# Galactic chemical evolution of the solar neighborhood, solar twins and
exoplanet indicators

Charles R. Cowley,1 and Kutluay Yüce2
1Department of Astronomy, University of Michigan, Ann Arbor, MI 48109-1107,
USA
2Department of Astronomy and Space Sciences, Faculty of Science, University of
Ankara, TR-06100 Tandoğan, Ankara, Turkey
E-mail<EMAIL_ADDRESS>(CRC)

(Accepted 2022 mar 4; Received 2022.Feb 17; in original form 2022 Jan 11)

###### Abstract

Galactic chemical evolution (GCE), solar analogues or twins, and peculiarities
of the solar composition with respect to the twins are inextricably related.
We examine GCE parameters from the literature and present newly derived values
using a quadratic fit that gives zero for a Solar age (i.e., 4.6 Gyr). We show
how the GCE parameters may be used not only to “correct” abundances to the
solar age, but to predict relative elemental abundances as a function of age.
We address the question of whether the solar abundances are depleted in
refractories and enhanced in volatiles and find that the answer is sensitive
to the selection of a representative standard. The best quality data sets do
not support the notion that the Sun is depleted in refractories and enhanced
in volatiles. A simple model allows us to estimate the amount of refractory-
rich material missing from the Sun or alternately added to the average solar
twin. The model gives between zero and 1.4 earth masses.

###### keywords:

Galaxy: solar neighborhood – stars: abundances – astrochemistry – techniques:
spectroscopy

††pubyear: 2015††pagerange: Galactic chemical evolution of the solar
neighborhood, solar twins and exoplanet indicators–6

##  1 Introduction

Galactic chemical evolution (GCE) refers to the relation between the
abundances of chemical elements and ages of the stars. Elemental abundances of
the Sun and its twins are the keystone to galactic chemical evolution in the
solar neighborhood–a volume defined by stars with mean distances of $\leq$
some 100 pc (Bedell et al., 2018). The elemental abundance is typically taken
as the logarithmic ratio of an element to a reference element, which is
usually hydrogen or iron: $[El/H]$ or $[El/Fe]$. The bracket notation
indicates that the stellar ratio is relative to a standard. That standard is
usually the Sun, although we explore the possibility of using some average of
nearby solar-type stars.

In the small volume within some 100 pc of the Sun, secular changes due to
stellar nucleosynthesis are modified by the diffusion of stars from the
interior and exterior of the Galaxy (Manea et al., 2022). Possible stream
remnants further complicate the picture (Matsuno et al., 2021), adding random
or stochastic components to any overall smooth relation between mean abundance
and age. Nevertheless a number of studies have described the behavior of
$[El/Fe]$ with stellar age by a simple linear relation (cf. Bedell et al.
(2018) and references below). We explore below the use of GCE parameters which
give zero for a star of the solar age.

##  2 The data

The present work is entirely based on published results of five studies of
stellar abundances and ages. Three of these, Bedell et al. (2018, henceforth,
BD18) which includes data from Spina et al. (2018), Nissen et al. (2020,
henceforth, NIS), and Liu et al. (2020, henceforth, LIU), give data for fewer
than 100 stars. Two, Brewer et al. (2016, henceforth, BR) and Delgado Mena et
al. (2017, 2019, henceforth, DM), have data for more than 1000 stars. Stellar
ages have been taken directly from the cited references.

All five works use the method of precision differential abundances (PDAs, cf.
Monroe et al. (2013). Abundance differences are based on ratios of abundances
from line pairs for each spectrum (e.g., Fe i or Fe ii). Individual stellar
properties do enter, as model atmospheres are used. However, the abundance
differences are more accurate than individual abundances determined separately
for each star, because uncertain factors such as oscillator strengths cancel.

Many papers relevant to PDAs, GCE, solar twins, and $T_{\rm c}$ correlations
are not examined here. A sample list includes: Adibekyan et al. (2012),
Adibekyan, et al. (2014), Adibekyan et al. (2015),Bensby et al. (2014),
Battistini & Bensby (2016), Casali et al. (2020), Gonzalez et al. (2010),
González Hernández et al. (2013), Ramirez et al. (2009), Ramírez et al.
(2014a), Spina et al. (2016b), and Ramírez et al. (2014b). The latter
reference is of interest because it lists all but one of the BD18 stars in a
solar twin search. Additional references may be found in the exhaustive study
of the solar twin problem by Yana Galarza et al. (2021).

We focus here on the BD18 work for two reasons. First, that study includes 30
elements from carbon to dysprosium, significantly more elements than in the
other studies considered here. Second, we find the accuracy of the abundances
is at least as good as the best of the others. The NIS survey has 13 elements,
C-Ni, Sr and Y. LIU has 17 elements C-Zn, Sr, Y, Zr, Ba-Dy. BR and DM are
discussed below.

Tab. 1 will give some insight into the relative accuracy of the five studies.
We give averages for the absolute value of the pairwise abundance differences
for carbon, iron, aluminium, and silicon for stars in common. The mean and/or
standard deviations of the differences in Tab.1 indicate that uncertainties
range from the 0.01 dex level to somewhat more than 0.1 dex for some elements.
The BD18 and NIS differences are the smallest.

Table 1: Abundance absolute magnitude differences for $[C/H],[Fe/H],[Al/H]$,
and $[Si/H]$ from five studies

|BD18-BR|
---
| C | Fe | Al | Si
Mean | 0.044 | 0.013 | 0.067 | 0.018
St. Dev. | 0.032 | 0.009 | 0.138 | 0.023
No. Stars | 15 | 15 | 15 | 15
|BD18-DM|
Mean |  | 0.112 | 0.127 | 0.107
St. Dev. |  | 0.053 | 0.093 | 0.020
No. Stars |  | 8 | 8 | 8
|LIU-NIS|
Mean | 0.073 | 0.040 | 0.080 | 0.025
St. Dev. | 0.084 | 0.056 | 0.085 | 0.025
No. Stars | 3 | 4 | 4 | 4
|BD18-NIS|
Mean | 0.022 | 0.007 | 0.010 | 0.008
St. Dev. | 0.011 | 0.005 | 0.009 | 0.005
No. Stars | 22 | 22 | 22 | 22

  * •

No carbon abundances in DM

  * •

Carbon is an outlier for LIU-NIS in HD 4915 and is omitted

The BR and DM stellar samples are an order of magnitude more numerous (BR:
1615; DM: 1059) than BD18 and significantly more diverse in $T_{\rm eff}$,
$\log(g)$, and composition. The BR stars have 15 elements, including nitrogen,
missing from the other surveys. DM also has 15 elements: the refractories Mg -
Ti and the intermediate volatiles Cu and Zn as well as Ba, Ce, Nd, and Eu.

The relation between stellar abundance and age for these surveys is complex.
Plots of abundance versus age reveal “families” of stars defined by common
spatial (thin or thick disk) or chemical (metal- or alpha-rich) properties
(see Fig. 7 of DM). The BD18 data constitute a spatially and chemically more
homogeneous group than the larger samples. Similar remarks apply to the NIS
and LIU data.

##  3 Galactic Chemical Evolution–plots and parameters

Four of the five studies considered here, excepting BR, have obtained linear
relations between $[El/Fe]$ and stellar age. We report values of GCE
parameters for the BR stars based on a selected set of 160 stars described in
Tab. 3 (see note to Col. 11).

The most common GCE parameters are the slope and intercept of a linear fit, or
sometimes, a broken linear fit to a plots of $[El/Fe]$ vs. age such as in Fig.
1.

The elemental abundances of a star of a given age may considered as due to
smooth, secular changes (GCE) and other, peculiar, causes such as accretion
from a nearby supernova or the ingestion of an earthlike planet. These
parameters may be used to compare the peculiar abundances of stars that are
not due to the secular effect of GCE. If we want to compare the peculiar
abundances of solar twins to the Sun, we must remove the differences due to
GCE and obtain the abundances the star would have (had) if its age were 4.6
Gyr, the age of the Sun (Bonanno et al., 2002). We refer to these adjustments
as “corrections”. Since the current linear fits do not give precisely zero
corrections for 4.6 Gyr, we have explored two relations that do give precisely
zero for the solar age. The first is a constrained linear (CL) relation with
only one parameter. Its fits have significantly larger variances than the two-
parameter unconstrained linear (UL) fits; it is therefore not useful. The
second is a constrained quadratic (CQ), which like the UL fit has two free
parameters.

The CQ fit, as well as the broken linear fits, allow for a change of slope
with age. This can be of importance when attempting to make a correction for
GCE, which depends on the slope of $[El/Fe]$ vs. age. The corrections add (or
subtract) an amount from the $[El/Fe]$. For example, the BD18 GCE parameters,
$m(age)$ and $b$ are used in

$[El/Fe]_{\rm corrected}=[El/Fe]-[m(age)+b]$ (1)

This linear correction does not allow for a change in slope $m$, so if the
true slope does change the "correction" could be in the wrong direction for
some age range!

The GCE plots of silicon and sodium are shown in Fig. 1. They represent the
typical monotonic behavior of the abundances with age, and the occasional
element for which the plot changes slope. A liberal uncertainty estimate for
the points in the vertical direction is 0.022 dex, compatible with entry in
Tab. 1, the mean of $|BD18-NIS|$ for carbon. The symbol in the upper left of
the diagrams of Fig. 1 represents these abundance uncertainties. The age
uncertainty is taken as 1.4 Gyr which is based on the standard deviation of
independent age determinations of NIS and Spina et al. (2018). We present
similar plots for all 30 elements and 79 stars of the BD18 study in the files
PlotsofEloFe* at https://doi.org/10.5281/zenodo.6077735. The archive contains
both PowerPoint (pptx) and pdf versions.

Figure 1: $[Si/Fe]$ and $[Na/Fe]$ vs. Age for 68 BD18 stars in Gyr. Blue:
unconstrained linear (UL) fit; Black: constrained linear (CL) fit; Orange:
constrained quadratic (CQ) fit. In these plots, the standard abundance is not
that of the Sun but the average of the 12 BD18 twins of table 2. Selected
“twins” are plotted as red stars. See text for further discussion.

The CQ parameters are derived by least squares fits to plots such as those of
Fig. 1 and have the form

$[El/Fe]=aa\cdot({\rm age}-4.6)+bb\cdot({\rm age}-4.6)^{2},$ (2)

where $aa$ and $bb$ are constants for individual elements. Age is in units of
Gyr. The constant $aa$ is the slope at age 4.6 Gyr.

Values of the variance ratios for the UL and CQ fits are remarkably close to
unity for the BD18, NIS, and LIU samples. Most have values between 0.95 and
1.1, and none indicate a strong preference for the UL or CQ fits.

Plots similar to those of Fig. 1 were shown by da Silva et al. (2012) and
Spina et al. (2016a). In the latter paper, three fitting techniques were
employed: a simple linear and a broken linear fit as well as a hyperbolic fit
(see their Fig. 1). Most of those hyperbolic fits are in qualitative agreement
with those in PlotsofEloFe on zenodo: Na, Ca, Co, Ni, Cu, and Zn.

Table 2: Selected twins from BD18 stars and parameters HD | HIP | Te (K) | $\log{g}$ | $[Fe/H]$ | age (Gyr)
---|---|---|---|---|---
13357 | 10175 | 5719 | 4.485 | -0.028 | 3.2
16008 | 11915 | 5769 | 4.48 | -0.067 | 3.4
36152 | 25670 | 5760 | 4.42 | 0.054 | 4.9
88072 | 49756 | 5789 | 4.435 | 0.023 | 4.5
115169 | 64713 | 5788 | 4.435 | -0.043 | 5.3
124523 | 69645 | 5751 | 4.435 | -0.026 | 5.3
138573 | 76114 | 5740 | 4.41 | -0.024 | 6.6
146233 | 79672 | 5808 | 4.44 | 0.041 | 4.0
145927 | 79715 | 5816 | 4.38 | -0.037 | 6.1
167060 | 89650 | 5851 | 4.415 | -0.015 | 4.6
183658 | 95962 | 5805 | 4.38 | 0.029 | 6.0
200633 | 104045 | 5826 | 4.41 | 0.051 | 4.5
BD18 stars | averages | 5785. | 4.427 | -0.0035 | 4.86

NIS (see their Fig. 4) has plots of $[El/Fe]$ for 12 elements vs. age similar
to our Fig. 1. The color coding of these plots emphasizes their conclusion
that the sample consists of two groups with distinct chemical history. The
first group consists mostly of stars that are younger than the Sun, while
stars of the second group are mostly older. This is consistent with the
morphology of our plots of $[El/H]$ vs. atomic number, $Z$, which may be
downloaded from zenodo. An examination of those plots of the BD18 data shows
that most of the stars with positive overall shapes (POS) are younger than the
Sun, while those with the V-shape are older (see Fig. 2).

The overall shapes of the NIS plots (see their Fig. 4) generally agree with
our CQ fits, but not entirely. For example, the plot for $[Na/Fe]$ vs. age
would support our negative curvature, except for high outliers which are
unusually sodium rich: HD 1461, HD 66428, HD 111031, HD 134487, HD 134606, HD
169691, and HD 204313. These same stars lie below the mean of the
$[O/Fe]$-plot; alternately, they are part of a separate population if broken
GCE lines are adopted.

Figs. 4, 5, and 6 of LIU show $[El/Fe]$ vs age. We examined similar plots for
their 68 “comparison” stars only, finding little convincing evidence of
curvature, even for Na and Cu. Only 4 of the LIU stars were in the BD18 and
NIS surveys.

The larger BR and DM samples are broader in the temperature-gravity spread.
The DM stars are classified into populations. Most, 882, are thin disk. There
are 108 thick disk, 60 high-$\alpha$, and 9 halo stars. This mixture causes
apparent clustering in certain age ranges, often somewhat displaced from the
general trend. In addition to DM’s published GCE parameters, we determined new
UL ones based on severely limited samples (see Tab. 3 footnote to Col. 6). The
73 DM stars had all been classified as thin disk. BR does not explicitly
classify its stars into populations, but notes the occurrence of $\alpha$-rich
stars at sub-solar metalicities. Selection criterior for the BR stars are in
the table footnotes to Cols. 11 and 12.

We made GCE fits to $[El/Fe]$ vs. age for the BR and DM samples. Without
careful sample selection, the results show a considerable scatter due to the
inclusion of populations that were purged in the BD18 sample. Different
populations are readily seen in DM’s colorful plots of $[El/Fe]$ vs age (see
Delgado Mena et al., 2019), Fig. 7.

Unlike the other four surveys, BR concentrated on the techniques and results
of abundance determinations; GCE parameters were not obtained. We obtain them
here for the limited sample of 160 BR stars.

In Tab.3 we compare published and newly computed slopes of the GCE relations
from BD18, NIS, LIU and selections of stars from BR and DM. All newly derived
parameters are from unweighted least squares fits. All of the stars are from
the thin disk. We followed BD18 and excluded 11 of their 79 stars that are
old, $\alpha$-rich or of the thick disk population. Thus the plots of Fig. 1
and in the zenodo archive are of the 68 thin disk stars. Similarly in Nissen
(2015)’s study of 21 stars, they excluded 3 $\alpha$-enhanced stars from their
GCE fits. In addition to the slopes from the 2015 paper, we present newly
derived UL slopes based on the more recent Nissen et al. (2020) results, one
determined using 71 stars excluding those with $[Fe/H]\geq 0.15$, (Col. 8),
and 40 (Col. 9) stars also excluding those with $|[Fe/H]|\geq 0.15$. The Liu
et al. (2020) figures are for the 68 “comparison stars” taken from that paper.

New UL coefficients for 73 DM stars are based on the criteria specified in 3,
Col. 6. The age accuracy condition was adopted from DM and also applies to
Cols. 5 and 6 of Tab. 3.

UL and CQ coefficients were derived in the present study from 160 stars of the
BR study (Cols. 11 and 12). The 1615 BR stars were filtered as described in
the supplement to Tab. 3.

The BR survey was the only one of the five discussed here to determine an
abundance for the element nitrogen. This element is critical to the search for
correlations of $[El/Fe]$ with $T_{\rm c}$, as there are only two other highly
volatile elements that are spectroscopically available, carbon and oxygen. The
$[N/Fe]$ ratio was included in the extensive examination of carbon isotopes
and elemental nitrogen by Botelho et al. (2020). Fig. 5 of that study shows
$[C/Fe]$ and $[N/Fe]$ vs. age; their broken-linear fit for nitrogen is in
agreement with our CQ fit to the BR data–positive slope for the younger stars,
and negative for the older.

Although the slopes of Tab. 3 differ numerically, they generally agree as
summarized here:

  1. 1.

For the lighter elements, the slopes are positive or small (e.g. Ca). Positive
slopes indicate an increase in the $[El/Fe]$-ratio for the older stars.

  2. 2.

Beyond the iron-group (3d) elements the slopes are mostly negative, indicating
an increase in $[El/Fe]$ for the youngest stars.

  3. 3.

The slope parameter for exceptions to items (1) and (2), such as for Ca or Cr
are small.

  4. 4.

The triplet of neutron-addition (nA) elements, Sr, Y, and Zr has very steep
slopes in the BD18 data, and NIS data for Y.

  5. 5.

The heaviest nA elements, from Ba to Dy all have negative slopes in the 68
BD18 stars. The DM slopes from Ba to Dy are mixed.

Table 3: GCE slopes for elements. CQ constants in Columns 2 and 3

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---
el | aa | bb | BD18, m | Fig.7(DM) | DM(UL) | Nissen | Nissen | Nissen | Liu | Brew(CQ) | Brew(UL)
C | 0.01481 | -0.00227 | 0.0115 | 0.0187 |  | 0.0139 | 0.0162 | 0.0147 | 0.013 | 0.0091 | 0.0084
N |  |  |  |  |  |  |  |  |  | 0.0093 | 0.0090
O | 0.00849 | 0.00002 | 0.0088 |  |  | 0.0161 | 0.0201 | 0.0172 | 0.002 | 0.0141 | 0.0137
Na | 0.00651 | -0.00326 | 0.0086 |  |  | 0.0098 | 0.0064 | 0.0071 | 0.001 | 0.0161 | 0.0176
Mg | 0.01027 | -0.00054 | 0.0099 | 0.02059 | 0.0149 | 0.0089 | 0.0126 | 0.01 | 0.011 | 0.0145 | 0.0149
Al | 0.01424 | 0.00007 | 0.0139 | 0.0126 | 0.0113 | 0.0167 | 0.0168 | 0.0163 | 0.033 | 0.0255 | 0.0272
Si | 0.00578 | -0.00086 | 0.0063 | 0.0075 | 0.0073 | 0.0006 | 0.0079 | 0.0065 | 0.000 | 0.0098 | 0.0104
S | 0.00757 | -0.00194 | 0.0098 |  |  | 0.0007 |  |  | 0.003 |  |
Ca | -0.00045 | 0.00105 | -0.0011 | 0.00616 | -0.0005 | -0.0001 | 0.0026 | 0.0002 | -0.002 | -0.0016 | -0.0016
Sc | 0.00605 | -0.00015 | 0.0059 |  |  |  |  |  | 0.024 |  |
Ti | 0.00418 | 0.00053 | 0.0036 | 0.01136 | 0.0048 | 0.0063 | 0.0087 | 0.0062 | 0.007 | 0.0088 | 0.0079
V | 0.00122 | 0.00010 | 0.0013 |  |  |  |  |  | -0.013 | 0.0058 | 0.0043
Cr | -0.00131 | 0.00012 | -0.0016 |  |  | -0.0026 | -0.0014 | -0.0018 | 0.001 | -0.0023 | -0.0031
Mn | 0.00075 | -0.00235 | 0.0023 |  |  |  |  |  | -0.012 | 0.0066 | 0.0072
Co | 0.00628 | -0.00176 | 0.0074 |  |  |  |  |  | 0.007 |  |
Ni | 0.00575 | -0.00218 | 0.0071 |  |  | 0.0048 | 0.0033 | 0.0041 | 0.007 | 0.0116 | 0.0124
Cu | 0.01442 | -0.00286 | 0.0149 | 0.00609 | 0.0112 |  |  |  |  |  |
Zn | 0.00932 | -0.00161 | 0.0102 | 0.01413 | 0.0121 | 0.0117 |  |  | 0.027 |  |
Sr | -0.02749 | -0.00096 | -0.0251 | -0.01922 | -0.0255 |  | -0.0172 | -0.0211 |  |  |
Y | -0.02581 | -0.00106 | -0.0238 | -0.01658 | -0.0223 | -0.033 | -0.0207 | -0.0233 | -0.007 | -0.0159 | -0.0149
Zr | -0.02359 | -0.00088 | -0.0219 | -0.00787 | -0.0165 |  |  |  |  |  |
Ba | -0.02898 | 0.00441 | -0.0317 | -0.01193 | -0.0196 |  |  |  |  |  |
La | -0.02196 | 0.00155 | -0.0227 |  |  |  |  |  |  |  |
Ce | -0.02007 | 0.00282 | -0.0220 | -0.00525 | -0.0079 |  | -0.0021 |  |  |  |
Pr | -0.01020 | 0.00028 | -0.0103 |  |  |  |  |  |  |  |
Nd | -0.01800 | 0.00271 | -0.0198 | 0.00767 | -0.0021 |  |  |  |  |  |
Sm | -0.00727 | 0.00149 | -0.0077 |  |  |  |  |  |  |  |
Eu | -0.00583 | -0.00076 | -0.0056 | 0.01058 | 0.0085 |  |  |  |  |  |
Gd | -0.00616 | -0.00013 | -0.0060 |  |  |  |  |  |  |  |
Dy | -0.00780 | 0.00234 | -0.0073 |  |  |  |  |  |  |  |

  * •

(Col. 1) Element

  * •

(Col. 2) $aa$, CQ slope at 4.6 Gyr, Eq. 3, 68 stars, this work

  * •

(Col. 3) $bb$, CQ constant, Eq. 3, 68 stars, this work

  * •

(Col. 4) UL slope $m(t)$ taken directly from BD18 Table 3, 68 stars

  * •

(Col. 5) DM GCE parameters a and b from Figure 7, for about 270 thin disk
stars with age uncertainties $\leq$ 1.5 Gyr

  * •

(Col. 6) 73 selected DM stars; age uncertainty $\leq 1.5$ Gyr, age < 9 Gyr,
5480 K < $T_{\rm eff}$< 6080 K, 4.24 < log($g$)< 4.64

  * •

(Col. 7) Nissen et al. (2015) Table 6, 18 stars

  * •

(Col. 8) Nissen et al. (2020), Vizier Cat., 71 stars

  * •

(Col. 9) Same as 8, but -0.15 < [Fe/H] < 0.15, 40 stars

  * •

(Col. 10) Liu, et al. (2020), Figs. 4, 5, & 6, 68 stars

  * •

(Col. 11) Brewer, et al. (2016), 160 stars, CQ fit; age < 9.0 Gyr, 5400K <
$T_{\rm eff}$< 6400K, 4.24 < log($g$), and -0.15 < $[Fe/H]$ < 0.15.

  * •

(Col. 12) Brewer, et al. (2016), 160 stars, UL fit, same restrictions as CQ
fit.

In Sec. 4 below, we consider the question of whether the abundance deviations
from solar values are correlated with elemental condensation temperatures,
$T_{\rm c}$ (Lodders, 2003). Many studies have looked for such correlations
(see references in Sec. 2).

The chemical elements with stellar abundances have an obvious correlation of
$[El/Fe]$ with GCE (Adibekyan, et al., 2014). Basically, the volatile elements
carbon and oxygen are more abundant relative to iron, silicon and other
refractories in older stars. Thus one should correct a star’s abundance for
GCE prior to looking for a correlation with $T_{\rm c}$: The first part of Eq.
3 describes the UL fit, while the second gives tha CQ fit, where the
correction vanishes for 4.6 Gyr.

$\displaystyle[El/Fe]_{\rm corr}$ $\displaystyle=$ $\displaystyle[El/Fe]_{\rm
obs}-[m*{\rm age}+b],{\rm or}$ (3) $\displaystyle=$ $\displaystyle[El/Fe]_{\rm
obs}-[aa\cdot({\rm age}-4.6)+$ $\displaystyle bb\cdot({\rm age}-4.6)^{2}].$

The constants $m$, $b$, $aa$, and $bb$ belong to each element.

But these GCE coefficients have additional meaning. They give the mean
abundance for the sample as a function of age. A motion picture display may be
seen in the iPoster from the 235th meeting of the American Astronomical
Society
111https://aas235-aas.ipostersessions.com/?s=58-BE-9F-2E-6D-57-C6-85-88-1E-B6-1D-29-37-EC-8E
that shows the average abundance patterns $[El/H]$ as a function of atomic
number. Ten epochs are shown in steps of 1 Gyr based on the BD18 UL parameters
of their Tab. 3. They clearly illustrate the transition from a positive
overall slope (POS) in young stars to a V-shaped configuration in the oldest
stars. Those two basic shapes are illustrated by the squares (and stars) in
Fig. 2.

A dominant feature of these plots is the migration of the triplet Sr, Y, Zr
from high values in young to low values in old stars. While Sr, Y, Zr move
downward with age, the lighter refractories, Mg, Al, Si (black dots between Z
= 11, Na, and Z = 16, S) move up slightly. This accounts for the ratio
$[Y/Mg]$ used as a Galactic clock (da Silva et al., 2012; Magrini et al.,
2021), but also shows that the behavior of these two elements is closely
related to their neighbors in Z.

Most of the downward migration of the Sr, Y, Zr triplet takes place within 2
Gyr of the Sun’s age. This could be related to the “bi-modality” of s-process
production discussed by Kamath et al. (2021), as well as the two populations
discussed by NIS.

In the young stars, with POS-type slopes the elements Mg-Zn align nearly
perpendicular to the overall slope. In the older stars, this line rotates
slightly counter clockwise, nearly as a whole. In the oldest stars, the
heaviest elements of this group, Cu and Zn, disconnect, and move upward. This
is due primarily to eleven stars that were purged from the original BD18 list
of 79 stars but are included in the file BD18stars_plots in the zenodo
archive.

We now compare the differential abundances of individual stars with the mean
values computed from the CQ GCE coefficients. Fig. 2 shows two examples–of
relatively young (HIP 101905) and old (HIP 15527) stars. The star symbols
represent the elemental differential abundances for individual stars.
Ordinates (y-values) for the squares are given by Eq. 3; we show the
$Z$-dependence of the GCE coefficients explicitly:

$\displaystyle y(Z,{\rm age})$ $\displaystyle=$ $\displaystyle<[El/Fe]>_{\rm
ave}$ $\displaystyle=$ $\displaystyle aa(Z)\cdot({\rm
age}-4.6)+bb(Z)\cdot({\rm age}-4.6)^{2}$

The young-star abundances display the expected pattern– POS, both observed
(stars) and predicted (squares). The older star shows the typical V-shape, due
to the strong drop for the intermediate s-process-dominated elements Sr, Y,
and Zr.

Figure 2: [El/Fe] vs. Z for 2 BD18 stars. Stars represent observed abundances
while squares come from the CQ “average” fits. Alpha elements, C, O, Mg, Si,
S, and Ca are shown in red. The black mark in the upper-left corner represents
the uncertainty in $[El/Fe]$ for the star points, roughly 0.02 dex. The
squares are computed directly from the CQ formula and do not have a similar
uncertainty. See text for further discussion.

In Fig. 2 (left), the iron group (3d) elements follow the means (squares). The
alpha elements scatter from the means (squares): C, O and S are about 0.04 dex
low, while Mg, Si, and Ca are approximately fit. All of the neutron addition
(nA) elements are significantly higher than the means.

Abundances in the older star fit the means more closely. Individual stars have
their own usually small departures from the mean abundances for their age.
Plots of this kind may be seen for all the BD18 stars in ObsandPrePlots in the
zenodo archive. It will be seen that deviations from the average patterns are
common, and occasionally as large as 0.1 dex.

##  4 Solar twins, condensation, and engulfment

Meléndez et al. (2009, henceforth, MEL09) showed that the Sun may not be the
best chemical sample of its neighborhood. Their Fig. 3 is a good illustration,
where logarithmic differential abundances–Sun minus twins–are plotted against
condensation temperature, $T_{\rm c}$. This plot shows the highly volatile
element C ($T_{\rm c}$ = 40K) with an excess of about 0.05 dex. The highly
refractive Al ($T_{\rm c}$ = 1641K) has a depletion of about 0.03 dex. The
extremes of about 0.08 dex, are comparable to the differences of Tab. 1,
excepting BD18 and NIS. The corresponding slope, $-5\cdot 10^{-5}{\rm dex\
deg}^{-1}$ is similar in magnitude to those of (suspected and) confirmed
planet host stars (see LIU Table 5, Column 2).

Numerous workers have sought an optimum sample of representative stars for the
solar neighborhood. The topic was reviewed in detail by Yana Galarza et al.
(2021), who discuss new criteria based on Gaia data, geochronology, and
magnetic activity.

Our “twins” were selected with four parameters: $T_{\rm eff}$, $\log{(g)}$,
$[Fe/H]$, and age, taken from the original papers (BD18, NIS, LIU, BR, and
DM). A dozen stars were were chosen based on minimum Euclidian distances from
the Sun in these parameters. Prior to calculating the distances, the variables
were normalized to fall between -1 and +1. We selected the 12 stars with the
smallest distances in hyperspace from the Sun for each of the surveys.

Average values of $[El/Fe]$ for the nA elements from carbon to zinc were
obtained. This average of the logarithms is not the same as the logarithm of
the (non-logarithmic) average (Bedell et al., 2018, see Eq. 1), but differs,
typically, by one or two units in the third decimal, which is unimportant for
our purposes. Note that averaging logs rather than taking an average of
antilogs has the advantage of weakening the effect of outliers.

The BD18 twins are given in Tab. 2; the remaining sets are in Tab 4.

A number of very impressive abundance surveys are related to correlations of
differential abundances with $T_{\rm c}$ as an indication of planetary
engulfment (see references in Sec. 2).

The study by Spina et al. (2021) argues strongly that engulfment is the most
likely explanation for many such correlations. However, they present an
argument independent of $T_{\rm c}$, essentially based on the probability of
abundance anomalies due to engulfment being more prominant in stars with
higher effective temperatures with thinner convection zones.

Relative elemental abundance averages for each of the five surveys were
plotted as a function of condensation temperature, $T_{\rm eff}$ (Lodders,
2003). Here, the relative abundances are for the Sun, with respect to the mean
for the twins of Tab. 2 and Tab. 4. The average twin $[El/Fe]$ relative to the
Sun and corrected with CQ for GCE we designati $[El/Fe]_{\rm twc}$, for "twin,
corrected":

$\displaystyle[El/Fe]_{\rm twc}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\sum_{i=1}^{12}[El/H]_{\rm BD18}-[Fe/H]_{\rm BD18}$
(5) $\displaystyle-$ $\displaystyle\left(aa\cdot({\rm age}-4.6)+bb\cdot({\rm
age}-4.6)^{2}\right)$

The plots of Fig. 3 follow the orientation of MEL09, so that deficiencies of
refractory elements in the Sun result in a downward slope with increasing
$T_{\rm c}$.

Only elements from carbon to zinc were used. We excluded the neutron-addition
(nA) elements from consideration in the search for $T_{\rm c}$ correlations
for reasons noted by Meléndez et al. (2014), and adopted by Cowley et al.
(2020) in their survey of the BD18 sample. Many examples may be seen in the
plots of the BD18 abundances vs. $T_{\rm c}$, where the nA abundances scatter
sometimes above (HIP 8507) and sometimes below (HIP 9349) the trends for the
lighter elements (see TccorGIF1 in the zenodo archive). This is evidence that
the neutron-addition elements have a distinct GCE from C-Zn, as argued by
Meléndez et al. (2014).

The BD18 survey has abundances for 18 elements, from C to Zn, plus 12 nA
elements not used here. BR has one such, element, Y, and 14 from C to Ni,
including the ultravolatile, N, which is unavailable in the other two surveys.
DM has only the intermediate volatiles Cu and Zn, and the refractories Mg -
Fe.

Twins for BR, DM, NIS, and LIU in Tab. 4 are listed in order of their distance
in hyperspace from the Sun. Thus, of the BR twins, HD 9407 is closer to the
Sun in $T_{\rm eff}$, $log(g)$, $[Fe/H]$, and age than HD 157347.

Table 4: Twins from four PDA surveys; last row gives minimum and maximum ages in Gyr BR(160*) | DM(73*) | NIS(71*) | LIU(68*)
---|---|---|---
HD | HD | HD | HD
9407 | 150437 | 2071 | 146233
157347 | 29137 | 1461 | 140538
152391 | 134987 | 4915 | 138004
203030 | 115585 | 361 | 68168
50692 | 90722 | 7134 | 10307
147750 | 33822 | 6204 | 190406
103828 | 109271 | 13724 | 106116
85689 | 134664 | 196390 | 219542
157338 | 147513 | 189625 | 141937
34411 | 76151 | 202628 | 159222
24496 | 155968 | 12264 | 4915
10307 | 189625 | 59967 | 217014
4.3–4.9 | 1.6–7.2 | 1.5–7 | 2.5–6.8
Figure 3: $[El/Fe]_{\rm twc\ or\ 68twc}$ = $[El/Fe]_{\rm Sun}-[El/Fe]_{\rm
Average\,twins}$. vs. condensation temperature, $T_{\rm c}$. Filled circles
are BD18 points; grey (blue online) points are volatiles C (40K), O (179K), Na
(953K), S (655K), and Zn (723K) and Cu (1033K). Black filled circles are
refractories. Error bars are the lengths of the standard deviations of BD18
minus NIS Left: GCE CQ corrections are used. Right: UL GCE corrections used.
The straight lines are an unweighted least square fits. Note the different
vertical scales, left and right.

Fig. 3 uses GCE-corrected abundances for 12 twins, by CQ (left) and 68 twins
UL (right), numerical values for the latter are from BD18’s Tab. 3. Error bars
are given for elements in common for BD18 and NIS surveys. The bar lengths are
equal to the standard deviations of $[El/H]_{\rm BD18}-[El/H]_{\rm NIS}$ for
the 11 non neutron-addition elements in common. S, Zn, Cu, and Mn (1158 K)
were not determined in NIS.

Neither fit in Fig. 3 is significant222“Significance” or “sig” is the
probability that the correlation is due to chance. If sig ¿ 0.05, the
correlation is not considered significant. The larger of the absolute value of
the two tiny slopes is $-0.21\cdot 10^{-5}$ for the 12 twins (left). This
“lack” of an indication of depletion of the solar refractories was
sufficiently surprising that a calculation was made using all 68 stars from
BD18 as twins. We again excluded 11 of their 79 stars. The resulting plot,
Fig. 3 (right) supported the normality of the solar refractory abundances
relative to these two sets of twins.

A further confirmation of the normality of solar refractories in BD18 are the
abundances of a subset of 25 stars with ages within $\pm$ 1.5 Gyr of the Sun’s
age. Because of their proximity to the Sun’s age, GCE corrections are small.
Averages of $[El/Fe]$ are very small from a minimum of $-$0.026 for carbon and
a maximum of +0.017 for calcium. Values for the refractory Ti (1573 K) and Al
(1641 K) were respectively 0.013 and 0.006.

Scatter for the other surveys is larger than for BD18. If we combine the
recent NIS and LIU into 26 points, unweighted linear fits (not shown) have
slopes of $-1.39\times 10^{-5}{\rm dex\ deg}^{-1}$ (uncorrected for GCE), sig
= 0.054, and $-0.95\times 10^{-5}{\rm dex\ deg}^{-1}$ (corrected), sig = 0.20,
giving weak support to the depletion of solar refractories. An examination of
60 NIS stars with ages < 9 Gyr shows small depletion of all 10 elements from
carbon to nickel. Depletions of refractories magnesium – nickel are all < 0.02
dex if GCE corrections are applied. The greatest depletion is for oxygen,
-0.04 dex.

It is possible to relate the slopes of abundance vs $T_{\rm c}$ plots to an
engulfed mass using the toy model of Cowley et al. (2021). The model gives the
slope, $q$, of the $[El/H]$ vs. $T_{\rm c}$plot. Because
$[El/H]=[El/Fe]-[Fe/H]$ there will be no difference in the slope of plots of
$[El/H]$ and $[El/Fe]$ for constant $[Fe/H]$.

We assume solar abundances (Asplund et al., 2021) for the mass and composition
of the convection zone (SCZ) as well as the composition of the mass ($q\times
M_{\rm BE}$) added. Here, we take $M_{\rm SCZ}=0.025M_{\odot}$, having the
Sun’s composition (Asplund et al., 2021). For negative slopes, the mass is
added to the average solar twin and missing from the Sun.

The added mass is $q$ times the earth’s mass and has the earth’s composition
(Wang et al., 2018). For other convection zone masses, the added mass must be
adjusted accordingly–e.g. in a linear regime, doubling the $M_{\rm SCZ}$ would
require doubling the added mass.

Quantitative values for the $q$ parameters and the corresponding slopes are
given in Tab. 5. The coefficient of $10^{-5}$ is fortuitously nearly equal to
$q$ because of the correlation of $M_{\rm BE}/M_{\rm SCZ}$ with $T_{\rm c}$.

Table 5: Slopes of $[El/H]$ for different values of $q$; last row gives minimum and maximum ages in Gyr $q$ | slope*105 | $q$ | slope*105
---|---|---|---
0.02 | 0.02 | 1.0 | 1.08
0.10 | 0.11 | 3.0 | 3.13
0.33 | 0.36 | 10.0 | 9.35
0.50 | 0.54 | 30.0 | 22.1

The negative slopes of the abundances vs. $T_{\rm c}$ indicate 0.21 to 1.4
$M_{\rm BE}$ was added to the average solar twin but is missing from the Sun.
The tiny slopes of Fig. 3 corresponds to no refractive mass difference in the
Sun and average twin.

##  5 Summary

We have examined precision differential abundances from five surveys. We
obtained new GCE parameters from the BD18 set, using an algorithm that gives
zero correction for stars with solar age. The new constrained quadratic (CQ)
algorithm allows for curvature that is in agreement for a number of elements
with previous studies and in particular supports the conclusion of NIS of two
populations with different abundance patterns for stars younger and older than
the Sun. For most elements, curvature described by our CQ fits was small; the
overall agreement BD18 parameters is good.

Sets of twelve stars were chosen from each of five surveys of F, G, and K
stars as solar analogues or “twins.” The choice was based on proximity in a
hyperspace consisting of $T_{\rm eff}$, $\log{(g)}$, [Fe/H], and age.
Additionally, we considered a subgroup of 68 BD18 stars as twins.

Similar parameters for the NIS, LIU, and the larger DM and BR were obtained.
Results depend sensitively on the selection of subsets of stars from the
samples. Abundances from each set were averaged, and plotted against
condensation temperature. The peak-to-peak scatter is nearly 0.1 dex, In the
case of the BD18 data, that scatter is about 0.025 dex. The BD18 data does not
support the thesis that the Sun is depleted in refractory elements.
Examination of several subsets of the data shows generally very small
depletions of solar refractories but also depletions of oxygen.

Depletions of the refractories in the Sun are probably < 0.02 dex, but depend
on the choise of comparison stars (twins).

A simple model of addition of bulk earth material to a solar convection zone
connects the slopes of $[El/Fe]$ vs. $T_{\rm c}$ plots to the amount of
ingested (or missing) refractory-rich material. We estimate the amount of
ingested (or missing) material is between zero and 1.4 earth masses.

## Acknowledgements

This research has made use of the NASA Exoplanet Archive, which is operated by
the California Institute of Technology, under contract with the National
Aeronautics and Space Administration under the Exoplanet Exploration Program.
We also used the SIMBAD database operated at CDS, Strasbourg, France (Wenger
et al., 2000). Thanks are due to D. J. Bord and to the referee for many useful
comments and suggestions.

C. Cowley @ https://orcid.org/0000-0001-9837-3662
K. Yüce @ https://orcid.org/0000-0003-1910-3344

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##  6 Data Availability

This work is based entirely on data that may be obtained from the CDS in
Strasbourg France.