthetic processes, including stellar
nucleosynthesis, SNe II, SNe Ia, and AGB stars. With this framework, we are
able to estimate the Sun’s birth radius ($R_{\rm birth,\odot}$) within the
Milky Way galaxy through a multi-elemental chemical evolution analysis.
Additionally, we employ a stoichiometric model to derive the chemical
compositions of the PBBs, enabling us to establish constraints on the initial
conditions of the Solar system. Through our analysis, we explore the temporal
and spatial distribution of PBBs and examine their significant implications
for the architectural structure and habitability of the Solar system.

The paper is structured as follows: In Section 2, we present our models for
the chemical evolution of the Milky Way and the stoichiometric model for
estimating the chemical compositions of the PBBs. In Section 3, we present the
GCE model results and compare them with observed Galactic chemical evolution
trends and gradients. Furthermore, we estimate the Sun’s birth radius within
the Milky Way galaxy. Section 4 delves into the predicted chemical
compositions of PBBs throughout the Galactic disc. Finally, in Section 5, we
summarize our main findings and give implications for the origin of the Solar
system in the Milky Way galaxy.

##  2 Models

In order to investigate the origin of the Sun in the Milky Way galaxy and to
gain insight into the distribution of the chemical composition of planet-
building blocks (PBBs) at a galactic scale, we have computed the mass and
chemical evolution of the Milky Way galaxy using a standard one-dimensional
treatment (Talbot & Arnett, 1971) with the inclusion of the gas infall term
(Chiosi, 1980). The Galactic disc is divided into concentric cylindrical
rings, each evolving independently and containing a homogeneous mixture of gas
and stars. Time $t$ is the only independent variable for each ring, with no
consideration given to radial gas flow or stellar migration (see Section 2.1).
Subsequently, we employ the ‘stoichiometric model’ (Santos et al., 2015, 2017)
on the results of the Galactic chemical evolution (GCE) to predict the
chemical compositions of PBBs based on equilibrium condensation models of the
protoplanetary disc (see Section 2.2).

Throughout this study, we adopted the solar abundance pattern of Asplund et
al. (2009), with solar metallicity $Z_{\odot}=0.0134$, and for comparisons
with observations, we rescale the observed abundances to those of the solar
mixture adopted in this study. We also adopted the Kroupa (2001) stellar
initial mass function (IMF), assumed to be constant in time and space. The
Galactocentric distance of the Sun from the Galactic centre is adopted as
$R_{\odot}=8.2$ kpc.

###  2.1 Inside-out formation model of Galactic discs

Gas infall from the circumgalactic region into a dark matter halo and
eventually onto a disc plane is crucial in disc galaxy formation (Fall &
Efstathiou, 1980; Mo et al., 1998). We used the two-infall model (e.g.
Chiappini et al., 1997; Noguchi, 2018; Spitoni et al., 2019), for the gas
infall rate from the halo. In this model, the Milky Way formed through two
major episodes of gas infall, the first creating the stellar halo and thick
disc, and the second forming the thin disc over a longer time scale.
Cosmological simulations of galaxy formation confirmed such two infall
episodes (Colavitti et al., 2008; Grand et al., 2018). In this study, we adopt
a simple functional form of the two-infall model as described below:

$\displaystyle\frac{\Sigma_{\rm infall}(R,t)}{{\rm M_{\rm\odot}}~{}{\rm
pc}^{-2}~{}{\rm yr}^{-1}}$ $\displaystyle=\frac{\Sigma^{\rm
thick}_{0}(R)}{\tau_{\rm d}^{\rm thick}}\frac{\exp(-t/\tau_{\rm d}^{\rm
thick})}{1-\exp(-T_{\rm end}/\tau_{\rm d}^{\rm thick})}$ $\displaystyle\
+\frac{\Sigma^{\rm thin}_{0}(R)}{\tau_{\rm d}^{\rm thin}}\frac{(t/\tau_{\rm
d}^{\rm thin})\exp(-t/\tau_{\rm d}^{\rm thin})}{1-(1+T_{\rm end}/\tau_{\rm
d}^{\rm thin})\exp(-T_{\rm end}/\tau_{\rm d}^{\rm thin})},$ (1)

where $\tau_{\rm d}^{\rm thick}$ and $\tau_{\rm d}^{\rm thin}$ indicate the
timescales for gas infall onto the halo/thick-disc and thin-disc components,
respectively. $\Sigma_{{\rm 0}}^{\rm thin}(R)$ and $\Sigma_{{\rm 0}}^{\rm
thick}(R)$ express the surface mass densities of stellar thin and thick discs,
respectively. Specifically, we express the surface mass density of the $i$-th
disc (where $i=$ thin or thick disc) as $\Sigma_{{\rm
0}}^{i}(R)=\Sigma_{\odot}^{i}\exp(-(R-R_{\odot})/R_{{\rm d}}^{i})$. Here,
$R_{{\rm d}}^{i}=3.5$ and 2.8 kpc, respectively, while $\Sigma_{\odot}^{i}=40$
$M_{\rm\odot}~{}{\rm pc}^{-2}$ (including gas surface density of $13.7$
$M_{\rm\odot}~{}{\rm pc}^{-2}$) and 7 $M_{\rm\odot}~{}{\rm pc}^{-2}$ are the
current values at solar vicinity. These values of scale lengths are carefully
chosen such that, with the employment of the star formation law (see eq.(3)),
the stellar discs acquire scale lengths $R_{\rm d,*}^{\rm thin}=3.5/1.4=2.5$
and $R_{\rm d,*}^{\rm thick}=2.8/1.4=2.0$ kpc, respectively (Bland-Hawthorn &
Gerhard, 2016). Additionally, the value of the scale-length of the gas disc to
form the stellar thin disc agrees well with the measured value (3.75 kpc) of
the HI distribution in the outskirts (Kalberla & Kerp, 2009).

It is well-established that the abundance of metals decreases with $R$ along
the thin disc (e.g. Anders et al., 2014). Such negative abundance gradients
suggest that star formation has been more efficient in the inner regions of
the Galactic disc compared to the outer regions. To reproduce these gradients,
an inside-out formation of the disc is one possibility (e.g. Larson, 1976). In
this scenario, the disc forms through gas infall from the halo, which occurs
more quickly in the inner disc than in the outer disc, resulting in a gradient
in the star formation rate (SFR; Matteucci & Francois, 1989; Prantzos &
Aubert, 1995; Chiappini et al., 2001; Grisoni et al., 2018; Grand et al.,
2018; Palla et al., 2020; Spitoni et al., 2021). This inside-out mechanism of
the thin disc can be achieved by assuming a time scale of gas infall
increasing $R$ as described below:

$\frac{\tau_{\rm d}^{\rm thin}(R)}{{\rm Gyr}}=\alpha_{\rm
infall}\left(\frac{R}{{\rm kpc}}-R_{\odot}\right)+\tau_{\rm infall,\odot},$
(2)

where $\alpha_{\rm infall}$ is the radial gradient of the infall timescale,
which controls the radial gradient of mass assembly and elemental abundances;
$\tau_{\rm infall,\odot}$ is the infall timescale at solar vicinity. On the
other hand, since observations suggest that the thick disc does not form in
the inside-out manner but was formed from a well-mixed material (e.g. Haywood
et al., 2018), we fixed $\tau_{\rm d}^{\rm thick}=0.1$ Gyr (Grisoni et al.,
2017; Spitoni et al., 2021). Fig. 1(a) shows the time evolution of the infall
rate surface density ($\Sigma_{\rm infall}$) at the solar vicinity for
$\tau_{\rm infall,\odot}=7$ Gyr. This long timescale for the assembly of the
thin disc at the solar vicinity is consistent with the GCE models (Yoshii et
al., 1996; Boissier & Prantzos, 1999; Alibés et al., 2001) and cosmological
hydrodynamic simulations (Nuza et al., 2019; Iza et al., 2022).

The metallicity of the infalling gas is unknown. It is widely accepted that
the continuous infall of the metal-poor gas is necessary to match the
observations of the metallicity distribution of the local disc stars (Larson,
1972; Tosi, 1988). Observations suggest that the current infalling gas
generally has low metallicities ($Z_{\rm in}\sim 0.1~{}{\rm Z_{\rm\odot}}$;
e.g. Wakker et al., 1999), but they provide no information on the past
chemical composition of such clouds. In this study, we adopt the simplest
possible assumption, namely that the infalling gas always has primordial
composition, i.e. $Z_{\rm in}=0$. While this assumption hardly changes the
results, it allows for the existence of disc stars with metallicities lower
than [Fe/H] $=-1$ (Tosi, 1988).

We have to include a model for star formation in the galaxy formation model to
track the evolution of the stellar and gaseous phases separately. Once the gas
within a halo has settled into a disc, it begins to form stars. We adopt a
model the star formation, whose rate (SFR) is parameterized according to the
Schmidt-Kennicutt relation:

$\frac{\Sigma_{\rm SFR}(R,t)}{{\rm M_{\rm\odot}}~{}{\rm yr}^{-1}~{}{\rm
kpc}^{-2}}=\epsilon_{\rm SFE}~{}\left(\frac{\Sigma_{\rm gas}(R,t)}{{\rm
M_{\rm\odot}}~{}{\rm pc}^{-2}}\right)^{1.4},$ (3)

where $\epsilon_{\rm SFE}$ is the star formation efficiency (SFE) and is set
to $1.3\times 10^{-4}$ (Kennicutt & Evans, 2012).

To track the evolution of the abundance of the individual elements, we use the
CELib (Saitoh, 2017), a chemical evolution library that incorporates
literature-derived stellar yield tables. Specifically, we employ the Type II
SNe yields of Nomoto et al. (2013) for massive stars and the AGB stars’ yields
of Karakas (2010) and Doherty et al. (2014) for low- and intermediate-mass
stars. The plateau observed in $\alpha$ elements (O, Mg, Si) at [Fe/H]$\sim$-1
is widely believed to result from the SNe II, followed by a decrease caused by
SNe Ia (e.g. Matteucci & Greggio, 1986; Weinberg et al., 2017). Non-LTE
observations show the plateau values of [C/Fe], [O/Fe], [Mg/Fe] and [Si/Fe] at
[Fe/H]$\lesssim$-1 is approximately 0.1, 0.6, 0.4–0.5, and 0.3–0.4,
respectively (Zhao et al., 2016; Amarsi et al., 2019). In order to reproduce
these observed plateaus, we have artificially modified the theoretical C, O,
Mg, and Si yields from massive stars by a factor of 1.25, 0.6, 0.7, and 0.52,
respectively (see details in Appendix A). The use of such ad-hoc yield
prescriptions has been a common practice in GCE studies (e.g. François et al.,
2004). For the Type Ia SNe, we use the Iwamoto et al. (1999) theoretical
yields (their model W7), and a power-law delay time distribution (DTD) such
that $\propto t_{\rm delay}^{-1}$ within the range of $t_{\rm min}\leq t_{\rm
delay}\leq 10~{}{\rm Gyr}$ (Totani et al., 2008). We set $t_{\rm min}=150$
Myr, approximately the median value among the observed values ($t_{\rm
min}\approx 40$–300 Myr; Maoz et al., 2014). The normalization of the DTD,
$N_{\rm Ia}$, is determined by the observational prediction that the
cumulative number of Type Ia SNe becomes $0.8\times 10^{-3}~{}{\rm
M_{\rm\odot}}^{-1}$ at $10~{}{\rm Gyr}$ from the birth of the population
(Tsujimoto & Bekki, 2012; Maoz & Graur, 2017; Weinberg et al., 2023). For
stellar evolutionary models, we adopt the metallicity-dependent stellar
lifetime table of Portinari et al. (1998) implemented in CELib.

Figure 1:  Left: Local surface densities of (a) gas infall rate ($\Sigma_{\rm
infall}$) and SFR ($\Sigma_{\rm SFR}$), (b) gas mass ($\Sigma_{\rm gas}$) and
stellar mass ($\Sigma_{\rm star}$) at $R=8$ kpc, are functions of the look-
back time $t_{\rm bk}(\equiv 13~{}{\rm Gyr}-t)$. It is assumed that $\tau_{\rm
infall,\odot}=7$ Gyr. The open circle with an error bar indicates the observed
value of the SFR density is 2–5 $M_{\rm\odot}~{}{\rm Gyr^{-1}}~{}{\rm
pc}^{-2}$ (Rana, 1991; Kennicutt & Evans, 2012), gas and star density are
10–16 $M_{\rm\odot}~{}{\rm pc}^{-2}$ and $33\pm 3$$M_{\rm\odot}~{}{\rm
pc}^{-2}$ at solar vicinity (McKee et al., 2015), respectively. The vertical
solid line (green) indicates the Sun’s birth time, i.e. $t_{\rm bk}=t_{\rm
bk,\odot}\equiv 4.6$ Gyr. Right: Radial profiles of (c) $\Sigma_{\rm SFR}$,
(d) $\Sigma_{\rm gas}$ and $\Sigma_{\rm star}$ at $t_{\rm bk}=$ 0 (solid) and
$t_{\rm bk}=t_{\rm bk,\odot}$ (dashed), with assuming $\alpha_{\rm
infall}=0.7~{}\rm Gyr~{}kpc^{-1}$. Crosses with an error bar indicate the
observed values of the SFR density (Rana, 1991) and gas density (from Appendix
in Kubryk et al., 2015). Open circles with error bars indicated the observed
local values (Rana, 1991; Kennicutt & Evans, 2012; McKee et al., 2015).

###  2.2 Stoichiometric model for planet-building block compositions

To estimate the chemical compositions of planet-building blocks (PBBs), we
relied on the chemical abundances derived from the host stars using the
results of the GCE model. During the process of gas cloud collapse, which
leads to the formation of stars and their protoplanetary discs, a diverse
range of condensed materials is generated. Understanding the composition of
these materials is primarily guided by the $N_{\rm C}/N_{\rm O}$ and $N_{\rm
Mg}/N_{\rm Si}$ ratios (where $N_{X}$ represents the number of atoms of each
species $X$). When the $N_{\rm C}/N_{\rm O}$ ratio exceeds a specific
threshold, the availability of free oxygen for silicate formation becomes
severely limited, resulting in geological compositions dominated by carbonates
(e.g. Kuchner & Seager, 2005; Bond et al., 2010). Conversely, when the $N_{\rm
C}/N_{\rm O}$ ratio falls below this threshold, planetesimal geology is
primarily characterized by primary magnesium silicates. It is noteworthy that
our GCE model consistently points to $N_{\rm C}/N_{\rm O}<1$, signifying
oxygen-rich systems. Thus, throughout this study, we have centred our
investigation on the scenario of $N_{\rm C}/N_{\rm O}<1$.

In oxygen-rich systems, the mineral composition of silicon (Si) is notably
influenced by the $N_{\rm Mg}/N_{\rm Si}$ ratio, as discussed in Unterborn &
Panero (2017). When $N_{\rm Mg}/N_{\rm Si}<1$, the major portion of magnesium
(Mg) is allocated to the production of $\rm MgSiO_{3}$ (enstatite). In this
scenario, the remaining Si contributes to the formation of quartz ($\rm
SiO_{2}$) and feldspars ($\rm CaAl_{2}Si_{2}O_{8}$). Conversely, systems where
$N_{\rm Mg}/N_{\rm Si}>1$ exhibit a combination of pyroxene and olivine ($\rm
Mg_{2}SiO_{4}$; forsterite).222In cases where $N_{\rm Mg}/N_{\rm Si}>2$,
olivine and other Mg compounds like MgO and MgS dominate (Carter-Bond et al.,
2012). Importantly, certain molecules, including $\rm H_{2}$, He, $\rm H_{2}O$
(water), $\rm CH_{4}$ (methane), Fe, $\rm MgSiO_{3}$, $\rm Mg_{2}SiO_{4}$,
$\rm SiO_{2}$, are anticipated to have a significant role within
protoplanetary discs (e.g. Lodders, 2003; Unterborn & Panero, 2017; Bitsch &
Battistini, 2020). Consequently, within the framework of this study, we have
opted to utilize the ‘stoichiometric’ model developed by Santos et al. (2015,
2017). This approach is based on the idea that the composition of condensed
materials forming within protoplanetary discs is primarily determined by the
availability of specific chemical elements.

The stoichiometric model yields the subsequent relationships, elucidating the
connection between the elemental composition of host stars and the condensed
materials encountered within protoplanetary discs:

$\displaystyle\begin{pmatrix}N_{\rm O}\\\ N_{\rm Mg}\\\ N_{\rm Si}\\\ N_{\rm
C}\end{pmatrix}=\begin{pmatrix}1&3&4&0\\\ 0&1&2&0\\\ 0&1&1&0\\\
0&0&0&1\end{pmatrix}\begin{pmatrix}N_{\rm H_{2}O}\\\ N_{\rm MgSiO_{3}}\\\
N_{\rm Mg_{2}SiO_{4}}\\\ N_{\rm CH_{4}}\end{pmatrix},$ (4)

for $N_{\rm Mg}>N_{\rm Si}$, otherwise:

$\displaystyle\begin{pmatrix}N_{\rm O}\\\ N_{\rm Mg}\\\ N_{\rm Si}\\\ N_{\rm
C}\end{pmatrix}=\begin{pmatrix}1&3&2&0\\\ 0&1&0&0\\\ 0&1&1&0\\\
0&0&0&1\end{pmatrix}\begin{pmatrix}N_{\rm H_{2}O}\\\ N_{\rm MgSiO_{3}}\\\
N_{\rm SiO_{2}}\\\ N_{\rm CH_{4}}\end{pmatrix}.$ (5)

Inverting these equations and adding the stellar abundances ($N_{\rm C}$,
$N_{\rm O}$, $N_{\rm Mg}$, and $N_{\rm Si}$) predicted by the GCE model allows
us to derive the numbers of the key condensed molecules, $N_{\rm H_{2}O}$,
$N_{\rm MgSiO_{3}}$, $N_{\rm Mg_{2}SiO_{4}}$, $N_{\rm SiO_{2}}$, and $N_{\rm
CH_{4}}$, in the planet-building blocks.

Then we compute the condensation mass fraction of all heavy elements ($f_{\rm
cond}$) expected for the planetary building blocks:

$f_{\rm cond}=\frac{m_{\rm CH_{4}}+m_{\rm H_{2}O}+m_{\rm Fe}+m_{\rm
MgSiO_{3}}+m_{\rm Mg_{2}SiO_{4}}+m_{\rm SiO_{2}}}{M_{\rm tot}},$ (6)

the iron-to-silicon mass fraction ($f_{\rm iron}$):

$f_{\rm iron}=\frac{m_{\rm Fe}}{m_{\rm Fe}+m_{\rm MgSiO_{3}}+m_{\rm
Mg_{2}SiO_{4}}+m_{\rm SiO_{2}}},$ (7)

and the water mass fraction ($f_{\rm water}$):

$f_{\rm water}=\frac{m_{\rm H_{2}O}}{m_{\rm H_{2}O}+m_{\rm Fe}+m_{\rm
MgSiO_{3}}+m_{\rm Mg_{2}SiO_{4}}+m_{\rm SiO_{2}}},$ (8)

where $m_{\rm X}=N_{\rm X}*\mu_{\rm X}$ and $\mu_{\rm X}$ their mean molecular
weights; $M_{\rm tot}=N_{\rm H}*\mu_{\rm H}+N_{\rm He}*\mu_{\rm He}+N_{\rm
C}*\mu_{\rm C}+N_{\rm O}*\mu_{\rm O}+N_{\rm Mg}*\mu_{\rm Mg}+N_{\rm
Si}*\mu_{\rm Si}+N_{\rm Fe}*\mu_{\rm Fe}$. Though the above equations are
mathematically defined as fractions, we discuss these values in per cent units
(%) for convenience.

The stoichiometric model predicts the values for the Sun, which are compatible
with the values derived in Lodders (2003). However, the model value of $f_{\rm
water}$ is about 3% lower than the actual value. This is largely due to the
fact that the gaseous CO and ${\rm CO_{2}}$ are not taken into account in the
model.333Bitsch & Battistini (2020) pointed out that including the gaseous CO
and ${\rm CO_{2}}$ binds much O, which is then not available to form water
ice, in contrast to the stoichiometric model. This binding of O can greatly
reduce the amount of available water around central stars with high C/O ratios
(see Appendix B of Bitsch & Battistini, 2020).

The distribution of organic molecules within the Milky Way galaxy holds
significant implications, especially when we discuss habitability on a
galactic scale. Recent observations have revealed the presence of complex
organic molecules (COM) throughout the Galaxy. For instance, methanol has been
detected even in low-metallicity environments in the outer regions of the
Galaxy (e.g. Shimonishi et al., 2021; Bernal et al., 2021; Fontani et al.,
2022). These COMs are fundamental for prebiotic chemistry and present new
challenges to the conventional Galactic Habitable Zone (GHZ) concept, which
primarily relies on metallicity (e.g. Lineweaver et al., 2004). While our
study does not directly address the abundance of COMs, future research could
elucidate the distribution of these compounds within the Milky Way, drawing
upon the elemental abundance insights from the GCE model presented here.
Nonetheless, such a discussion is beyond the scope of this study.

##  3 Estimation of the Sun’s birth radius from Chemical Evolution of Galactic
discs

We present the results on the chemical evolution of the Galactic disc(s) using
the GCE model described in Section 2.1, and deduce the Sun’s birth radius
through a multi-elemental chemical evolution analysis. To trace the evolution
of the surface mass densities of gas ($\Sigma_{\rm gas}$), stars ($\Sigma_{\rm
star}$), and individual elements (H, He, C, O, Mg, Si, and Fe), we solve the
classical set of GCE equations for each zone as a function of Galactocentric
radius $R$ (in the range $4~{}{\rm kpc}\leq R\leq 16~{}{\rm kpc}$) and time
$t$ (in the range $0\leq t\leq 13~{}{\rm Gyr}$).

The region within $R<4$ kpc of the Milky Way galaxy contains various galactic
structures, such as the bulge, bar, and disc(s). These components are known to
exhibit distinct star formation/chemical evolution histories, which introduce
complexity to GCE models (e.g. Bovy et al., 2019; Wegg et al., 2019; Lian et
al., 2021; Eilers et al., 2022). Hence, we intentionally exclude the innermost
4 kpc from our GCE modelling. Conversely, our study computes models up to
$R=16$ kpc. However, it is a subject of active debate both observationally and
theoretically that the structure and star formation activity in the outer
Galactic disc ($R\gtrsim 12$ kpc) could be significantly perturbed by
disturbances caused by the Sagittarius dwarf galaxy (e.g. Laporte et al.,
2019; Ruiz-Lara et al., 2020). Consequently, it is essential to exercise
caution when comparing our models with observational data in the $R\gtrsim 12$
kpc region.

In the following, we present only the results of our fiducial model, and the
effects of model parameter dependencies on our results are displayed in
Appendix B.

###  3.1 Star formation history

Fig. 1(a) displays the time evolution of the local surface densities of infall
rate ($\Sigma_{\rm infall}$) and SFR ($\Sigma_{\rm SFR}$) at the solar
vicinity. We define the look-back time $t_{\rm bk}\equiv 13~{}{\rm Gyr}-t$. At
$t_{\rm bk}=0$, the present-day value of $\Sigma_{\rm SFR}$ is approximately
4.5 $M_{\rm\odot}~{}{\rm Gyr^{-1}}~{}{\rm pc}^{-2}$, which is within the range
of observated value (Rana, 1991; Kennicutt & Evans, 2012). The SFR had a
maximum value at $t_{\rm bk}\approx 3$–5 Gyr ago at the solar vicinity, in
reasonable agreement with star formation history inferred from observations
(Mor et al., 2019; Dal Tio et al., 2021; Sahlholdt et al., 2022).

As shown in Fig. 1(b), the gas density at the solar vicinity begins to
increase after $t_{\rm bk}\approx 11$ Gyr and remains almost constant at its
preset-day value of 13 $M_{\rm\odot}~{}{\rm pc}^{-2}$ after $t_{\rm bk}\approx
5$ Gyr, in good agreement with the observed range of 10–16
$M_{\rm\odot}~{}{\rm pc}^{-2}$ (McKee et al., 2015; Nakanishi & Sofue, 2016).
The stars at the solar vicinity steadily increase to the present-day value of
35 $M_{\rm\odot}~{}{\rm pc}^{-2}$, which is consistent with the range of 30–36
$M_{\rm\odot}~{}{\rm pc}^{-2}$ derived from the observations (e.g. Bland-
Hawthorn & Gerhard, 2016).

Our GCE model provides a good fit to the observed present-day radial profiles
of $\Sigma_{\rm SFR}$, $\Sigma_{\rm gas}$, and $\Sigma_{\rm star}$ in the
Galactic disc(s). In Fig. 1(c) and 1(d), we compare the predicted radial
profiles of $\Sigma_{\rm SFR}$, $\Sigma_{\rm gas}$, and $\Sigma_{\rm star}$
(solid lines) to the observations. We assumed $\alpha_{\rm infall}=0.7~{}\rm
Gyr~{}kpc^{-1}$ in these figures, similar to the models of Hou et al. (2000).
Our model reproduces the observed present-day profiles well. The dashed lines
in these figures indicate the profiles at $t_{\rm bk}=t_{\rm bk,\odot}$, i.e.
the Sun’s birth time. Our model suggests that the SFR and gas densities in the
regions $R\lesssim 6$ kpc were about twice as high at 4.6 Gyr ago as they are
currently.

Note that previous GCE models have predicted star formation histories
following this general trend, although specific predictions may vary depending
on the model’s assumptions and parameters (e.g. Chiappini et al., 1997;
Boissier & Prantzos, 1999; Naab & Ostriker, 2006; Minchev et al., 2013). Some
models may predict a slightly different peak time or a different overall shape
of the star formation history curve. However, the flattening of the profiles
is a general result of an inside-out formation scenario of the Galactic disc.
Therefore, our model is consistent with previous GCE models and observations,
where the SFR and gas densities in the inner regions of the disc were higher
in the past.

###  3.2 Evolution of Fe and $R_{\rm birth,\odot}$ estimation

Figure 2:  Evolution of [Fe/H] in the fiducial model with a parameter set of
$(N_{\rm Ia},\tau_{\rm infall,\odot},\alpha_{\rm infall})=(0.8\times
10^{-3}~{}{\rm M_{\rm\odot}}^{-1},7~{}{\rm Gyr},0.7~{}{\rm Gyr~{}kpc^{-1}}).$
Panel (a) Time evolution of [Fe/H] for four different radial zones, at 4
(red), 6 (blue), 8 (green), and 10 (yellow) kpc. The crosses (light blue) and
small circles (orange) represent the observed solar neighbourhood stars taken
from Casagrande et al. (2011) and Bensby et al. (2014), respectively. Panel
(b) The present-day MDFs for four different radial zones, at 4 (red), 6
(blue), 8 (green), and 10 (yellow) kpc. Observational data for the local stars
(i.e. around 8 kpc) are indicated with filled circles with error bars, which
are binned results using observational data sets taken from Table A2 in Mollá
et al. (2016). Panel (c) Radial profiles of the ISM elemental abundance [Fe/H]
at $t_{\rm bk}=0$ (solid) and $t_{\rm bk}=t_{\rm bk,\odot}$ (dashed). The
observational data sources are green diamonds, classical Cepheids from Luck
(2018); red circles, classical Cepheid from Genovali et al. (2015); light blue
squares, young open clusters from Netopil et al. (2016); grey triangles, young
($<1$ Gyr) open clusters from Yong et al. (2012). The yellow circle represents
the inferred Sun’s birth radius, $R_{\rm birth,\odot}\approx 5$ kpc.

In this subsection, we focus on estimating the Sun’s birth radius, $R_{\rm
birth,\odot}$, based on the chemical evolution of Fe. In Fig. 2(a), we show
the time evolution of [Fe/H] of the ISM, which is equivalent to the
age–metallicity relationship (AMR) of the stars in our fiducial model. The red
circle represents the Sun with [Fe/H]=0 and $t_{\rm bk}=t_{\rm bk,\odot}$
(=4.6 Gyr). The light blue crosses and orange small circles in the figure
represent the observed AMR taken from Casagrande et al. (2011) and Bensby et
al. (2014), respectively. Our model agrees well with the observed AMR.

At $R=8$ kpc (shown in green), [Fe/H] rapidly increases during the early
stages ($t_{\rm bk}\gtrsim$ 10 Gyr) and gradually after reaching [Fe/H]
$\approx-0.5$ dex at $t_{\rm bk}\approx 10$ Gyr. As time progresses, [Fe/H]
continues to increase, but by $t_{\rm bk}\approx t_{\rm bk,\odot}$, it only
reaches around [Fe/H]$\approx-0.2$ dex. Remarkably, at $R=8$ kpc, [Fe/H]
reaches zero around $t_{\rm bk}\approx 0$ Gyr (i.e. present-day), indicating
consistency with the established local ISM metallicity, which is slightly
subsolar based on B star population studies (Nieva & Przybilla, 2012; Feltzing
& Chiba, 2013). In contrast, the model value at $R\lesssim 6$ kpc reaches
[Fe/H] = 0 at $t_{\rm bk}\approx t_{\rm bk,\odot}$, suggesting that the Sun
was formed in this region.

To assess the accuracy of our Fe chemical evolution model, we compare the
metallicity distribution functions (MDFs) of stars at different radii, $R$,
with observational data. Fig. 2(b) illustrates the present-day MDF of stars at
$R=$ 4, 6, 8, and 10 kpc of the fiducial model. The local MDF in our model
(i.e. $R=8$ kpc; shown as the green line) has a peak around
[Fe/H]$\approx$-0.1 dex, which agrees well with the observed data of the local
disc stars (the black circles with error bars).

Fig. 2(b) also reveals that the MDFs at the inner radii (e.g. $R=4$ kpc; shown
as the red line) have peaks at richer metallicities, while the peaks at the
outer radii (e.g. $R=10$ kpc; shown as the yellow line) shift towards the more
metal-poor side. This is a natural outcome of the metallicity gradient shown
in Fig. 2(c), where the solid line represents the present-day [Fe/H] of the
ISM, while the symbols with error bars represent the observations of classical
Cepheids (Genovali et al., 2015; Luck, 2018) and young open clusters (Yong et
al., 2012; Netopil et al., 2016). Classical Cepheids, being younger stars with
an age of approximately 100 Myr or less, are thought to have abundances that
are representative of the current ISM. From this comparison, it is clear that
our model reproduces the observed [Fe/H] gradient with a negative gradient of
${\rm d[Fe/H]/d}R\approx-0.06~{}\rm dex~{}kpc^{-1}$ (e.g. Luck, 2018; Netopil
et al., 2016).

Nonetheless, the local MDF of stars in our models (green line in Fig. 2(b))
predicts fewer metal-rich stars than are observed. This is due to the lack of
incorporation of stellar radial migration, which is essential for reproducing
the metal-rich stars in the solar vicinity (e.g. Kordopatis et al., 2015;
Hayden et al., 2020; Tsujimoto, 2021). Additionally, while our model exhibits
skewed-negative MDFs at all radii, observations reveal a shift from a skew-
negative form in the inner region to a skew-positive form in the outer region,
with a weak skew-negative form in the solar neighbourhood (Hayden et al.,
2015). This behaviour is also explained by the effect of radial migration
(Hayden et al., 2015; Loebman et al., 2016; Johnson et al., 2021). Although
stellar migration has minimal impact on the means of the MDFs, it
significantly affects the spreads of the MDFs (e.g. Minchev et al., 2013;
Kubryk et al., 2015; Grand et al., 2015; Feuillet et al., 2018). Moreover, the
region with $R\gtrsim R_{\odot}$ is dominated by outward migrators from the
inner disc (i.e. metal-rich stars) than inward migrators from the outer disc
(e.g. Minchev et al., 2014; Prantzos et al., 2023). Despite the lack of
stellar migration in our model, the modelled local MDF successfully reproduces
the peak of the lower-[Fe/H] side of the observed local MDF, thus providing a
reasonable approximation of the observed MDFs.

Given that our model successfully satisfies the observational constraints on
the chemical evolution of Fe, we proceed to estimate the Sun’s birth radius,
$R_{\rm birth,\odot}$, based on our model. As shown Fig. 2(c), the dashed line
represents the [Fe/H] at $t_{\rm bk}=t_{\rm bk,\odot}$, indicating that at the
time of the Sun’s formation, [Fe/H] was zero at $R\approx 5$ kpc. Thus, we
infer that $R_{\rm birth,\odot}\approx 5$ kpc (depicted by the yellow circle).
This estimation is consistent with recent studies such as Tsujimoto & Baba
(2020) and Lu et al. (2022). Notably, Lu et al. (2022) employed a different
approach by estimating the temporal evolution of the radial profile of [Fe/H]
in the ISM based on observational constraints using the method proposed by
Minchev et al. (2018), without employing a GCE model similar to ours. Their
estimation resulted in $R_{\rm birth,\odot}\lesssim 5$ kpc (see also Ratcliffe
et al., 2023). Consequently, irrespective of the specific modelling approach
employed for the evolution of [Fe/H], a consistent conclusion of $R_{\rm
birth,\odot}\approx 5$ kpc has been obtained.

###  3.3 Estimating $R_{\rm birth,\odot}$ from evolution of O, Mg, Si and C

Figure 3:  Evolution of abundance ratios in the fiducial model as shown in
Fig. 2. Left column: Time evolution of the elemental abundances [X/Fe] for O,
Mg, Si, and C against [Fe/H], for four different radial zones, at 4 (red), 6
(blue), 8 (green), and 10 (yellow) kpc. The observational data sources are:
orange circles, fields stars from Zhao et al. (2016) and Sitnova et al.
(2015); blue triangles, classical cepheids from Luck (2018); red stars, fields
stars from Amarsi et al. (2019); brown squares, field stars from Takeda &
Honda (2005). The large circles (red) represent the Sun’s values. Centre
column: Time evolution of [X/Fe] for four different radial zones. Right
column: Radial profiles of the ISM elemental abundance [X/Fe] at $t_{\rm bk}=$
0 (solid) and $t_{\rm bk}=t_{\rm bk,\odot}$ (dashed). The yellow circles
represent the inferred Sun’s birth radius. The observational data sources are:
green diamonds, classical Cepheids from Luck (2018); red circles, classical
Cepheid from Genovali et al. (2015); grey triangles, young ($<1$ Gyr) open
clusters from Yong et al. (2012).

We next focus on estimating the Sun’s birth radius, $R_{\rm birth,\odot}$,
based on the chemical evolution of other elements, such as O, Mg, Si, and C.
The left panels of Fig. 3 exhibit the evolutional tracks of [Fe/H] vs. [X/Fe]
for O, Mg, Si and C at $R=$ 4, 6, 8, and 10 kpc in our fiducial model. During
the early stage of galaxy formation, only SNe II contribute, resulting in a
plateau of [X/Fe] ratios for O, Mg, and Si with a broad range of [Fe/H]. The
initiation of SNe Ia, characterized by ${\rm[Fe/H]}\sim-1$, results in an
increased production of iron compared to $\alpha$ elements, O, Mg, and Si,
causing a decline in [X/Fe] with increasing [Fe/H] (Matteucci & Greggio, 1986;
Andrews et al., 2017). The symbols indicate the observational data obtained
from homogeneous analysis of a substantial sample of nearby field stars
(Takeda & Honda, 2005; Zhao et al., 2016; Amarsi et al., 2019), classical
Cepheids (Genovali et al., 2015; Luck, 2018), and young open clusters (Yong et
al., 2012; Magrini et al., 2017). In regards to O, Mg, and Si, the observed
abundance ratios form a plateau of around 0.6, 0.3, and 0.4, respectively,
continuing until [Fe/H]$\sim-1$, where they sharply decrease. Our fiducial
model accurately reproduces these observational features.

In contrast, the evolutionary tracks of [C/Fe] vs. [Fe/H] in our fiducial
model are different from those of the $\alpha$ elements, because half of C in
the universe is produced by SNe II, while the rest is mainly produced by AGB
stars (e.g. Kobayashi et al., 2011). The [C/Fe] ratio is efficiently enhanced
by AGB stars because they do not produce Fe, and the contribution from these
stars creates a bump around ${\rm[Fe/H]}=-1$. We compare the model results
with the observations of nearby field stars and classical Cepheids. Note that,
according to Luck (2018), we use [(C+N)/Fe] instead of [C/Fe] for the
classical Cepheids, as the first dredge-up results in an increase in the
surface N abundance and a reduction in the surface C abundance, but the sum of
C+N is preserved (Iben, 1967). Our fiducial model is in good agreement with
these observational features.

Fig. 4 illustrates the [C/O] vs. [O/H] diagram. Our fiducial model reproduces
the upturn trend in [C/O] within the range of $\rm-1\lesssim[O/H]\lesssim 0$
satisfactorily. This pronounced increase in [C/O] is attributed to the onset
of feedback from AGB stars, which enhances C production compared to O as [O/H]
increases (e.g. Chiappini et al., 2003; Carigi et al., 2005). However, it is
worth noting that in the model, there is a trend of decreasing [C/O] for
$\rm[O/H]\gtrsim-0.5$, whereas this decline is not distinctly observed in the
data. If one were to consider this difference significant, it may suggest a
deficiency in the C supply process in our model for $\rm[O/H]\gtrsim 0$. As a
potential source of C in such metal-rich conditions, processes related to
stellar winds from rotating massive stars have been proposed (e.g. Romano et
al., 2020), but these processes have not been incorporated into our model. The
significance of carbon contributions from stellar winds, particularly in
high-z objects such as GN-z11, is expected to become more apparent through
observations using the James Webb Space Telescope (JWST) in the future (e.g.
Cameron et al., 2023; Bekki & Tsujimoto, 2023; Kobayashi & Ferrara, 2023).
However, it is crucial to acknowledge the considerable uncertainties
associated with C yields and the significant observational errors in
determining C abundance. As a result, the reliability of the model’s
predictions for C evolution is considered to be lower compared to the models
for O, Mg, and Si, which should be taken into account when interpreting the
results.

The central column of Fig. 3 depicts the time evolutions of the [X/Fe] ratios
in the ISM, representing the age-[X/Fe] relationship of stars in our model.
Our model demonstrates an increasing trend of these ratios with age (i.e.
$t_{\rm bk}$), which is expected as O, Mg, and Si are predominantly produced
in SNe II, while Fe is primarily synthesized in SNe Ia. Consequently, higher
[$\alpha$/Fe] ratios are observed at earlier stages of the Milky Way’s
evolution. These age–[$\alpha$/Fe] trends are consistent with the observed
trends (e.g. Haywood et al., 2013; Bensby et al., 2014; Lagarde et al., 2021;
Hayden et al., 2022). Furthermore, the [C/Fe] ratio also increases with age
until $t_{\rm bk}\lesssim 11$ Gyr, after which it exhibits a decreasing trend.
This behaviour can be attributed to the efficient production of C relative to
Fe by delayed AGB stars during the earlier stages of the Milky Way’s evolution
(e.g. Kobayashi et al., 2020) Consequently, [C/Fe] follows a similar
increasing trend with age as the $\alpha$-elements due to the lesser
contribution from SNe Ia.

Furthermore, the central column of Fig. 3 indicates that the [X/Fe] ratios in
the ISM are generally lower in the inner regions and higher in the outer
regions, suggesting the presence of a radial gradient in [X/Fe]. This is
further demonstrated in the right column of Fig. 3, which presents the radial
profiles of [X/Fe] ratios at $t_{\rm bk}=0$ (solid lines). Unlike the
behaviour of [Fe/H], the radial profiles of [X/Fe] in the model display
positive gradients with about 0.03, 0.03, 0.02, and 0.05 $\rm dex~{}kpc^{-1}$
for O, Mg, Si, and C, respectively. These positive gradients align with
expectations based on the inside-out formation scenario (e.g. Prantzos et al.,
2023) and are consistent with findings in observations of classical Cepheids
(Genovali et al., 2015; Luck, 2018) and young open clusters (Yong et al.,
2012).

Based on these radial distributions of [X/Fe], we estimate the Sun’s birth
radius ($R_{\rm birth,\odot}$), inferred as the radius where [X/Fe]$\approx$0
at $t_{\rm bk}=t_{\rm bk,\odot}$. In the central column of Fig. 3, the
vertical thick line in these panels represents the Sun’s birth time ($t_{\rm
bk,\odot}$), and it is evident that the model values at $R=8$ kpc (green) are
slightly higher than the solar ratio ([X/H] $=0$) at the Sun’s birth time,
while the model values at $R\lesssim 5$ kpc equal [X/Fe] $=0$ around 4.5 Gyr
ago. In the right column in Fig. 3, we display dashed lines indicating [X/Fe]
at $t_{\rm bk}=t_{\rm bk,\odot}$ and place yellow circles at the radii where
[X/Fe]=0. The inferred values for $R_{\rm birth,\odot}$ are 4.9 kpc, 5.3 kpc,
4.9 kpc, and 6.3 kpc for O, Mg, Si, and C, respectively. These estimates based
on [O/Fe], [Mg/Fe], and [Si/Fe] are consistent with the inferred value of 5
kpc obtained from [Fe/H] (see Section 3.2). However, the estimate based on
[C/Fe] suggests a slightly larger value of 6.3 kpc compared to other elements.
This discrepancy is not surprising given the lower reliability of the carbon
chemical evolution model results, as mentioned earlier.

Figure 4:  Evolution of [C/O] vs. [O/H] in the fiducial model as shown in Fig.
2. The solid lines represent time evolution for four different radial zones,
at 4 (red), 6 (blue), 8 (green), and 10 (yellow) kpc. The large circles (red)
represent the Sun’s value. The observational data sources are the same as
those used in Fig. 3.

##  4 Distributions of Planet Building Blocks in the Milky Way

Figure 5:  Central panel: $R$–$t_{\rm bk}$ plot of $f_{\rm cond}$ of the
fiducial model. The yellow star mark represents the expected birthplace of the
Sun, and the red star mark represents the current position of the Sun. Right
panel: Time evolution of $f_{\rm cond}$ for $R=4$ (dashed), 6 (solid), 8 (dot-
dashed), and 10 kpc (dotted). Top panel: Radial profiles of $f_{\rm cond}$ for
$t_{\rm bk}=0$ (dashed), 4.5 (solid), 6.5 (dot-dashed), and 8.5 Gyr (dotted).
Figure 6:  Same as Fig. 5, but for $f_{\rm iron}$ of the fiducial model.
Figure 7:  Same as Fig. 5, but for $f_{\rm water}$ of the fiducial model.

As described in Section 3, our fiducial GCE model successfully reproduces
various observational constraints on the chemical evolution of the Milky Way
galaxy, such as the AMR, the present-day MDFs, the abundance gradients, and
the [X/Fe] ratio trends. The consistent agreement between different elemental
ratios [X/Fe] and [Fe/H] further supports the notion that the Sun originated
from a region located within approximately 5 kpc of the Galactic disc. In this
section, we present the results of analyzing the temporal evolution of PBBs’
galactic-scale distribution using the stoichiometric model applied to our GCE
fiducial model.

###  4.1 Amount of PBBs in the Galactic-scale

In Fig. 5, we show the condensation (solid) mass fraction, $f_{\rm cond}$, as
function of the Galactocentric distance ($R$ in the $x$-axis) and look-back
time ($t_{\rm bk}$ in the $y$-axis). This quantity essentially scales with the
host star’s metallicity, [Fe/H]. The right panel displays the temporal
evolution of $f_{\rm cond}$ at $R=4$ (dashed), 6 (solid), 8 (dot-dased) and 10
kpc (dotted), while the upper panel shows the radial profiles of $f_{\rm
cond}$ at $t_{\rm bk}=0$ (dashed), 4.5 (solid), 6.5 (dot-dashed), and 8.5 Gyr
(dotted). We find that $f_{\rm cond}$ increases with time since the onset of
the formation of the Galactic disc (i.e. decreases with $t_{\rm bk}$),
reflecting the chemical evolution of the Galactic disc and the enrichment of
heavier elements necessary for planetary material formation. Furthermore, we
observe that $f_{\rm cond}$ decreases with increasing $R$, which is the
general outcome of the inside-out formation scenario of the Galactic disc
where the inner regions possess more planetary material.

Observationally, a positive correlation between the occurrence of giant gas
planets and the metallicity, [Fe/H], of the central star (e.g. Santos et al.,
2004; Fischer & Valenti, 2005) has been established. This is because stars
with greater [Fe/H] are more likely to produce more condensed material, which
is composed of the PBBs, and therefore, have a greater tendency to form
planets (e.g. Gáspár et al., 2016). The core accretion model (e.g. Pollack et
al., 1996; Kokubo & Ida, 2002) explains this scenario of planet formation.
Within the framework of the core accretion scenario in planet formation, our
model predicts that the occurrence of giant planets in the Milky Way will be
higher in the inner Galactic disc (or possibly bulge) than the outer regions
due to the distribution of planetary material.

###  4.2 Chemical compositions of PBBs in the Galactic-scale

We next investigate the expected chemical composition of the PBBs. Fig. 6
presents the iron-to-silicon mass fraction, $f_{\rm iron}$, as a function of
$R$ and $t_{\rm bk}$. Similar to $f_{\rm cond}$, $f_{\rm iron}$ exhibits an
increasing trend with time (i.e. decreases with $t_{\rm bk}$). In contrast to
$f_{\rm cont}$ and $f_{\rm iron}$, interestingly, the value of $f_{\rm water}$
is higher in the past and outer regions (Fig. 7).

These trends can be explained by eq.(5) and (6). Eq.(5) states that $f_{\rm
iron}\approx 1/(1+m_{\rm O+Mg+Si}/m_{\rm Fe})$, while eq.(6) approximates
$f_{\rm water}$ as $(m_{\rm O}/m_{\rm Fe})/(1+m_{\rm O+Mg+Si}/m_{\rm Fe})$.
Hence, $f_{\rm iron}$ decreases with [$\alpha$/Fe], whereas $f_{\rm water}$
increases with [$\alpha$/Fe]. On the other hand, observations of the
[$\alpha$/Fe] gradients in the Galactic disc are rather uncertain but
generally indicate a slightly positive gradient (typically $\sim 0.01~{}\rm
dex~{}kpc^{-1}$) based on studies of classical Cepheids and young open
clusters (e.g. Genovali et al., 2015; Luck, 2018; Yong et al., 2012). This is
primarily attributed to the slower progression of chemical evolution in the
outer regions compared to the inner regions, where the contribution from SNe
Ia is insufficient to significantly decrease [$\alpha$/Fe]. Consequently,
$f_{\rm iron}$ exhibits a negative radial gradient, while $f_{\rm water}$
shows a positive radial gradient.

Our model suggests that the iron2013silicate mass fraction, $f_{\rm iron}$,
increases towards the present (Fig. 6), with a greater value in the inner
region of the Milky Way. $f_{\rm iron}$ can serve as an indicator for the
metallic iron core2013to2013silicate mantle ratio in rocky planets (Dorn et
al., 2015; Unterborn & Panero, 2017). According to the internal evolution
scenario of rocky planets (Rubie et al., 2003), early differentiation led to
the formation of metallic iron cores and silicate mantles because metallic
iron is denser than any mantle silicate mineral at any pressure. The migration
of metallic iron towards the planetary core involves the melting of both the
metal and the silicate portion of the planet. Therefore, a higher or lower Fe
content observed in the spectrum of a star could suggest the presence of
larger or smaller metallic iron cores in rocky planets orbiting that star.
Additionally, our model suggests that planets in the outer Galactic disc are
more water-rich (Fig. 7), assuming $f_{\rm water}$ is a proxy of the total
amount of water available in the entire protoplanetary disc. Consequently, as
we move towards the present-day and inner regions of the Milky Way, it becomes
more likely for a planet to have a larger core but lower water content.

It is important to note that the stoichiometric model used here provides only
an estimate of the composition of PBBs based on the chemical composition of
the host star. It cannot directly determine the final composition or internal
structure of a final differentiated planet, which depends on various factors
such as the planet’s formation processes, as well as subsequent physical and
geological processes (e.g. Schneider & Bitsch, 2021). Additionally, $f_{\rm
water}$ represents the total amount of water available in the entire
protoplanetary disc, and not necessarily the amount present on individual
planets (e.g. Raymond et al., 2004; Raymond et al., 2018). Indeed, the water
mass fraction in the terrestrial planets of our Solar system varies widely
from planet to planet, and external processes such as giant impacts can
significantly alter the amount of water on a planet’s surface (e.g. Genda &
Abe, 2005; Genda, 2016). However, we expect a correlation, albeit with a large
scatter, between the iron-to-silicon ratio and water content of the entire
protoplanetary disc and the core-mantle ratio and water content of individual
planets.

###  4.3 Implications for origins of the Solar system

In this subsection, we compare our model predictions with observed solar
values, as shown in Figs. 5–7. The yellow star mark indicates the birth time
($t_{\rm bk,\odot}$) and Galactocentric radius ($R_{\rm birth,\odot}$) of the
Sun, while the green-coloured horizontal and vertical regions illustrate their
range. The red star mark denotes the current position of the Sun. Our results
indicate that if the Sun was born at a distance of approximately 5 kpc, our
predicted values of $f_{\rm cond}$, $f_{\rm iron}$ and $f_{\rm water}$ align
with observed values. This outcome is not surprising, as $R_{\rm
birth,\odot}\approx 5$ kpc achieved [Fe/H] $=0$ before 4.6 Gyr, and the
elemental abundance pattern corresponds to the solar composition.

The birth location of the Sun within the Milky Way galaxy has significant
implications for the structure and habitability of the Solar System. To
further explore the origin of the Solar System’s architecture in the context
of the Milky Way’s chemical evolution, we consider the possibility of the
Solar system forming at a location significantly different from $R_{\rm
birth,\odot}\approx 5$ kpc around 4.6 Gyr ago. If the Solar system had formed
at its current location of 8 kpc, our model predicts that $f_{\rm cond}$ would
be much lower than the actual value (Fig. 5). While the reduced amount of
planetary material in the Solar System may have an impact, other factors such
as protoplanetary disc conditions and planetary interactions also influence
the resulting structure of planetary systems (Raymond & Morbidelli, 2022, and
references therein). Therefore, the specific outcome relies on the initial
conditions of the Solar system and the disc’s evolution, and cannot be
definitively stated. However, this suggests that a protoplanetary disc with
low solid content could have impeded the formation of gas giant planets in the
Solar system (Ikoma et al., 2000; Kokubo & Ida, 2002; Mordasini et al., 2012;
Coleman & Nelson, 2016). Conversely, if the Solar system had formed well
within 5 kpc, an abundance of condensed solid matter could have led to the
formation of numerous gas giant planets, including hot Jupiters, due to inward
migration (Lin et al., 1996; Alibert et al., 2005; Ida & Lin, 2008). Such
migrations could significantly impact terrestrial planet formation and might
even lead to scenarios where Earth would not have existed, as it could have
been swallowed by the Sun (e.g. Coleman & Nelson, 2016)444 In the scenario of
terrestrial planet growth during and after giant planet migration, such
migration can serve as an accelerant for the formation of terrestrial planets,
and engender considerable radial homogenization of material throughout the
system (e.g. Raymond et al., 2006). Nonetheless, such planetary systems
exhibit a markedly disparate architectural structure compared to the solar
system. .

Regarding the iron-to-silicate mass fraction and water mass fraction, if the
Solar system had formed around its current location of 8 kpc, the predicted
$f_{\rm iron}$ and $f_{\rm water}$ would be significantly lower and higher,
respectively (Fig. 6 and 7). This suggests that rocky planets may have a small
metallic iron core and more water content if the Solar system were born around
its current location. The core-to-mantle ratio on rocky planets could
influence habitability by modifying mantle convection patterns, which in turn
could affect geological and plate tectonic activity, atmospheric composition,
and the stability of the long-term global carbon cycle (Noack et al., 2014;
Noack et al., 2017). However, a smaller core is unlikely to significantly
impact plate tectonic activity and greenhouse gas outgassing (see their Figs.7
and 8 in Noack et al., 2014). On the other hand, the presence of water in the
mantle significantly reduces silicate melting temperatures and mantle
viscosity, thereby influencing the mantle’s chemistry and promoting plate
tectonics (Korenaga, 2010). In the case of water-rich rocky planets, active
plate tectonics and abundant greenhouse gas emissions result in increased
surface temperatures, which could decrease the planet’s habitability for life.

Ultimately, the specific consequences depend on the details of the core and
mantle composition, as well as the internal structure and dynamics of the
planets. Nonetheless, our findings highlight the importance of considering the
broader Galactic context when examining the origins of the Solar system and
the habitability of planets. Future research should continue to explore the
connections between chemical compositions, planet formation processes, and
habitability to provide a more comprehensive understanding of planetary
systems within the Milky Way galaxy.

##  5 Discussion and Summary

In this study, we explored the origin of the Solar system within the context
of the Milky Way galaxy’s chemical evolution. For this purpose, we developed a
multi-zone Galactic chemical evolution (GCE) model, which takes into account
key elements for planet mineralogy, such as C, O, Mg, Si, and Fe. Our GCE
model successfully reproduced various observational constraints including the
stellar age-metallicity relation (AMR), metallicity distribution functions of
stars (MDFs), abundance gradients, and [X/Fe] ratio trends (Figs. 2 and 3).
Our GCE model led us to conclude that the Sun was formed in the inner Galactic
disc, approximately 5 kpc from the Galactic centre, which is in agreement with
recent studies (Tsujimoto & Baba, 2020; Lu et al., 2022; Ratcliffe et al.,
2023).

Our model suggests that the high star formation rate in the inner Galactic
disc played an important role in the formation and evolution of the Solar
system. Detailed analysis of meteorites reveals that the early Solar system
contained large amounts of short-lived radioactive isotopes, such as 60Fe and
26Al, produced mainly by supernova explosions and Wolf-Rayet winds. The
abundance ratios of these isotopes are much higher than those estimated from
average Galactic nucleosynthesis models (e.g. Nittler & Ciesla, 2016).
Interestingly, Fujimoto et al. (2020) provided an interpretation for these
high isotope abundance ratios by performing hydrodynamic simulations of the
entire Milky Way galaxy. They suggested that the Sun formed in an environment
with higher star formation rates than the galaxy-wide average (Desch et al.,
2023, for a review). This finding is consistent with our present results,
indicating that the Solar system formed in the inner Galactic disc within a
high star formation rate environment. Such insights have important
implications for understanding the formation and early history of the Solar
system.

Additionally, we combined a stoichiometric model with the GCE model to examine
the temporal evolution and spatial distribution of PBBs within the Milky Way
galaxy. Our models predicted higher occurrence rates of gas giant planets in
the inner Galactic disc and revealed trends in the iron-to-silicon mass
fraction and water mass fraction with time and towards the inner Galactic disc
regions (Figs. 5–7). Although the specific outcome relies on the initial
conditions and the protoplanetary disc’s evolution including the planet’s
formation processes, and subsequent physical and geological processes (e.g.
Schneider & Bitsch, 2021; Raymond et al., 2018; Raymond & Morbidelli, 2022),
our results suggest that the Sun’s birth location could influence the
architecture and habitability of the Solar system, demonstrating a strong
connection between variations in PBBs and resulting planetary systems in the
Sun’s birth location. Future research should continue exploring the
connections between chemical compositions, planet formation processes, and
habitability to provide a more comprehensive understanding of planetary
systems within the Milky Way galaxy.

However, our results on the Sun’s birth radius raise an essential question
about how the Sun migrated from its birthplace to its current location at
$R_{\odot}\approx 8$ kpc. For the sake of simplicity, by postulating that the
Milky Way galaxy maintains an axisymmetric structure, and through the backward
calculation of the Sun’s orbit over 4.6 Gyr predicated on the current velocity
vector of the Sun, the Sun’s birth radius is deduced to lie within the
approximate radial range of $R_{\odot}\pm 1$ kpc (e.g. Portegies Zwart, 2009).
This estimate is inconsistent with the $R_{\rm birth,\odot}\approx 5$ kpc
derived from our Galactic chemical evolution model. Consequently, for the Sun
to reach its current location from its birth radius, it must have undergone
orbital migration influenced by resonant interactions with the Galactic
spirals (Sellwood & Binney, 2002; Grand et al., 2012; Baba et al., 2013) and
the Galactic bar (Minchev & Famaey, 2010; Grand et al., 2015; Martínez-Barbosa
et al., 2015), as well as encounters with giant molecular clouds (Fujimoto et
al., 2023).

Nevertheless, it is not straightforward for the Sun, which was born around 5
kpc, to migrate to its current location around 8 kpc within 4.6 Gyr. Recent
estimates suggest that the co-rotation (CR) radius of the Galactic bar,
$R_{\rm CR,0}$, based on the bar pattern speed of around 35–40 $\rm
km\,s^{-1}\,kpc^{-1}$, is located around 6–6.5 kpc from the Galactic centre
(e.g. Portail et al., 2017; Sanders et al., 2019; Bovy et al., 2019; Asano et
al., 2020; Clarke & Gerhard, 2022; Li et al., 2022). If the Milky Way galaxy
already had a bar when the Sun was formed and its pattern speed is the same as
today, then the Sun was born well inside the current CR radius of the bar,
meaning $R_{\rm birth,\odot}<R_{\rm CR}(t_{\rm bk,\odot})=R_{\rm CR,0}$. In
such a scenario, the Sun would face significant challenges in moving beyond
the bar’s CR due to the low Jacobi energy (Binney & Tremaine, 2008). However,
if the Galactic bar formed after the Sun’s birth, the Sun’s orbital migration
may have been significantly influenced by the bar formation. $N$-body
simulations suggest that strong orbital migration of stars can be induced
during bar formation (Di Matteo et al., 2013; Khoperskov et al., 2020).
Alternatively, if the bar’s pattern speed was faster and its CR radius was
less than 5 kpc when the Sun was born (i.e., $R_{\rm CR}(t_{\rm
bk,\odot})<R_{\rm birth,\odot}$), the Sun might have migrated due to the
sweeping out caused by the bar’s slowdown (Ceverino & Klypin, 2007; Halle et
al., 2015, 2018; Khoperskov et al., 2020; Chiba et al., 2021; Tsujimoto &
Baba, 2020). Further investigation can provide a more detailed understanding
of the Sun’s migration by using numerical simulations that incorporate the
influences of the Galactic bar, spiral arms, and other relevant dynamical
processes. By comparing these simulations with observational data, we can
obtain a more precise understanding of the Sun’s migration history and its
implications for the architecture and habitability of the Solar system.

## Acknowledgements

We thank the anonymous referee for his/her constructive and helpful comments
which have improved the manuscript. We thank Yusuke Fujimoto for his valuable
comments. This work was supported by the Japan Society for the Promotion of
Science (JSPS) Grant Numbers 21H00054 and 21K03633. JB is supported by JSPS
KAKENHI Grant Number 22H01259. T. T. acknowledges the support by JSPS KAKENHI
Grant Nos. 18H01258, 19H05811, 22K18280, and 23H00132. T.R.S acknowledges the
support by JSPS KAKENHI Grant Nos. 21K03614, 22K03688, and 22H01259, and MEXT
as ‘Program for Promoting Researches on the Supercomputer Fugaku’ (Structure
and Evolution of the Universe Unraveled by Fusion of Simulation and AI; Grant
Number JPMXP1020230406).

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

##  Appendix A Yield corrections

In GCE studies, one of the primary sources of uncertainty stems from the
selection of stellar nucleosynthesis yields (e.g. Romano et al., 2010; Mollá
et al., 2015; Côté et al., 2017). While some earlier studies have utilized
theoretical yields derived from self-consistent stellar evolution calculations
(e.g. Woosley & Weaver, 1995; Kobayashi et al., 2006; Nomoto et al., 2013;
Chieffi & Limongi, 2004; Limongi & Chieffi, 2018; Prantzos et al., 2018;
Kobayashi et al., 2020, for SNe II), it is noteworthy that these results have
not converged, resulting in substantial uncertainty regarding the values of
these theoretical yields. This uncertainty arises from variations in how
certain processes, such as internal convection, rotation, and stellar winds,
are treated in stellar evolution models, as well as uncertainties in
nucleosynthesis reaction rates.

Fig. 8 illustrates the [Fe/H] vs. [X/Fe] evolution using different theoretical
SNe II yields while maintaining consistent AGB stars/SNIa yields with our
fiducial models. The solid green line represents the results obtained from our
fiducial model, which involves applying empirical correction factors to the
Nomoto et al. (2013) yields (see below). The dashed red lines represent the
results with the original Nomoto et al. (2013) yields. The dash-dotted blue
lines depict the outcomes when considering hypernova yields (with a
fraction=0.5) in addition to the Nomoto et al. (2013) yields. The dotted
magenta line corresponds to results using Chieffi & Limongi (2004) yields,
while the dot-dash-dotted orange lines represent results Limongi & Chieffi
(2018) yields. When theoretical yields are employed for GCE calculations, it
is commonly observed that not all elements can effectively replicate
observational data. Specifically, when considering the Solar System as a
typical reference, one would naturally expect solutions where [Fe/H]=0 to also
satisfy [X/Fe]=0. However, when relying on theoretical yields, such solutions
often do not align with observations.

In light of this, our study has taken a different approach. We have applied
empirical adjustments, utilizing constant factors, to the yields of each
element based on theoretical yields (Nomoto et al., 2013). The practice of
using such empirical yields is commonplace in GCE studies (e.g. Portinari et
al., 1998; François et al., 2004; Wiersma et al., 2009). As shown by the solid
green lines in Fig. 8, this has been done to construct a chemical evolution
pathway that aligns with observations of halo stars (i.e.
$\rm[Fe/H]\lesssim-2$) and the solar composition (i.e.
$\rm[Fe/H]\approx[X/Fe]\approx 0$). While François et al. (2004) introduced
mass-dependent correction factors rather than constants (e.g. as seen in their
fig.8), we have chosen to employ constant factors to maintain simplicity and
avoid unnecessary complexity. Additionally, it is worth noting that AGB stars
and SNIa yields also carry uncertainties, but for simplification purposes, we
have applied correction factors to SNe II yields only. Hence, in order to
minimize the influence of uncertainties stemming from theoretical yields and
observational errors, we have embraced this approach in our study.

Figure 8:  Time evolution of the elemental abundances [X/Fe] for O, Mg, Si,
and C against [Fe/H] for $R=8$ kpc in the models with different SNeII yields.
Solid (green) lines represent our fiducial model with the ad hoc prescription;
dashes (red) lines Nomoto et al. (2013) yields (N13); dash-dotted (blue) lines
Nomoto et al. (2013) with hypernovae (fraction is 0.5) yields (N13+HN); dotted
(magenta) line Chieffi & Limongi (2004) yields (CL04); dosh-dot-dotted
(orange) lines Limongi & Chieffi (2018) yields (LC18). The observational data
sources are the same as those used in Fig. 3. The large circles (red)
represent the Sun’s values.

##  Appendix B Parameter dependences

This section investigates the impact of uncertainties in the input parameters
on the GCE model results. Fig. 9 demonstrates the sensitivity of the model
parameters, $N_{\rm Ia}$ and $\tau_{\rm infall,\odot}$. The upper panels
display the evolution of the local AMR. The AMR is mainly influenced by the
value of $N_{\rm Ia}$ (left panel), but $\tau_{\rm infall,\odot}$ has a minor
effect (right panel). For the case of $N_{\rm Ia}\gtrsim 1.3\times
10^{-3}~{}{\rm M_{\rm\odot}}^{-1}$, [Fe/H]$\approx$0 at $R=8$ kpc around 4.5
Gyr ago. However, for the case of $N_{\rm Ia}=1.3\times 10^{-3}~{}{\rm
M_{\rm\odot}}^{-1}$, the MDF peaks at [Fe/H]$>0$, thereby indicating an overly
metal-rich environment (second left panel). Similarly, for $\tau_{\rm
infall,\odot}=5$ Gyr, the MDF peak exceeds zero, signalling an excessively
metal-rich condition.

The lower panels in Fig. 9 display the parameter dependencies of [Fe/H] vs.
[X/Fe] tracks for $R=8$ kpc. Considering [Mg/Fe] and [Si/Fe] does not
significantly deviate from the [O/Fe] evolution, we present only the [O/Fe]
evolution here. The results indicate that the [Fe/H] is most sensitive to
$N_{\rm Ia}$, while $\tau_{\rm infall,\odot}$ has a negligible effect.

The upper three rows of Fig. 10 demonstrate the radial dependencies of the
model parameters in the GCE model at $t_{\rm bk}=0$. In addition to $N_{\rm
Ia}$ and $\tau_{\rm infall,\odot}$, the influence of $\alpha_{\rm infall}$,
which substantially affects the radial gradient, is also represented. The
solid lines represent the outcomes of the fiducial model discussed in the main
text. The leftmost column illustrates the dependency on $N_{\rm Ia}$. As
expected, increasing the value of $N_{\rm Ia}$ does not modify the gradients
of [Fe/H], [O/Fe], and [C/Fe], but rather provokes a uniform shift along the
$y$-axis. Implementing $N_{\rm Ia}$ values of $1.0\times 10^{-3}$ or
$1.3\times 10^{-3}~{}{\rm M_{\rm\odot}}^{-1}$ leads to systematic deviations
from the observed values.

The centre column demonstrates the dependence on $\tau_{\rm infall,\odot}$.
When $\tau_{\rm infall,\odot}=5$ Gyr, the gradients of [Fe/H], [O/Fe], and
[C/Fe] become steeper, suggesting an accelerated chemical evolution in the
inner Galaxy. Although the cases with $\tau_{\rm infall,\odot}=5$ and 9 Gyr
align with the span of the observed [Fe/H] distribution, they also encompass
the distributions of [O/Fe] and [C/Fe] with satisfactory accuracy. The
rightmost column portrays the dependence on $\alpha_{\rm infall}$. Altering
$\alpha_{\rm infall}$ affects the gradient for the inner Galaxy ($R\lesssim 6$
kpc), while the outer regions remain relatively unaffected.

The lower panel of Fig. 10 illustrates the radial distribution of $f_{\rm
cond}$ and $f_{\rm water}$ at $t_{\rm bk}=t_{\rm bk,\odot}$, i.e. the Sun’s
birth time. The vertical axis represents the relative residuals from the
estimated values of the Solar system. Hence, the radii at which the relative
residuals of $f_{\rm cond}$ and $f_{\rm water}$ approach zero correspond to
the inferred radii for the birth of the Solar system. It is evident from the
figure that the fiducial model predicts radii of approximately $R_{\rm
birth,\odot}\approx 4.5$–5.5 kpc, where both $f_{\rm cond}$ and $f_{\rm
water}$ residuals approach zero.

For the cases of $N_{\rm Ia}=1.0\times 10^{-3}$ and $1.3\times 10^{-3}~{}{\rm
M_{\rm\odot}}^{-1}$ (bottom left panel of Fig. 10), the radii where $f_{\rm
cond}$ residuals cancel outfall around 5.5–6 kpc, while the radii where
$f_{\rm water}$ residuals nullify are distributed around 7–9.5 kpc. Hence, a
unified solution that nullifies both is absent. In the bottom centre panel of
Fig. 10, the variation of $\tau_{\rm infall}$ accelerates the chemical
evolution within the inner Galaxy, leading to significant changes in $f_{\rm
cond}$. Specifically, for $\tau_{\rm infall}=9$ Gyr, $R_{\rm
birth,\odot}\approx 4.5$ kpc, wheareas for $\tau_{\rm infall}=5$ Gyr, $R_{\rm
birth,\odot}\approx 6.5$ kpc. However, the MDF does not fit the observations
for $\tau_{\rm infall}=5$ Gyr, as shown in Fig. 9. The variations of
$\alpha_{\rm infall}$ have a minimal effect on the birth radius, with $R_{\rm
birth,\odot}\approx 5$–6 kpc (bottom right panel).

In conclusion, considering $N_{\rm Ia}=0.8\times 10^{-3}~{}{\rm
M_{\rm\odot}}^{-1}$ $\tau_{\rm infall}\gtrsim 7$ Gyr, and $\alpha_{\rm
infall}=0.5$–1.2 Gyr kpc-1, our model suggests that the Sun migrated from the
inner disc with a birth radius in the range of $4.5\lesssim R_{\rm
birth,\odot}\lesssim 6$ kpc. These results are consistent with the recent
estimations (e.g. Tsujimoto & Baba, 2020; Lu et al., 2022; Ratcliffe et al.,
2023).

Figure 9:  Dependence of $N_{\rm Ia}$ (left) and $\tau_{\rm infall,\odot}$
(right). The solid (red) lines indicate the fiducial model.  Figure 10:
Dependence of $N_{\rm Ia}$ (left), $\tau_{\rm infall,\odot}$ (center) and
$\alpha_{\rm infall}$ (right). The solid lines indicate the outcomes of the
fiducial model with $(N_{\rm Ia},\tau_{\rm infall},\alpha_{\rm
infall})=(0.8\times 10^{-3}~{}{\rm M_{\rm\odot}}^{-1},7~{}{\rm Gyr},0.7~{}{\rm
Gyr~{}kpc^{-1}})$.

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