\section{Introduction}

Members of our group have been involved in long-term studies of abundances in
Galactic halo stars.
These studies have been designed to address a number of important
issues, including: the synthesis mechanisms of
the heavy,
specifically, neutron capture ($n$-capture)
elements, early in the history of the
Galaxy; the identities of the earliest stellar generations,
the progenitors of the
halo stars; the site or sites for the synthesis of the
rapid $n$-capture ({\it i.e.}, $r$-process) material
throughout the Galaxy; the Galactic Chemical Evolution (GCE) of
the elements; and by employing the abundances of the
radioactive elements (Th and U) as
chronometers, the ages of the oldest stars, and hence the lower limit
on the age of the Galaxy and the Universe. (See \citealt{truran02},
\citealt{snedencowan03}, \citealt{cowan04}, and \citealt{sneden08}
for discussions of these related and significant topics.)

In the following paper we review some of the results of our studies,
starting with new stellar abundance determinations arising from
more accurate laboratory atomic data
in \S II, followed by  abundance comparisons
of the
lighter and heavier $n$-capture elements in the $r$-process-rich stars
in \S III, with new species  detections in the star BD+17{$^{\circ}$} 3248 \
and the ubiquitous nature of the $r$-process throughout the
Galaxy described in
sections \S IV
and \S V, respectively. We end with our Conclusions in \S VI.

\section{Atomic Data Improvements and Abundance Determinations}

Stellar abundance determinations of the $n$-capture elements in
Galactic halo stars have become increasingly more accurate
over the last decade with typical errors now
of less than 10\% \citep{sneden08}.
Much of that improvement in the precision of the stellar abundances
has been due to increasingly more accurate laboratory atomic data.
New measurements of the transition probabilities
have been published  for the rare earth elements (REE) and several
others, including:
La~II \citep{lawler01a};
Ce~II (\citealt{palmeri00};
   and recently transition probabilities for 921 lines for Ce~II,
   \citealt{lawler09});
Pr~II \citep{ivarsson01};
Nd~II (transition probabilities for  more than 700
   Nd~II lines, \citealt{denhartog03});
Sm~II (\citealt{xu03};
   and recently transition probabilities for more
   than 900 Sm~II lines, \citealt{lawler06});
Eu~I, II, and III (\citealt{lawler01c}; \citealt{denhartog02});
Gd~II \citep{denhartog06};
Tb~II (\citealt{denhartog01}; \citealt{lawler01b});
Dy~I and II \citep{wickliffe00};
Ho~II \citep{lawler04};
Er~II (transition probabilities for 418
lines of Er II, \citealt{lawler08});
Tm~I and II (\citealt{anderson96}; \citealt{wickliffe97});
Lu~I, II, and III (\citealt{denhartog98}; \citealt{quinet99},
   \citealt{fedchak00});
Hf~II \citep{lawler07};
Os~I and II (\citealt{ivarsson03,ivarsson04}; \citealt{quinet06});
Ir~I and II (\citealt{ivarsson03,ivarsson04}; \citealt{xu07});
Pt~I \citep{denhartog05};
Au~I and II (\citealt{fivet06}; \citealt{biemont07});
Pb~I \citep{biemont00};
Th~II \citep{nilsson02a};
U~II \citep{nilsson02b};
and finally  in new, more precise solar and stellar abundances of
Pr, Dy, Tm, Yb, and Lu \citep{sneden09}.

These new atomic data have been employed to redetermine the solar
and stellar abundances.
We show in  Figure~\ref{f8} (from \citealt{sneden09})
the relative REE, and Hf,
abundances in five $r$-process rich stars:  BD+17{$^{\circ}$} 3248, \mbox{CS~22892-052}, \mbox{CS~31082-001},
HD~115444 and HD~221170,
where
the abundance distributions have been scaled to the element Eu for
these comparisons. Also shown in Figure~\ref{f8}
are two  Solar System $r$-process-only
abundance predictions from \citet{arlandini99} (based upon a
stellar model calculation)  and
\citet{simmerer04}   (based upon the ``classical'' $r$-process residual
method) that are also matched to the Eu abundances.
What is clear from the figure is that all of the
REE abundances---as well as Hf, which is a heavier interpeak element---are
in the same relative proportions from
star-to-star and with respect to the solar $r$-process abundances.
This agreement between the heavier $n$-capture elements and the
Solar System $r$-process abundance distribution
has been noted in the past (see, {\it e.g.}, \citealt{sneden03}), but
the overall agreement has become much more
precise, and convincing,  as a result of the new atomic laboratory data.

\begin{figure*}
\plotone{f8.eps}
\caption{Recent
abundance determinations in five $r$-process rich
stars, based upon
new atomic lab data, compared with two solar system $r$-process only
predictions.  The abundances in each star have been normalized to the
element Eu. After \citet{sneden09}.
Reproduced by permission of the AAS.
}
\label{f8}
\end{figure*}

\section{Abundance Comparisons}

We can also compare more comprehensive---not just the
REE---elemental abundance determinations for the
$r$-process-rich halo stars. This is potentially
a more rewarding enterprise, as it can illuminate the complex
nucleosynthetic
origin of the lightest $n$-capture elements, and can provide new ways of
looking at the age of the Galactic halo.

\subsection{Heavy $n$-capture Elements}

We show in Figure~\ref{compnew4} abundance comparisons with extensive
elemental  data for
10 $r$-process-rich stars
(from the top):
filled (red) circles, CS~22892-052 \citep{sneden03,sneden09};
filled (green) squares, HD~115444 \citep{westin00,sneden09,hansen11};
filled (purple) diamonds, BD+17{$^{\circ}$} 3248\ \citep{cowan02,roederer10b};
(black) stars, CS~31082-001 \citep{hill02,plez04};
solid (turquoise) left-pointing triangles, HD~221170 \citep{ivans06,sneden09};
solid (orange) right-pointing triangles, HE~1523-0901 \citep{frebel07};
(green) crosses, CS~22953-003 \citep{francois07};
open (maroon) squares, HE~2327-5642 \citep{mashonkina10};
open (brown) circles, CS~29491-069 \citep{hayek09}; and
open (magenta) triangles, HE~1219-0312 \citep{hayek09}.
The abundances of all the stars except
\mbox{CS~22892-052}\ have been vertically displaced
downwards for display purposes.
In each case the
solid lines are (scaled) solar system $r$-process only predictions from
\citet{simmerer04} that have been matched to the Eu abundances.

The figure indicates that for the  ten stars plotted, the
abundances of {\it all} of the heavier stable $n$-capture elements
({\it i.e.}, Ba and above) are
consistent with the relative solar system $r$-process abundance distribution
(see also \citealt{sneden09}).
Earlier work had demonstrated this agreement for
several $r$-process rich stars (where [Eu/Fe] $\simeq$ 1), including \mbox{CS~22892-052},
and the addition of still more such $r$-process-rich stars supports that
conclusion.

\begin{figure*}
    \centering
\includegraphics[angle=-90,width=7.00in]{compnew4.ps}
\caption{
Abundance comparisons between 10 $r$-process rich stars and the Solar
System $r$-process values.
See text for references.
Adapted from  \citet{sneden11}.
}
\label{compnew4}
\end{figure*}

\subsection{Light $n$-capture Elements}

While the heavier $n$-capture elements appear to be consistent with
the scaled solar system $r$-process curve, the lighter $n$-capture
elements (Z $<$ 56) seem to fall below that same solar curve.
One problem in analyzing this region of interest
is that there have been relatively few stellar observations of these
lighter $n$-capture elements until now.
With the limited amount of data it is not yet clear if the pattern
is the same from star-to-star for the lighter $n$-capture elements in
these $r$-process rich stars.

There has been extensive work on trying to understand the
synthesis of these elements.
Observations of 4 metal-poor $r$-enriched stars
by \citet{crawford98}
suggested that Ag (Z = 47) was produced in rough proportion
to the heavy elements in stars with $-$2.2~$<$~[Fe/H]~$< -$~1.2.
\citet{wasserburg96} and \citet{mcwilliam98} pointed out
that multiple sources of heavy elements (other than the $s$-process)
were required to account for the observed abundances
in the solar system and extremely metal-poor stars, respectively.
\citet{travaglio04} quantized this effect, noting that Sr-Zr
Solar System abundances
could not be totally accounted for from traditional sources, such as
the $r$-process, the (main) $s$-process and the weak $s$-process.
They suggested that
the remaining (missing) abundances---8\% for Sr to 18\%
for Y and Zr---came from
a light element primary process (LEPP).
Travaglio et al.\ also noted,
``The discrepancy in the $r$-fraction of Sr-Y-Zr between the
$r$-residuals method and the \mbox{CS~22892-052}\ abundances
becomes even larger for elements from Ru to Cd: the weak
$s$-process does not contribute to elements from Ru to Cd. As
noted [previously], this discrepancy suggests an even
more complex multisource nucleosynthetic origin for elements
like Ru, Rh, Pd, Ag, and Cd.''

\citet{montes07} extended studies of the LEPP and suggested
that a range of $n$-capture elements, perhaps even including heavier
elements such as Ba,  might have a contribution
from this primary process. (Since, however, Ba in $r$-process
rich stars is consistent with
the solar $r$-process abundances, such contributions
for these heavier
elements must be quite small.)
They noted, in particular, that this LEPP might
have been important in  synthesizing the abundances in the $r$-process poor
star HD~122563.

Further insight into the (complicated)
origin of the lighter $n$-capture elements is
provided by the detections of Ge (Z = 32) in a few stars.
\citet{cowan05} noted a correlation of Ge with the
iron abundances in the halo
stars with $-$3.0~$\lesssim$~[Fe/H]~$\lesssim -$1.5,
suggesting that the Ge is being produced along with the Fe-group elements
at these low metallicities.
To produce the protons needed to satisfy
such a correlation, a new neutrino ({\it i.e.}, $\nu$-p) process that might
occur in supernovae was suggested \citep{frohlich06}.
We note that for higher ({\it i.e.}, at  solar)
metallicities,  Ge is considered a neutron-capture element,
synthesized in the $r$-process (52\%) and the $s$-process (48\%)
(Simmerer et al. 2004; Sneden et al. 2008). Thus, there should be
a change in the slope  of the Ge abundances from low metallicities to
higher metallicities, a behavior  that has not yet been observed.

\begin{figure*}
\centering
\plotone{bdp17.eps}
\caption{$r$-process abundance predictions for light $n$-capture elements
compared with observations of  BD+17{$^{\circ}$} 3248 \ from \citet{roederer10b}.
See text for further details.}
\label{bdp17}
\end{figure*}

We show in
Figure~\ref{bdp17} several $r$-process predictions for the
lighter $n$-capture element abundances compared with observations of
those elements in BD+17{$^{\circ}$} 3248 \ from \citet{roederer10b}.
The two Solar System $r$-process models (``classical'' and ``stellar model'')
reproduce some of these elements but begin to diverge from the
observed abundances at Rh (Z = 45).
Also shown in
Figure~\ref{bdp17} are predictions from
a High Entropy Wind (HEW) model, that might be
typical in a core-collapse (or Type II) supernova  (\citealt{farouqi09};
K.-L.~Kratz, private communication.)
This model gives a better fit to the abundances, but does not reproduce
the observed odd-even effects in Ag (Z = 47)and Cd (Z = 48) in this star
(resembling a trend discovered in other
$r$-enriched stars by \citealt{johnson02}).
Recent work by \citet{hansen11} to study Pd (Z = 46) and Ag
abundances in stars with $-$3.2~$\lesssim$~[Fe/H]~$\lesssim -$0.6
confirms the divergence between observations and simulation predictions.

These comparisons between calculations and observations do in fact
argue for a combination of processes to reproduce
the observed stellar abundances of some of these light $n$-capture elements.
This combination of processes might include (contributions from)
the main $r$-process, the LEPP, the $\nu$-p process,
charged-particle reactions
accompanied by $\beta$-delayed fission
and the weak $r$-process ({\it e.g.}, \citealt{kratz07}).
(See, {\it e.g.},
\citealt{farouqi09,farouqi10}, \citealt{roederer10a,roederer10b},
and \citealt{arcones11}
for further discussion.)
It may also be that during the synthesis
the main $r$-process  and the LEPP are separate processes,
and that the abundance patterns in all metal-poor stars could be
reproduced by mixing their yields \citep{montes07}.
Alternatively, it may be
that the $r$-process and the LEPP
can be produced in the same events, but sometimes
only the lower neutron density components  are present
\citep{kratz07,farouqi09}.
It has also been suggested that the
heavier and lighter
$n$-capture elements are synthesized in separate sites (see {\it e.g.},
\citealt{qian08}).

New observations of heavy elements in metal-poor globular cluster
stars reaffirm the abundance patterns seen in field stars.
In the globular cluster M92, \citet{roederer11b} found that the
observed star-to-star dispersion in Y (Z = 39) and Zr (Z = 40)
is the same as for the Fe-group elements ({\it i.e.}, consistent
with observational uncertainty only).
Yet, the Ba \citep{sneden00}, La, Eu, and Ho abundances exhibit
significantly larger star-to-star dispersion that cannot be
attributed to observational uncertainty alone.
Furthermore, the Ba and heavier elements were produced by $r$-process
nucleosynthesis without any $s$-process contributions.
This indicates that, as in the field stars,
these two groups of elements could not have
formed entirely in the same nucleosynthetic process in M92.

\section{New Species Detections}

\citet{roederer10b} reanalyzed near-UV spectra obtained with
HST/STIS of the
star BD+17{$^{\circ}$} 3248.
(See also \citealt{cowan02,cowan05} for earlier HST observations of BD+17{$^{\circ}$} 3248.)
We show in
Figure~\ref{f1} (from \citealt{roederer10b})
spectral regions around Os~II and Cd~I lines in the
stars BD+17{$^{\circ}$} 3248, HD~122563 and HD~115444.  There is a clear detection of Os~II
in both BD+17{$^{\circ}$} 3248\ and HD~115444 but not in HD~122563. The star HD~115444 is
similar in metallicity and atmospheric parameters to HD~122563
(see \citealt{westin00}), but much more
$r$-process rich: [Eu/Fe] = 0.7 versus $-$0.5, respectively.
In the lower panel of
Figure~\ref{f1} we see the presence of Cd~I in BD+17{$^{\circ}$} 3248 \ and HD~115444, as well
as a weak detection in HD~122563.
Synthetic fits to these spectra in BD+17{$^{\circ}$} 3248\ and HD~122563
indicate the presence of
Cd~I and Lu~II lines in both stars,
as well as the detection (and upper limit of) Os~II in the same
two stars, respectively.
This work was
the first to detect Cd~I, Lu~II, and Os~II
in metal-poor halo stars.

\begin{figure}
\centering
\plotone{f1.eps}
\vskip0pt
\caption{
HST (near-UV)
spectral regions containing Os~II and Cd~I lines  in BD+17{$^{\circ}$} 3248,  HD~122563,
and HD~115444 from \citet{roederer10b}.
Reproduced by permission of the AAS.
}
\label{f1}
\end{figure}

In addition to these new detections,
\citet{roederer10b} employed Keck/HIRES spectra
to derive new abundances of Mo I, Ru I and Rh I in this star.
Combining these abundance determinations led to the detection of a total of
32 $n$-capture species  ---the most of any metal-poor halo star.
(Previously, CS~22892-052 had the most such detections.)
Further, we note that
the total detections in BD+17{$^{\circ}$} 3248 \ did not count the element Ge.
And while Ge may be
synthesized in proton-rich processes early in the history of the
Galaxy,
it is classified as a $n$-capture element in Solar System material
(see \citealt{simmerer04} and \citealt{cowan05}).
We illustrate this total abundance distribution
in Figure~\ref{bdfourthb} compared with the
two Solar System $r$-process curves from \citet{simmerer04} and
\citet{arlandini99}. We again see the close agreement between the heavier
$n$-capture elements and (both of) the predictions for
the Solar System $r$-process curve, as well as
the deviation between the abundances of the
lighter $n$-capture elements and that
same curve.

  \begin{figure*}
    \centering
\vskip 0.55in
\includegraphics[width=5.00in]{bdfourthb.eps}
\vskip 0.35in
\caption{
The total observed abundance distribution in BD+17{$^{\circ}$} 3248.
There are a total of 32---not including Ge---detections of $n$-capture elements,
the most in any metal-poor halo star. This distribution is
compared with the
two Solar System $r$-process curves from \citet{simmerer04} and
\citet{arlandini99}.
}
\label{bdfourthb}
\end{figure*}

\section{The $r$-process Throughout the Galaxy}

The results of \citet{roederer10b} also confirm earlier work indicating
significant  differences in the abundances between $r$-process rich stars, such
as BD+17{$^{\circ}$} 3248, and $r$-process poor stars, such as HD~122563.
This difference is shown clearly in Figure~\ref{f4}. The abundance
distribution for BD+17{$^{\circ}$} 3248 \ (shown in the top panel) is relatively flat---compare
the abundance of Sr with Ba---and
is consistent with the scaled Solar System $r$-process abundances for
the heavy $n$-capture elements.
In contrast the lower panel of this figure indicates that the abundances
in the $r$-process poor HD~122563 fall off dramatically with increasing
atomic number---again compare the abundance of Sr with Ba.

  \begin{figure*}
    \centering
\includegraphics[width=5.45in]{f4.eps}
\caption{
Abundance distributions in BD+17{$^{\circ}$} 3248\ and HD~122563 with detections indicated by
filled symbols and upper limits by downward-pointing open triangles.
The new measurements of Os, Cd, and Lu illustrated in  Figure~\ref{f1}
are labeled.
In the top panel (BD+17{$^{\circ}$} 3248)
the bold curve is an HEW calculation from \citet{farouqi09}
normalized to Sr, while the solid line is the Solar System $r$-process
curve \citep{sneden08} normalized to Eu.
In the bottom panel (HD~122563) the solar curve is normalized both to Eu
(solid line) and
Sr (dotted line).
Abundances were obtained from \citet{cowan02,cowan05}, \citet{honda06},
\citet{roederer09,roederer10b}, and \citet{sneden09}.
Figure from \citet{roederer10b}.
Reproduced by permission of the AAS.
}
\label{f4}
\end{figure*}

  \begin{figure*}
    \centering
\includegraphics[angle=-90,width=6.00in]{f11b.eps}
\caption{
Differences between Solar System $r$-process abundances and stellar abundances
for 16 metal-poor stars, normalized to Sr.
The stars are listed in order of descending [Eu/Fe], and that value and
[Sr/Fe] are listed in the box to the  right in the figure.
A value for a typical uncertainty is illustrated in the lower left.
Note the difference in the abundance pattern between the $r$-process
rich star \mbox{CS~22892-052}\ and that of HD~122563, with the other stars falling
between those extremes.
Abundance references are as follows:
S.S. $r$-process  abundances \citep{sneden08};
\mbox{HE~1523-0901} (\citealt{frebel07} and A.\ Frebel,
2009, private communication);
\mbox{CS~31082-001} \citep{hill02,plez04,sneden09};
\mbox{CS~22892-052} \citep{sneden03,sneden09};
\mbox{HE~1219-0312} \citep{hayek09,roederer09};
\mbox{UMi-COS~82} \citep{aoki07};
\mbox{CS~31078-018} \citep{lai08};
\mbox{CS~30306-132} \citep{honda04};
\mbox{BD$+$17~3248} \citep{cowan02,sneden09,roederer10b};
\mbox{HD~221170} \citep{ivans06,sneden09};
\mbox{HD~115444} \citep{westin00,roederer09,sneden09};
\mbox{HD~175305} \citep{roederer10c};
\mbox{BD$+$29~2356} \citep{roederer10c};
\mbox{BD$+$10~2495} \citep{roederer10c};
\mbox{CS~22891-209} \citep{francois07};
\mbox{HD~128279} \citep{roederer10c};
\mbox{HD~13979} (I.\ Roederer et al., in preparation);
\mbox{CS~29518-051} \citep{francois07};
\mbox{CS~22873-166} \citep{francois07};
\mbox{HD~88609} \citep{honda07};
\mbox{CS~29491-053} \citep{francois07};
\mbox{HD~122563} \citep{honda06,roederer10b}; and
\mbox{CS~22949-037} \citep{depagne02}.
Figure from \citet{roederer10a}.
Reproduced by permission of the AAS.
}
\label{f11b}
\end{figure*}

It is clear from much work
({\it e.g.}, Honda et al. 2006, 2007) that the abundances even in a
star such as HD~122563 do come from the
$r$-process---the source for the $s$-process,
low- or intermediate-mass stars on the AGB with longer evolutionary
timescales, have not had sufficient time to evolve prior to the formation
of this metal-poor halo star (cf.\ \citealt{truran81}).
Instead,  one can think of the abundance distribution
in HD~122563,  illustrated in Figure~\ref{f4},  as the result of
an ``incomplete $r$-process''---there were not sufficient numbers of
neutrons to form all of the heavier $n$-capture elements,
particularly the ``third-peak'' elements of Os, Ir, and Pt.
In the classical ``waiting point approximation'' the
lighter $n$-capture elements are synthesized from lower neutron number
density (n$_n$)
fluxes, typically 10$^{20}$--10$^{24}$,  with the heavier $n$-capture elements
(and the total $r$-process abundance distribution) requiring
values of n$_n$ = 10$^{23}$--10$^{28}$ cm$^{-3}$
(see Figures 5 and 6 of \citealt{kratz07}).
Physically in this ``incomplete'' or ``weak $r$-process,''
the neutron flux was too low to push the $r$-process ``path''
far enough away from the valley of $\beta$-stability
to reach the higher mass numbers after
$\alpha$ and $\beta$-decays back to stable nuclides.
Instead the lower neutron number densities result in the
$r$-process path being too close to the valley of stability
leading to a diminution in the
abundances of the heavier $n$-capture elements.
The lighter $n$-capture elements, such as Sr, in this star
may have formed as a result of this incomplete or
weak $r$-process, or the LEPP,  or combinations as described previously for the
$r$-process rich stars.

This analysis was extended to a larger sample by \citet{roederer10a}
and is illustrated in Figure~\ref{f11b}.
We show the differences between the abundance distributions of 16
metal-poor stars, normalized to Sr,
compared with the Solar System $r$-process distribution \citep{sneden08}.
The stars are plotted in order of descending values of
[Eu/Fe], a measure of their $r$-process richness. Thus, we see near the
top \mbox{CS~22892-052}\ with a value of [Eu/Fe] = 1.6 and near the bottom, HD~122563 with
[Eu/Fe] = $-$0.5. The figure illustrates
 the relative flatness of the distributions of
the most $r$-process-rich stars ([Eu/Fe] $\simeq$ 1)
with respect to the solar curves, while the
$r$-process poor stars have abundances that fall off sharply with
increasing atomic number.
It is also clear from Figure~\ref{f11b}
that there are a range of abundance distributions
falling between
these two extreme examples.
(We note that Figure~\ref{f11b} should not be taken as an unbiased
distribution of stars at low metallicity.)

We emphasize four important points here.
First, not all of the metal-poor stars  have the same abundance pattern
as \mbox{CS~22892-052}, only those that are $r$-process rich.
Second, while the distributions
are different between the $r$-process rich and poor stars there
is no indication of $s$-process synthesis for these elements.
Thus, all of the
elements in these stars were synthesized in the $r$-process, at least for
the heavier $n$-capture elements, and $r$-process material was common in the
early Galaxy.
Third, the approximate downward displacement from the top to the bottom
of the Figure~\ref{f11b} (a measure of the decreasing [Eu/Sr] ratio)
roughly scales as the [Eu/Fe] ratio, listed
in the right-hand panel.
This can be understood as follows: since the abundance patterns are
normalized to Sr, and if Sr is roughly proportional to Fe in these stars
(with a moderate degree of scatter---cf.\ Figure~7 of \citealt{roederer10a}),
then {\it of course} the [Eu/Sr] ratio roughly follows [Eu/Fe].
(See also \citealt{aoki05}.)
Finally, we note that Ba has been detected in all of these stars
and the vast majority of low-metallicity field and globular cluster stars
studied to date.
Only in a few Local Group dwarf galaxies do Ba upper limits
hint that Ba (and, by inference, all heavier elements) may be
extremely deficient or absent
\citep{fulbright04,koch08,frebel10}.

\section{Conclusions}

Extensive studies have demonstrated the presence of
$n$-capture elements in the atmospheres of
metal-poor halo and globular cluster stars.
New detections of the $n$-capture elements Cd~I (Z = 48), Lu~II
(Z = 71) and Os~II (Z = 76),
derived from HST/STIS spectra,
have been made in several metal-poor halo stars.
These were the first detections of these species in such stars.
Supplementing these observations with Keck data
and new measurements of  Mo I, Ru I and Rh I,
we reported the detections of 32 $n$-capture
elements in BD+17{$^{\circ}$} 3248. This is currently the most detections of these
elements in any metal-poor halo star, supplanting the previous ``champion''
\mbox{CS~22892-052}.

Comparisons  among the most
$r$-process-rich stars ([Eu/Fe] $\simeq$ 1) demonstrate that  the heaver
stable elements (from Ba and above) are remarkably consistent from star-to-star
 and consistent with the (scaled) solar system $r$-process distribution.
Detailed comparisons of the REE (along with Hf)  among
a well-studied group of $r$-process-rich stars,  employing new
experimental atomic data,
strongly supports this finding.
The newly determined, and lab-based, stellar abundances  are more
precise and show very little scatter from star-to-star and with
respect to the Solar System $r$-process abundances.
This suggests that the $r$-process produced these elements
early in the history of the Galaxy and that the same type(s) of process
was responsible for the synthesis of the $r$-process elements at the
time of the formation of the Solar System.

While the heavier elements appear to have formed from the main
$r$-process and are
apparently consistent with the solar $r$-process abundances,
the lighter $n$-capture element abundances in
these stars do
not conform to the solar pattern.
There have been little data in these stars
until recently, but now with the new detections
of Cd and increasing Pd and Ag detections, some patterns are becoming clear.
First, the main $r$-process alone is not responsible for the synthesis of
these lighter $n$-capture elements. Instead, other processes, alone or in
combination, may have responsible for such formation.
These processes include a so-called
``weak'' $r$-process (with lower values of n$_n$), the LEPP,
the $\nu$-p process, or charged particle
reactions in the HEW of a core-collapse supernova.
It is also not clear whether different processes are responsible for
different mass regions with one for Ge, a different one for Sr-Zr and
still another for Pd, Ag, and Cd.
It is also not clear whether these processes operate separately from
each other or in the same site, or whether different mass ranges of
the $n$-capture elements are synthesized in different sites.
Clearly, much more work needs to be undertaken to understand the
formation of these lighter $n$-capture elements.

The stellar abundance signatures of the heaviest of these  elements,
i.e., Ba and above, are consistent with the rapid neutron
capture process, $r$-process, but not the $s$-process in these
old stars. Similar conclusions are found for stars in the ancient globular
clusters
with comparable abundance spreads in the $r$-process elements
(see, {\it e.g.}, \citealt{gratton04}, \citealt{sobeck11},
\citealt{roederer11a}, and \citealt{roederer11b}).
There is also a clear distinction between the abundance patterns of the
$r$-process rich stars such as \mbox{CS~22892-052}\ and the $r$-process poor stars like
HD~122563. The latter seem to have an element pattern that was
formed as a result of a ``weak'' or ``incomplete'' $r$-process.
Most of the old, metal-poor halo stars have abundance distributions
that fall between the extremes of \mbox{CS~22892-052}\ and HD~122563.
However, the very presence of
$n$-capture elements in the spectra of these stars
argues for $r$-process events being a common occurrence
early in the
history of
the Galaxy.

Finally, we note the need for additional stellar observations,
particularly of the UV regions of the spectra {\it only} accessible
using STIS or COS aboard HST.
These observations require
high signal-to-noise ratios and high resolution to identify faint lines
in crowded spectral regions. Also we will require more laboratory
atomic data for
elements that have not been well studied to improve the precision of
the stellar and solar abundances.
Additional experimental nuclear data,
not yet available,
 for the heaviest neutron-rich nuclei
that participate in the $r$-process,
will be critical to these studies. Until that time
new, more physically based,
 theoretical prescriptions for nuclear masses, half-lives, etc.\
for these $r$-process nuclei will be necessary.
New theoretical models of supernova explosions and detailed synthesis
scenarios, such as might occur in the HEW, will be very important to help to
identify the site or sites
for the $r$-process, a search that has been ongoing since 1957.

\section{Acknowledgments}
We thank our colleagues for all of their contributions and helpful
discussions. We particularly are grateful for all of the
contributions from George W.\ Preston,  as he celebrates his
80th birthday. Partial scientific support for this research was
provided by the NSF (grants AST~07-07447 to J.J.C., AST~09-08978 to C.S.,
and AST~09-07732 to J.E.L.).
I.U.R.\ is supported by the Carnegie Institution of Washington
through the Carnegie Observatories Fellowship.


\section{Introduction}

General relativity (GR) is currently the most established theory of gravitation. It correctly describes a number of observations, such as planetary orbits in the Solar System, the motion of masses in the Earth's gravitational field \cite{BETA.Will:2014kxa}, the recently discovered gravitational waves \cite{BETA.Ligo} or the $\Lambda$CDM model in cosmology \cite{BETA.Planck}. However successful on these scales, GR itself does not provide sufficient answers to fundamental open questions such as the reason for the accelerated expansion of the universe, the phase of inflation or the nature of dark matter. Further tension arises from the fact that so far no attempt to extend GR to a full quantum theory has succeeded.

GR is expected to be challenged by different upcoming experiments on ground and in space, such as high precision clocks \cite{BETA.cacciapuoti2011atomic} and atom interferometers in Earth orbits, pulsar timing experiments \cite{BETA.Pulsar} and direct observations of black hole shadows \cite{BETA.Goddi,BETA.Broderick}. This plethora of existing and expected experimental data, together with the tension with cosmological observations, motivates studying alternative theories of gravitation \cite{BETA.Nojiri}. In particular, the upcoming experiments are expected to give more stringent constraints on the parameter spaces of such theories or even find violations of GR's predictions.

One class of alternative theories are scalar-tensor theories of gravity (STG) - an extension to GR that contains a scalar degree of freedom in addition to the metric tensor.
The detection of the Higgs proved that scalar particles exist in nature \cite{BETA.higgs} and scalar fields are a popular explanation for inflation \cite{BETA.inflation.guth} and dark energy \cite{BETA.Quintessence.and.the.Rest.of.the.World}.
Further, effective scalar fields can arise, e.g., from compactified extra dimensions \cite{BETA.compactified.extra.dimensions} or string theory \cite{BETA.Damour}.

While the motivation for such alternative theories of gravitation is often related to cosmology, of course any such theory must also pass Solar System tests. The most prominent class of such tests is based on the post-Newtonian limit of the theory under consideration, which is usually discussed in the parametrized post-Newtonian (PPN) framework \cite{BETA.will.book,BETA.Will:2014kxa}.
It allows characterizing theories of gravitation in the weak field limit in terms of a number of parameters, that can be calculated from the field equations of the theory, and will, in general, deviate from the parameters predicted by general relativity. These parameters can be constrained using observational data and experiments \cite{BETA.Fomalont:2009zg,BETA.Bertotti:2003rm,BETA.Hofmann:2010,BETA.Verma:2013ata,BETA.Devi:2011zz}.
In this work we are interested in the parameters $\gamma$ and $\beta$ only, as these are the only parameters that may differ in fully conservative gravity theories, to which also STG belongs~\cite{BETA.will.book}.

The most thoroughly studied standard example of a scalar-tensor theory is Brans-Dicke theory \cite{BETA.Brans-Dicke.1961}, which contains a massless scalar field, whose non-minimal coupling to gravity is determined by a single parameter $\omega$. This theory predicts the PPN Parameter $\gamma = (1+\omega)/(2+\omega)$, in contrast to $\gamma=1$ in GR. Both theories predict $\beta = 1$. Adding a scalar potential gives the scalar field a mass, which means that its linearized field equation assumes the form of a Klein-Gordon equation, which is solved by a Yukawa potential $\sim e^{-m r}/r$ in the case of a point-like source. In this massive scalar field case, the PPN parameter $\gamma$ becomes a function of the radial coordinate $r$ \cite{BETA.Olmo1,*BETA.Olmo2,BETA.Perivolaropoulos}.

Scalar-tensor theories can be expressed in different but equivalent conformal frames. This means that the form of the general scalar-tensor action is invariant under conformal transformations of the metric, which depend on the value of the scalar field. There are two such frames that are most often considered:
In the Jordan frame, test particles move along geodesics of the frame metric while in the Einstein frame, the scalar field is minimally coupled to curvature.
The PPN parameters $\gamma$ and $\beta$ for scalar-tensor theories with a non-constant coupling have been calculated in the Jordan \cite{BETA.HohmannPPN2013,*BETA.HohmannPPN2013E} and in the Einstein frame \cite{BETA.SchaererPPN2014}.
These works consider a spacetime consisting of a point source surrounded by vacuum.
As will be elucidated below, this assumption leads to problems when it comes to the definition and calculation of the PPN parameter $\beta$.

Applying conformal transformations and scalar field redefinitions allows to transform STG actions, field equations and observable quantities between different frames. It is important to note that these different frames are physically equivalent, as they yield the same observable quantities~\cite{BETA.Postma:2014vaa,BETA.Flanagan}. Hence, STG actions which differ only by conformal transformations and field redefinitions should be regarded not as different theories, but as different descriptions of the same underlying theory.
This observation motivates the definition of quantities which are invariant under the aforementioned transformations, and to express observable quantities such as the PPN parameters or the slow roll parameters characteristic for models of inflation fully in terms of these invariants~\cite{BETA.JarvInvariants2015,BETA.KuuskInvariantsMSTG2016,BETA.Jarv:2016sow,BETA.Karam:2017zno}.

The PPN parameters $\gamma$ and $\beta$ were calculated for a point source \cite{BETA.HohmannPPN2013,*BETA.HohmannPPN2013E,BETA.SchaererPPN2014}, and later expressed in terms of invariants~\cite{BETA.JarvInvariants2015,BETA.KuuskInvariantsMSTG2016}. However, the assumption of a point source leads to a number of conceptual problems. The most important of these problems is the fact that, in terms of post-Newtonian potentials, the Newtonian gravitational potential becomes infinite at the location of the source, so that its gravitational self-energy diverges. It is therefore impossible to account for possible observable effects caused by a modified gravitational self-energy of the source in a theory that differs from GR. We therefore conclude that the assumption of a point source is not appropriate for a full application of the PPN formalism to STG. This has been realized earlier in the particular case of STG with screening mechanisms~\cite{BETA.SchaererPPN2014,BETA.Zhang:2016njn}.

The goal of this article is to improve on the previously obtained results for the PPN parameters $\gamma$ and $\beta$ for a general class of scalar-tensor theories, in which the divergent gravitational self-energy has been neglected. Instead of a point mass source, the gravitating mass source we consider in this article is given by a sphere with homogeneous density, pressure and internal energy that is surrounded by vacuum. In this case the gravitational self-energy remains finite, and can therefore be taken into account. During our calculation we do not fix a particular frame, but instead make use of the formalism of invariants mentioned above already from the beginning in order to calculate the effective gravitational constant as well as the PPN parameters $\gamma$ and $\beta$.

The article is structured as follows. In Sec. \ref{sec:theory} we discuss the scalar-tensor theory action, the field equations and the invariants.
The perturbative expansion of relevant terms is outlined in Sec. \ref{sec PPN Expansion} and the expanded field equations are provided in Sec. \ref{sec Expanded field equations}.
Next, in Sec. \ref{sec Massive field and spherical source}, these are solved explicitly for a non-rotating homogeneous sphere and the PPN parameters are derived.
Sec. \ref{sec Comparison to observations} applies our results to observations.
Finally, we conclude with a discussion and outlook in Sec.~\ref{sec Conclusion}.
The main part of our article is supplemented by Appendix~\ref{app coefficients}, in which we list the coefficients appearing in the post-Newtonian field equations and their solutions.

\section{Theory}\label{sec:theory}
We start our discussion with a brief review of the class of scalar-tensor tensor theories we consider. The most general form of the action, in a general frame, is displayed in section~\ref{ssec:action}. We then show the metric and scalar field equations derived from this action in section~\ref{ssec:feqns}. Finally, we provide the definition of the relevant invariant quantities, and express the field equations in terms of these, in section~\ref{ssec:invariants}.

\subsection{Action}\label{ssec:action}

We consider the class of scalar-tensor gravity theories with a single scalar field \(\Phi\) besides the metric tensor \(g_{\mu\nu}\), and no derivative couplings. Its action in a general conformal frame is given by~\cite{BETA.Flanagan}
\ba
\label{BETA.equ: action}
S = \frac{1}{2\kappa^2} \int d^4x \sqrt{-g}
\left\{ \mathcal{A}(\Phi) R - \mathcal{B}(\Phi) g^{\mu\nu} \partial_\mu \Phi \partial_\nu \Phi
- 2 \kappa^2 \mathcal{U}(\Phi)\right\}
+ S_m [e^{2\alpha(\Phi)} g_{\mu\nu},\chi] \,.
\ea
Any particular theory in this class is determined by a choice of the four free functions $\mathcal{A}, \mathcal{B}, \mathcal{U}$ and $\alpha$, each of which depends on the scalar field $\Phi$.
The function $\mathcal{B}$ determines the kinetic energy part of the action. The scalar potential is given by $\mathcal{U}$; a non-vanishing potential may be used to model inflation, a cosmological constant or give a mass to the scalar field.
The last part \(S_m\) is the matter part of the action. The matter fields, which we collectively denote by $\chi$, couple to the so-called Jordan frame metric $e^{2\alpha(\Phi)} g_{\mu\nu}$. It is conformally related to the general frame metric $g_{\mu\nu}$. The latter is used to raise and lower indices and determines the spacetime geometry in terms of its Christoffel symbols, Riemann tensor and further derived quantities.
In general, the scalar field is non-minimally coupled to curvature. This coupling is determined by the function $\mathcal{A}(\Phi)$.

There are different common choices of the conformal frame; see~\cite{BETA.JarvInvariants2015} for an overview. In the Jordan frame, one has \(\alpha = 0\) and the matter fields couple directly to the metric \(g_{\mu\nu}\). By a redefinition of the scalar field one may further set $\mathcal{A} \equiv \Phi$. Typically, one considers the coupling function $\omega(\Phi) \equiv \mathcal{B}(\Phi) \Phi$. This particular choice of the parametrization is also known as Brans-Dicke-Bergmann-Wagoner parametrization.

Another possible choice for the conformal frame is the Einstein frame, in which the field couples minimally to curvature, $\mathcal{A} \equiv 1$. However, in this case the matter fields in general do not couple to the frame metric directly, but through a non-vanishing coupling function $\alpha \neq 0$. In this case one may also choose the canonical parametrization $B \equiv 2$.

We call the scalar field minimally coupled if the Jordan and Einstein frames coincide, i.e., if one can achieve $\mathcal{A} \equiv 1$ and $\alpha \equiv 0$ simultaneously through a conformal transformation of the metric.

\subsection{Field equations}\label{ssec:feqns}

The metric field equations are obtained by varying the action \eqref{BETA.equ: action} with respect to the metric. Written in the trace-reversed form they are
\ba \bs
\label{BETA.equ: tensor field equation trace reversed long}
R_{\mu\nu}
&- \frac{\mathcal{A}'}{\mathcal{A}} \left( \nabla_\mu \nabla_\nu \Phi + \f12 g_{\mu\nu} \square \Phi \right)
- \left( \frac{\mathcal{A}''}{\mathcal{A}} + 2\mathcal{F} - \frac{3 {\mathcal{A}'}^2 }{2 \mathcal{A}^2} \right) \partial_\mu \Phi \partial_\nu \Phi
\\
&- \f12 g_{\mu\nu} \frac{\mathcal{A}''}{\mathcal{A}} g^{\rho\sigma} \partial_\rho \Phi \partial_\sigma \Phi
- \frac{1}{\mathcal{A}} g_{\mu\nu} \kappa^2 \mathcal{U}
= \frac{\kappa^2}{\mathcal{A}} \left( T_{\mu\nu} - \f12 g_{\mu\nu} T \right) \,,
\es \ea
where we use the d'Alembertian $\square X \equiv \nabla^2 X = g^{\mu\nu} \nabla_\mu \nabla_\nu X$ and the notation $X' \equiv \frac{\partial X}{\partial\Phi}$.
Taking the variation with respect to the scalar field gives the scalar field equation
\ba \bs
\label{BETA.equ: scalar field equation}
\mathcal{F} \, \square \Phi
&+ \f12 \left( \mathcal{F}' + 2 \mathcal{F} \frac{\mathcal{A}'}{\mathcal{A}} \right) g^{\mu\nu} \partial_\mu \Phi \partial_\nu \Phi
+ \frac{\mathcal{A}'}{\mathcal{A}^2} \kappa^2 \mathcal{U}
- \frac{1}{2 \mathcal{A}} \kappa^2 \mathcal{U}'
= \kappa^2 \frac{\mathcal{A}' - 2 \mathcal{A} \alpha' }{4 \mathcal{A}^2} T \,.
\es \ea
The function $\mathcal{F}$ introduced on the left hand side is defined by
\ba
\label{BETA.F}
\mathcal{F} \equiv \frac{2 \mathcal{A} \mathcal{B} + 3 {\mathcal{A}'}^2}{4 \mathcal{A}^2} \,.
\ea
Note that these equations simplify significantly in the Einstein frame $\mathcal{A} \equiv 1$ and $\alpha \equiv 0$. We will make use of this fact in the following, when we express the field equations in terms of invariant quantities.

Further, note that the functions $\mathcal{A}$ and $\mathcal{B}$ should be chosen such that $\mathcal{F} > 0$. A negative $\mathcal{F}$ would lead to a positive kinetic term in the Einstein frame, causing a ghost scalar field that should be avoided.

\subsection{Invariants}\label{ssec:invariants}

Given a scalar-tensor theory in a particular frame, it can equivalently be expressed in a different frame by applying a Weyl transformation of the metric tensor $g_{\mu\nu} \rightarrow \bar{g}_{\mu\nu}$
and a reparametrization of the scalar field $\Phi \rightarrow \bar{\Phi}$
\bsub
\label{BETA.equ: transformations}
\ba
\label{BETA.equ: Weyl reparametrization}
g_{\mu\nu} &= e^{2\bar{\gamma}(\bar{\Phi})} \bar{g}_{\mu\nu} \,,
\\
\label{BETA.equ: scalar field redefinition}
\Phi &= \bar{f} (\bar{\Phi}) \,.
\ea
\esub

We defined $\mathcal{F}$ in \eqref{BETA.F} since it transforms as a tensor under scalar field redefinition and is invariant under Weyl transformation,
\ba \bs
\mathcal{F} &= \left( \frac{\partial \bar{\Phi}}{\partial \Phi} \right)^2 \bar{\mathcal{F}} \,.
\es \ea

In order to have a frame independent description, we want to express everything in terms of invariants, i.e., quantities that are invariant under the transformations given above. The matter coupling and the scalar potential can be written in an invariant form by introducing the two invariants~\cite{BETA.JarvInvariants2015}
\bsub
\ba
\mathcal{I}_1(\Phi) = \frac{e^{2\alpha(\Phi)}}{\mathcal{A}(\Phi)} \,,
\\
\mathcal{I}_2(\Phi) = \frac{\mathcal{U}(\Phi)}{\mathcal{A}^2(\Phi)} \,.
\ea
\esub
Given the action in a general frame, we can define the invariant Einstein and Jordan frame metrics by
\bsub
\label{BETA equ: Einstein and Jordan frame metric}
\ba
\label{BETA equ: Einstein frame metric}
g^{\mathfrak{E}}_{\mu\nu} := \mathcal{A}(\Phi) g_{\mu\nu} \,,
\\
\label{BETA equ: Jordan frame metric}
g^{\mathfrak{J}}_{\mu\nu} := e^{2\alpha(\Phi)} g_{\mu\nu}\,,
\ea
\esub
which are related by
\ba
\label{BETA equ: Einstein Jordan frame metric relation}
g^{\mathfrak{J}}_{\mu\nu} = \mathcal{I}_1 g^{\mathfrak{E}}_{\mu\nu} \,.
\ea
Note that if the action is already given in the Einstein frame, the metric coincides with the Einstein frame metric defined above, $g_{\mu\nu} = g^{\mathfrak{E}}_{\mu\nu}$, and the same holds for the Jordan frame.
We define the Einstein frame metric \eqref{BETA equ: Einstein frame metric} as it significantly simplifies the field equations.
The metric field equations reduce to
\ba
\label{equ: full metric field equation E-frame}
R^{\mathfrak{E}}_{\mu\nu} - 2 \mathcal{F} \, \partial_{\mu}\Phi \partial_{\nu} \Phi - \kappa^{2}g^{\mathfrak{E}}_{\mu\nu}\mathcal{I}_2 = \kappa^2 \bar{T}^{\mathfrak{E}}_{\mu\nu}\,,
\ea
where
\ba
\bar{T}^{\mathfrak{E}}_{\mu\nu} = T^{\mathfrak{E}}_{\mu\nu} - \frac{1}{2}g^{\mathfrak{E}}_{\mu\nu}T^{\mathfrak{E}}\,, \quad T^{\mathfrak{E}} = g^{\mathfrak{E}\,\mu\nu}T^{\mathfrak{E}}_{\mu\nu} = \frac{T}{\mathcal{A}^2}\,, \quad T^{\mathfrak{E}}_{\mu\nu} = \frac{T_{\mu\nu}}{\mathcal{A}}\,.
\ea
is the trace-reversed energy-momentum tensor in the Einstein frame. It is invariant under conformal transformations and field redefinitions, since also the left hand side of the field equations~\eqref{equ: full metric field equation E-frame} is invariant. Note that we use the invariant Einstein metric \(g^{\mathfrak{E}}_{\mu\nu}\) for taking the trace and moving indices here, in order to retain the invariance of this tensor. For later use, we also define the invariant Jordan frame energy-momentum tensor
\ba
\bar{T}^{\mathfrak{J}}_{\mu\nu} = T^{\mathfrak{J}}_{\mu\nu} - \frac{1}{2}g^{\mathfrak{J}}_{\mu\nu}T^{\mathfrak{J}}\,, \quad T^{\mathfrak{J}} = g^{\mathfrak{J}\,\mu\nu}T^{\mathfrak{J}}_{\mu\nu} = \frac{T}{e^{4\alpha(\Phi)}}\,, \quad T^{\mathfrak{J}}_{\mu\nu} = \frac{T_{\mu\nu}}{e^{2\alpha(\Phi)}}\,.
\ea
Similarly to the metric field equations, we obtain the scalar field equation \eqref{BETA.equ: scalar field equation}
\ba
\label{equ: full scalar field equation}
\mathcal{F} g^{\mathfrak{E}\,\mu\nu} \partial_{\mu}\partial_{\nu}\Phi
- \mathcal{F} g^{\mathfrak{E}\,\mu\nu} \Gamma^{\mathfrak{E}\,\rho}{}_{\nu\mu}\partial_{\rho}\Phi
+ \frac{\mathcal{F}'}{2} g^{\mathfrak{E}\,\mu\nu} \partial_{\mu}\Phi \partial_{\nu}\Phi
- \frac{\kappa^2}{2}{\mathcal{I}_{2}}'
= -\f14 \kappa^2 {(\ln\mathcal{I}_1)}' T^{\mathfrak{E}}\,.
\ea
These are the field equations we will be working with. In order to solve them in a post-Newtonian approximation, we will perform a perturbative expansion of the dynamical fields around a flat background solution. This will be done in the following section.

\section{PPN formalism and expansion of terms}
\label{sec PPN Expansion}
In the preceding section we have expressed the field equations of scalar-tensor gravity completely in terms of invariant quantities. In order to solve these field equations in a post-Newtonian approximation, we make use of the well known PPN formalism. Since we are dealing with different invariant metrics and their corresponding conformal frames, we briefly review the relevant parts of the PPN formalism for this situation. We start by introducing velocity orders in section~\ref{ssec:velorder}. These are used to define the PPN expansions of the scalar field in section~\ref{ssec:ppnscalar}, the invariant metrics in section~\ref{ssec:ppnmetric}, the energy-momentum tensor in section~\ref{ssec:ppnenmom} and the Ricci tensor in section~\ref{ssec:ppnricci}.

\subsection{Slow-moving source matter and velocity orders}
\label{ssec:velorder}

Starting point of the PPN formalism is the assumption of perfect fluid matter, for which the (Jordan frame) energy-stress tensor is given by
\ba
T^{\mathfrak{J}\,\mu\nu} = \left( \rho + \rho \Pi + p \right) u^\mu u^\nu + p g^{\mathfrak{J}\,\mu\nu} \,.
\ea
Since test particles fall on geodesics of the Jordan frame metric, we consider this as the `physical metric' and we define mass density $\rho$, pressure $p$ and specific internal energy $\Pi$ in this frame.
By $u^\mu$ we denote the four-velocity, normalized such that $u^\mu u_\mu = -1$, where indices are raised and lowered using the Jordan frame metric $g^{\mathfrak{J}}_{\mu\nu}$.

We now consider the PPN framework to expand and solve the field equations up to the first post-Newtonian order. For this purpose we assume that the source matter is slow-moving, $v^i = u^i/u^0 \ll 1$. We use this assumption to expand all dynamical quantities in velocity orders $\mathcal{O}(n) \sim |\vec{v}|^n$.
Note that $\rho$ and $\Pi$ each contribute at order $\mathcal{O}(2)$, while $p$ contributes at $\mathcal{O}(4)$. The velocity terms $v^i$ are, obviously, of order $\mathcal{O}(1)$. We finally assume a quasi-static solution, where any time evolution is caused by the motion of the source matter. Hence, each time derivative $\partial_0 \sim \mathcal{O}(1)$ increases the velocity order of a term by one.

\subsection{PPN expansion of the scalar field}
\label{ssec:ppnscalar}

We now expand the scalar field around its cosmological background value $\Phi_0$ in terms of velocity orders,
\ba
\Phi = \Phi_0 + \phi
= \Phi_0 + \order{\phi}{2} + \order{\phi}{4} + \mathcal{O}{(6)}\,,
\ea
where $\order{\phi}{2}$ is of order $\mathcal{O}{(2)}$ and $\order{\phi}{4}$ is of order $\mathcal{O}{(4)}$. Other velocity orders either vanish due to conservation laws or are not relevant for the PPN calculation.
Any function of the scalar field $\mathcal{X}(\Phi)$ can then be expanded in a Taylor series as
\ba
\bs
\mathcal{X}(\Phi)
&= \mathcal{X}(\Phi_0) + \mathcal{X}'(\Phi_0) \phi + \f12 \mathcal{X}''(\Phi_0) \phi^2 + \mathcal{O}(6)
\\
&= \mathcal{X}(\Phi_0) + \mathcal{X}'(\Phi_0) \order{\phi}{2}
+ \left[ \mathcal{X}'(\Phi_0) \order{\phi}{4} + \f12 \mathcal{X}''(\Phi_0) \order{\phi}{2} \,\, \order{\phi}{2} \right]
+ \mathcal{O}{(6)} \,.
\es
\ea
For convenience, we denote the Taylor expansion coefficients, which are given by the values of the functions and their derivatives evaluated at the background value, in the form
$F \equiv \mathcal{F}(\Phi_0)$,
$F' \equiv \mathcal{F}'(\Phi_0)$,
$I_1 \equiv \mathcal{I}_1(\Phi_0)$,
$I_1' \equiv \mathcal{I}_1'(\Phi_0)$,
$I_1'' \equiv \mathcal{I}_1''(\Phi_0)$,
and similarly for all functions of the scalar field.

\subsection{PPN expansion of the metric tensors}
\label{ssec:ppnmetric}

In the next step, we assume that the Jordan frame metric, which governs the geodesic motion of test masses, is asymptotically flat, and can be expanded around a Minkowski vacuum solution in suitably chosen Cartesian coordinates. The expansion of the Jordan frame metric components up to the first post-Newtonian order is then given by
\begin{subequations}\label{eqn:metricjppn}
\begin{align}
g^{\mathfrak{J}}_{00} &= -1 + \order{h}{2}^{\mathfrak{J}}_{00} + \order{h}{4}^{\mathfrak{J}}_{00} + \mathcal{O}(6)\,,\\
g^{\mathfrak{J}}_{0i} &= \order{h}{3}^{\mathfrak{J}}_{0i} + \mathcal{O}(5)\,,\\
g^{\mathfrak{J}}_{ij} &= \delta_{ij} + \order{h}{2}^{\mathfrak{J}}_{ij} + \mathcal{O}(4)\,.
\end{align}
\end{subequations}
It can be shown that these are all relevant and non-vanishing components.
A similar expansion of the Einstein frame metric \(g^{\mathfrak{E}}_{\mu\nu}\) can be defined as
\begin{subequations}\label{eqn:metriceppn}
\begin{align}
I_1g^{\mathfrak{E}}_{00} &= -1 + \order{h}{2}^{\mathfrak{E}}_{00} + \order{h}{4}^{\mathfrak{E}}_{00} + \mathcal{O}(6)\,,\\
I_1g^{\mathfrak{E}}_{0i} &= \order{h}{3}^{\mathfrak{E}}_{0i} + \mathcal{O}(5)\,,\\
I_1g^{\mathfrak{E}}_{ij} &= \delta_{ij} + \order{h}{2}^{\mathfrak{E}}_{ij} + \mathcal{O}(4)\,.
\end{align}
\end{subequations}
The $I_1$'s on the left sides are required in order to satisfy \eqref{BETA equ: Einstein Jordan frame metric relation}.
The expansion coefficients in the two frames are then related by
\begin{subequations}
\begin{align}
\order{h}{2}^{\mathfrak{E}}_{00} &= \order{h}{2}^{\mathfrak{J}}_{00}
+ \frac{I_{1}'}{I_1}\order{\phi}{2}\,,\\
\order{h}{2}^{\mathfrak{E}}_{ij} &= \order{h}{2}^{\mathfrak{J}}_{ij}
- \frac{I_{1}'}{I_1}\order{\phi}{2} \delta_{ij}\,,\\
\order{h}{3}^{\mathfrak{E}}_{0i} &= \order{h}{3}^{\mathfrak{J}}_{0i}\,,\\
\order{h}{4}^{\mathfrak{E}}_{00} &= \order{h}{4}^{\mathfrak{J}}_{00}
+ \frac{I_{1}'}{I_1}\order{\phi}{4} + \frac{I_1I_{1}'' - 2I_{1}' I_{1}'}{2I_1^2}\order{\phi}{2} \, \order{\phi}{2}
- \frac{I_{1}'}{I_1}\order{\phi}{2} \,\order{h}{2}^{\mathfrak{J}}_{00}\,,
\end{align}
\end{subequations}
as one easily checks.
Conversely, one finds the inverse relations
\begin{subequations}
\label{BETA.equ:metric E to J frame}
\begin{align}
\order{h}{2}^{\mathfrak{J}}_{00} &= \order{h}{2}^{\mathfrak{E}}_{00}
- \frac{I_{1}'}{I_1}\order{\phi}{2} \,,\\
\order{h}{2}^{\mathfrak{J}}_{ij} &= \order{h}{2}^{\mathfrak{E}}_{ij}
+ \frac{I_{1}'}{I_1}\order{\phi}{2} \delta_{ij}\,,\\
\order{h}{3}^{\mathfrak{J}}_{0i} &= \order{h}{3}^{\mathfrak{E}}_{0i}\,,\\
\order{h}{4}^{\mathfrak{J}}_{00} &= \order{h}{4}^{\mathfrak{E}}_{00}
- \frac{I_{1}'}{I_1}\order{\phi}{4} - \frac{I_{1}''}{2I_1}\order{\phi}{2} \, \order{\phi}{2}
+ \frac{I_{1}'}{I_1}\order{\phi}{2} \, \order{h}{2}^{\mathfrak{E}}_{00}\,.
\end{align}
\end{subequations}

\subsection{PPN expansion of the energy-momentum tensors}
\label{ssec:ppnenmom}

We now come to the PPN expansion of the energy-momentum tensors. Here we restrict ourselves to displaying the expansion of the invariant energy-momentum tensor in the Einstein frame, since this is the frame we will be using for solving the field equations. It is related to the invariant Jordan frame energy-momentum tensor by
\(T^{\mathfrak{E}}_{\mu\nu} = \mathcal{I}_1T^{\mathfrak{J}}_{\mu\nu}\).
Its PPN expansion follows from the standard PPN expansion of the energy-momentum tensor in the Jordan frame~\cite{BETA.will.book} and is given by
\begin{subequations}
\begin{align}
T^{\mathfrak{E}}_{00} &= I_1\rho\left(1 + \frac{2 I_{1,A}}{I_1}\order{\phi}{2}^A - \order{h}{2}^{\mathfrak{E}}_{00} + v^2 + \Pi\right) + \mathcal{O}(6)\,,\\
T^{\mathfrak{E}}_{0i} &= - I_1\rho v_i + \mathcal{O}(5)\,,\\
T^{\mathfrak{E}}_{ij} &= I_1(\rho v_iv_j + p\delta_{ij}) + \mathcal{O}(6)\,.
\end{align}
\end{subequations}
Its trace, taken using the Einstein frame metric, has the PPN expansion
\begin{equation}
T^{\mathfrak{E}} = I_1^2\left(-\rho + 3p - \Pi\rho - 2 \frac{I_{1,A}}{I_1}\rho\order{\phi}{2}^A\right) \,.
\end{equation}
Consequently, the trace-reversed energy-momentum tensor is given by
\begin{subequations}
\begin{align}
\bar{T}^{\mathfrak{E}}_{00} &= I_1\rho\left(\frac{1}{2} + \frac{I_{1,A}}{I_1}\order{\phi}{2}^A - \frac{\order{h}{2}^{\mathfrak{E}}_{00}}{2} + v^2 + \frac{\Pi}{2} + \frac{3p}{2\rho}\right) + \mathcal{O}(6)\,,\\
\bar{T}^{\mathfrak{E}}_{0i} &= - I_1\rho v_i + \mathcal{O}(5)\,,\\
\bar{T}^{\mathfrak{E}}_{ij} &= I_1\rho\left[v_iv_j + \frac{\order{h}{2}^{\mathfrak{E}}_{ij}}{2} + \left(\frac{1}{2} + \frac{I_{1,A}}{I_1}\order{\phi}{2}^A + \frac{\Pi}{2} - \frac{p}{2\rho}\right)\delta_{ij}\right] + \mathcal{O}(6)\,.
\end{align}
\end{subequations}

\subsection{Invariant Ricci tensor}
\label{ssec:ppnricci}

Finally, we come to the PPN expansion of the Ricci tensor of the invariant Einstein metric. We will do this in a particular gauge, which is determined by the gauge conditions
\bsub
\ba
h^{\mathfrak{E}}_{ij,j} - h^{\mathfrak{E}}_{0i,0} - \frac{1}{2}h^{\mathfrak{E}}_{jj,i} + \frac{1}{2}h^{\mathfrak{E}}_{00,i} = 0 \,,
\\
h^{\mathfrak{E}}_{ii,0} = 2h^{\mathfrak{E}}_{0i,i} \,,
\ea
\esub
which will simplify the calculation. In this gauge, the components of the Ricci tensor to the orders that will be required are given by
\bsub
\ba
\order{R}{2}^{\mathfrak{E}}_{00} &= -\frac{1}{2}\triangle\order{h}{2}^{\mathfrak{E}}_{00}\,,
\\
\order{R}{2}^{\mathfrak{E}}_{ij} &= -\frac{1}{2}\triangle\order{h}{2}^{\mathfrak{E}}_{ij}\,,
\\
\order{R}{3}^{\mathfrak{E}}_{0i} &= -\frac{1}{2}\left(\triangle\order{h}{3}^{\mathfrak{E}}_{0i} + \frac{1}{2}\order{h}{2}^{\mathfrak{E}}_{jj,0i} - \order{h}{2}^{\mathfrak{E}}_{ij,0j}\right)\,,
\\
\order{R}{4}^{\mathfrak{E}}_{00} &= -\frac{1}{2}\triangle\order{h}{4}^{\mathfrak{E}}_{00} + \order{h}{3}^{\mathfrak{E}}_{0i,0i} - \frac{1}{2}\order{h}{2}^{\mathfrak{E}}_{ii,00} + \frac{1}{2}\order{h}{2}^{\mathfrak{E}}_{00,i}\left(\order{h}{2}^{\mathfrak{E}}_{ij,j} - \frac{1}{2}\order{h}{2}^{\mathfrak{E}}_{jj,i} - \frac{1}{2}\order{h}{2}^{\mathfrak{E}}_{00,i}\right) + \frac{1}{2}\order{h}{2}^{\mathfrak{E}}_{ij}\order{h}{2}^{\mathfrak{E}}_{00,ij}\,.
\ea
\esub
We now have expanded all dynamical quantities which appear in the field equations into velocity orders. By inserting these expansions into the field equations, we can perform a similar expansion of the field equations, and decompose them into different velocity orders. This will be done in the next section.

\section{Expanded field equations}
\label{sec Expanded field equations}
We will now make use of the PPN expansions displayed in the previous section and insert them into the field equations. This will yield us a system of equations, which are expressed in terms of the metric and scalar field perturbations that we aim to solve for. We start with the zeroth order field equations in section~\ref{ssec:eqns0}, which are the equations for the Minkowski background, and will give us conditions on the invariant potential \(\mathcal{I}_2\). We then proceed with the second order metric equation in section~\ref{ssec:eqnsh2}, the second order scalar equation in section~\ref{ssec:eqnsp2}, the third order metric equation in section~\ref{ssec:eqnsh3}, the fourth order metric equation in section~\ref{ssec:eqnsh4} and finally the fourth order scalar equation in section~\ref{ssec:eqnsp4}.

\subsection{Zeroth order metric and scalar equations}
\label{ssec:eqns0}

At the zeroth velocity order, the metric equations \eqref{equ: full metric field equation E-frame} are given by
\begin{equation}\label{eqn:h0mn}
-\kappa^2\frac{I_2}{I_1}\eta_{\mu\nu} = 0\,,
\end{equation}
which is satisfied only for \(I_2 = 0\), and hence restricts the choice of the invariant potential \(\mathcal{I}_2\).
At the same velocity order, the scalar equation reads
\begin{equation}\label{eqn:phi0}
-\frac{\kappa^2}{2}I_{2}'  = 0 \,,
\end{equation}
and is solved only by \(I_{2}' = 0\), so that we obtain another restriction on the allowed potential $\mathcal{I}_2$. In the following, we will only consider theories in which these conditions on $\mathcal{I}_2$ are satisfied.

\subsection{Second order metric $h^{\mathfrak{E}}_{00}$ and $h^{\mathfrak{E}}_{ij}$}
\label{ssec:eqnsh2}

At the second velocity order we find the $00$-metric field equation
\begin{equation}
\order{R}{2}^{\mathfrak{E}}_{00}
- \kappa^2\frac{I_2}{I_1}\order{h}{2}^{\mathfrak{E}}_{00}
+ \kappa^2\frac{I_{2}'}{I_1}\order{\phi}{2}^A
= \frac{\kappa^2}{2}I_1\rho \,.
\end{equation}
Inserting the expansion of the Ricci tensor shown in section~\ref{ssec:ppnricci} and using \(I_2 = 0\) and \(I_{2}' = 0\) we solve for \(\order{h}{2}^{\mathfrak{E}}_{00}\) and find the Poisson equation
\begin{equation}
\label{eqn:h200}
\triangle \order{h}{2}^{\mathfrak{E}}_{00} = -\kappa^2I_1\rho = -8\pi G\rho\,,
\end{equation}
where we introduced the Newtonian gravitational constant
\ba
\label{equ: Newtonian gravitational constant}
G = \frac{\kappa^2I_1}{8\pi}\,.
\ea
The $ij$-equations at the same order are given by
\ba
\order{R}{2}^{\mathfrak{E}}_{ij}
- \kappa^2\frac{I_2}{I_1}\order{h}{2}^{\mathfrak{E}}_{ij}
- \kappa^2\frac{I_{2}'}{I_1}\order{\phi}{2}^A\delta_{ij}
= \frac{\kappa^2}{2}I_1\rho\delta_{ij} \,,
\ea
which similarly reduces to
\ba
\label{eqn:h2ij}
\triangle\order{h}{2}^{\mathfrak{E}}_{ij} = -\kappa^2I_1\rho\delta_{ij} = -8\pi G\rho\delta_{ij}\,.
\ea
Note that the diagonal components $i=j$ satisfy the same equation~\eqref{eqn:h200} as \(\order{h}{2}^{\mathfrak{E}}_{00}\).

\subsection{Second order scalar field $\phi^A$}
\label{ssec:eqnsp2}

The second order scalar field equation is given by
\ba
I_1 F \triangle\order{\phi}{2}
- \frac{\kappa^2}{2}I_{2}''\order{\phi}{2}
= \frac{\kappa^2}{4}I_1I_{1}'\rho\,.
\ea
It is convenient to introduce the scalar field mass $m$ by
\ba
\label{equ: scalar mass}
m^2 &\equiv \frac{\kappa^2}{2} \frac{1}{I_1 F} I_{2}''
\ea
and
\ba
k &= \frac{\kappa^2}{4} \frac{1}{F} I_{1}' \,.
\ea
We assume that $m^2 > 0$, since otherwise the scalar field would be a tachyon.
Then, the second order scalar field equation takes the form of a screened Poisson equation,
\ba
\label{eqn:phi2}
\triangle\order{\phi}{2} - m^2 \order{\phi}{2} = k \rho\,.
\ea
We will see that $m$ can be interpreted as the mass of the scalar field, while $k$ is a measure for the non-minimal coupling of the scalar field at the linear level.  We finally remark that \(m\) is an invariant, while \(k\) transforms as a tangent vector to the real line of scalar field values~\cite{BETA.JarvInvariants2015}.

\subsection{Third order metric $h^{\mathfrak{E}}_{0i}$}
\label{ssec:eqnsh3}

The third order metric equation reads
\begin{equation}
\order{R}{3}^{\mathfrak{E}}_{0i} - \kappa^2\frac{I_2}{I_1}\order{h}{3}^{\mathfrak{E}}_{0i} = -\kappa^2I_1\rho v_i \,.
\end{equation}
Thus we can solve for the third order metric perturbation and obtain another Poisson equation,
\begin{equation}\label{eqn:h30i}
\triangle\order{h}{3}^{\mathfrak{E}}_{0i} = \order{h}{2}^{\mathfrak{E}}_{ij,0j} - \frac{1}{2}\order{h}{2}^{\mathfrak{E}}_{jj,0i} + 2\kappa^2I_1\rho v_i\,.
\end{equation}
Note that the source terms on the right hand side of this equation are given by time derivatives of other metric components and moving source matter, and hence vanish for static solutions and non-moving sources.

\subsection{Fourth order metric $h^{\mathfrak{E}}_{00}$}
\label{ssec:eqnsh4}

The fourth order metric field equation reads
\begin{equation}
\bs
\order{R}{4}^{\mathfrak{E}}_{00}
- \kappa^2\frac{I_2}{I_1}\order{h}{4}^{\mathfrak{E}}_{00}
+ \kappa^2\frac{I_{2}'}{I_1}\order{\phi}{4}
- \kappa^2\frac{I_{2}'}{I_1}\order{\phi}{2} \; \order{h}{2}^{\mathfrak{E}}_{00}
+ \frac{\kappa^2}{2}\frac{I_{2}''}{I_1}\order{\phi}{2} \; \order{\phi}{2}
\\
= \frac{\kappa^2}{2}I_1\rho\left(2\frac{I_{1}'}{I_1}\order{\phi}{2}
- \order{h}{2}^{\mathfrak{E}}_{00}
+ 2v^2 + \Pi + 3\frac{p}{\rho}\right)\,.
\es
\end{equation}
Solving for the fourth order metric perturbation then yields
\begin{equation}\label{eqn:h400}
\begin{split}
\triangle\order{h}{4}^{\mathfrak{E}}_{00}
&= 2\order{h}{3}^{\mathfrak{E}}_{0i,0i}
- \order{h}{2}^{\mathfrak{E}}_{ii,00}
+ \order{h}{2}^{\mathfrak{E}}_{00,i}\left(\order{h}{2}^{\mathfrak{E}}_{ij,j}
- \frac{1}{2}\order{h}{2}^{\mathfrak{E}}_{jj,i}
- \frac{1}{2}\order{h}{2}^{\mathfrak{E}}_{00,i}\right)
+ \order{h}{2}^{\mathfrak{E}}_{ij}\order{h}{2}^{\mathfrak{E}}_{00,ij}\\
&\phantom{=}+ \kappa^2\left(\frac{I_{2}''}{I_1}\order{\phi}{2} \; \order{\phi}{2}
- 2I_{1}'\order{\phi}{2} \rho
+ I_1\order{h}{2}^{\mathfrak{E}}_{00}\rho
- 2 I_1 v^2 \rho - I_1 \Pi \rho - 3 I_1 p \right)\,.
\end{split}
\end{equation}
Also this equation has the form of a Poisson equation.

\subsection{Fourth order scalar field $\phi^A$}
\label{ssec:eqnsp4}

Finally, for the scalar field we have the fourth order equation
\begin{multline}
I_1 F \triangle\order{\phi}{4}
- I_1 F \order{\phi}{2}_{,00}
- \frac{\kappa^2}{2}I_{2}''\order{\phi}{4}
- I_1 F \order{\phi}{2}_{,ij}\order{h}{2}^{\mathfrak{E}}_{ij}
+ I_1 F' \triangle\order{\phi}{2} \; \order{\phi}{2}
\\
+ \frac{I_1}{2} F' \order{\phi}{2}_{,i}\order{\phi}{2}_{,i}
+ \frac{I_1}{2} F  \order{\phi}{2}_{,i}\left(2\order{h}{2}^{\mathfrak{E}}_{ij,j}
- \order{h}{2}^{\mathfrak{E}}_{jj,i}
+ \order{h}{2}^{\mathfrak{E}}_{00,i}\right)
- \frac{\kappa^2}{4}I_{2}'''\order{\phi}{2} \; \order{\phi}{2} \\
= -\frac{\kappa^2}{4}\left[3I_1I_{1,A}p
- I_1 I_1' \Pi\rho
- (I_{1}' I_{1}'
+ I_1 I_1'')\order{\phi}{2} \rho \right] \,.
\end{multline}
Solving for the fourth order scalar perturbation then yields
\begin{equation}\label{eqn:phi4}
\begin{split}
\triangle\order{\phi}{4}
- m^2 \order{\phi}{4}
&= \order{\phi}{2}_{,00}
+ \order{\phi}{2}_{,ij} \order{h}{2}^{\mathfrak{E}}_{ij}
- \frac{1}{2}\order{\phi}{2}_{,i}\left(2\order{h}{2}^{\mathfrak{E}}_{ij,j}
- \order{h}{2}^{\mathfrak{E}}_{jj,i}
+ \order{h}{2}^{\mathfrak{E}}_{00,i}\right)
- \frac{F'}{F} \left[ \triangle\order{\phi}{2} \; \order{\phi}{2}
+ \frac{1}{2} \order{\phi}{2}_{,i} \order{\phi}{2}_{,i}\right]\\
&\phantom{=}+ \frac{\kappa^2}{4} \f1F \left[\frac{I_{2}'''}{I_1}
\order{\phi}{2} \; \order{\phi}{2}
- 3 I_{1}' p + I_{1}' \Pi \rho + \left(\frac{({I_1}')^2}{I_1}
+ {I_1}'' \right) \order{\phi}{2} \rho\right]\,.
\end{split}
\end{equation}
This is again a screened Poisson equation, which contains the same mass parameter \(m\) as the second order scalar field equation~\eqref{eqn:phi2}.

These are all necessary equations in order to determine the relevant perturbations of the invariant Einstein frame metric and the scalar field. We will solve them in the next section, under the assumption of a massive scalar field, \(m > 0\), and a static, homogeneous, spherically symmetric source mass.

\section{Massive field and spherical source}
\label{sec Massive field and spherical source}
In the previous section we derived the gravitational field equations up to the required post-Newtonian order. We will now solve these field equations for the special case of a homogeneous, non-rotating spherical mass distribution. This mass distribution, as well as the corresponding ansatz for the PPN metric perturbation and the PPN parameters, are defined in section~\ref{ssec:homosphere}. We then solve the field equations by increasing order. The second order equations for the invariant Einstein frame metric and the scalar field are solved in sections~\ref{ssec:solh2} and~\ref{ssec:solp2}, while the corresponding fourth order equations are solved in sections~\ref{ssec:solh4} and~\ref{ssec:solp4}. From these solutions we read off the effective gravitational constant as well as the PPN parameters \(\gamma\) and \(\beta\) in section~\ref{sec PPN parameters}. A few limiting cases of this result are discussed in section~\ref{ssec:limits}.

\subsection{Ansatz for homogeneous, spherical mass source}
\label{ssec:homosphere}

In the following we consider a static sphere of radius $R$ with homogeneous rest mass density, pressure and specific internal energy, surrounded by vacuum. Its density \(\rho\), pressure \(p\) and specific internal energy \(\Pi\) are then given by
\ba\label{eqn:homosource}
\rho(r) =
\begin{cases}
	\rho_0 & \text{if } r \leq R\\
	0, & \text{if } r > R\\
\end{cases} \,,
\quad
p(r) =
\begin{cases}
	p_0 & \text{if } r \leq R\\
	0, & \text{if } r > R\\
\end{cases} \,,
\quad
\Pi(r) =
\begin{cases}
	\Pi_0 & \text{if } r \leq R\\
	0, & \text{if } r > R\\
\end{cases} \,,
\ea
where \(r\) is the radial coordinate and we use isotropic spherical coordinates. We further assume that the mass source is non-rotating and at rest with respect to our chosen coordinate system, so that the velocity \(v^i\) vanishes.

For the metric perturbation corresponding to this matter distribution, which is likewise spherically symmetric, we now use the ansatz
\begin{subequations}
\label{BETA.equ:PPN metric ansatz}
\begin{align}
\label{BETA.equ:PPN metric ansatz h200}
\order{h}{2}^{\mathfrak{J}}_{00} &= 2 G_\text{eff} U
\,,\\
\label{BETA.equ:PPN metric ansatz h2ij}
\order{h}{2}^{\mathfrak{J}}_{ij} &= 2 \gamma G_\text{eff} U \delta_{ij}
\,,\\
\label{BETA.equ:PPN metric ansatz h30i}
\order{h}{3}^{\mathfrak{J}}_{0i} &= 0
\,,\\
\label{BETA.equ:PPN metric ansatz h400}
\order{h}{4}^{\mathfrak{J}}_{00} &= -2 \beta G_\text{eff}^2 U^2
+ 2 G_\text{eff}^2 (1+3 \gamma-2 \beta) \Phi_2
+ G_\text{eff}(2\Phi_3 +6 \gamma \Phi_4)
\,.
\end{align}
\end{subequations}
Here \(U, \Phi_2, \Phi_3, \Phi_4\) denote the standard PPN potentials, which satisfy the Poisson equations~\cite{BETA.will.book}
\bsub
\label{BETA.equ: Poisson equ potentials}
\ba
\label{BETA.equ: Poisson equ U}
\triangle U &= - 4 \pi \rho \,,
\\
\label{BETA.equ: Poisson equ Phi_2}
\triangle \Phi_2 &= - 4 \pi U \rho \,,
\\
\label{BETA.equ: Poisson equ Phi_3}
\triangle \Phi_3 &= - 4 \pi \rho \Pi \,,
\\
\label{BETA.equ: Poisson equ Phi_4}
\triangle \Phi_4 &= - 4 \pi p \,.
\ea
\esub
For the homogeneous, spherically symmetric mass source we consider they are given by
\bsub
\ba
U(r)
&= \begin{cases}
- \frac{M}{2 R^3}(r^2 - 3 R^2)
& \text{if } r \leq R
\\
\frac{M}{r}
& \text{if } r > R
\\
\end{cases} \,,
\\
\Phi_2 &=
	\begin{cases}
	\frac{3 M^2}{40 R^6}(r^2 - 5 R^2)^2 & \text{if } r \leq R
	\\
	\frac{6 M^2}{5 R r} & \text{if } r > R
	\\
	\end{cases} \,,
\\
\Phi_3 &=
	\begin{cases}
	-\frac{M \Pi_0}{2 R^3} (r^2 - 3 R^2) & \text{if } r \leq R
	\\
	\frac{M \Pi_0}{r} & \text{if } r > R
	\\
	\end{cases} \,,
\\
\Phi_4 &=
	\begin{cases}
	-\frac{2\pi p_0}{3} (r^2 - 3 R^2) & \text{if } r \leq R
	\\
	\frac{4 \pi p_0 R^3}{3 r} & \text{if } r > R
	\\
	\end{cases} \,,
\ea
\esub
where \(M = \frac{4\pi}{3}\rho_0R^3\) is the total mass. The metric ansatz~\eqref{BETA.equ:PPN metric ansatz} further depends on the effective gravitational constant \(G_{\text{eff}}\) and the PPN parameters \(\gamma\) and \(\beta\). These quantities, which are sufficient to describe the post-Newtonian limit of a fully conservative theory, i.e., a theory without preferred location or preferred frame effects, are determined by the particular theory under consideration. Note that these parameters are, in general, not constant, if one considers a massive scalar field, as we will do in the following.

We finally remark that in the ansatz~\eqref{BETA.equ:PPN metric ansatz} we have used the perturbations of the invariant Jordan frame metric \(g^{\mathfrak{J}}_{\mu\nu}\) defined in~\eqref{BETA equ: Jordan frame metric}. This choice is related to the fact that the matter coupling, and hence the geodesic motion of test particles from which the PPN parameters are determined, is given by \(g^{\mathfrak{J}}_{\mu\nu}\).

\subsection{Second order metric}
\label{ssec:solh2}

We start by solving the metric field equations at the second velocity order. Its temporal component~\eqref{eqn:h200} takes the form
\ba
\triangle \order{h}{2}^{\mathfrak{E}}_{00}
= \begin{cases}
-\frac{3 \text{I}_1 \kappa^2 M}{4 \pi  R^3}
& \text{if } r \leq R
\\
0
& \text{if } r > R
\\
\end{cases}\,.
\ea
The solution is given by
\ba
\order{h}{2}^{\mathfrak{E}}_{00}
= 2GU
= \begin{cases}
- \frac{I_1 \kappa^2 M}{8 \pi R^3}(r^2 - 3 R^2)
& \text{if } r \leq R
\\
\frac{\text{I}_1 \kappa ^2 M}{4 \pi  r}
& \text{if } r > R
\\
\end{cases} \,.
\ea
Since the spatial metric equations~\eqref{eqn:h2ij} at the same order are identical to the temporal equation, except for a Kronecker symbol, their solution immediately follows as
\ba
\order{h}{2}^{\mathfrak{E}}_{ij} = \order{h}{2}^{\mathfrak{E}}_{00} \delta_{ij} \,.
\ea

\subsection{Second order scalar}
\label{ssec:solp2}

We then continue with the scalar field equation~\eqref{eqn:phi2} at the second velocity order, which reads
\ba
\left(\triangle - m^2 \right) \order{\phi}{2}
= \begin{cases}
\frac{3 I_1' \kappa^2 M}{16 \pi F R^3}
& \text{if } r \leq R
\\
0
& \text{if } r > R
\\
\end{cases} \,.
\ea
The solution is then given by.
\ba
\order{\phi}{2}
= \begin{cases}
-\frac{3 I_1' \kappa^2 M}{16 \pi F m^2 R^3}+\frac{3 e^{-m R} I_1' \kappa^2 M \
(1+m R) }{16 \pi F m^3 R^3} \frac{\sinh (m r)}{r}
& \text{if } r \leq R
\\
-\frac{3 \kappa^2 M I_1' \left( e^{-m R}(1+m R) + e^{m R} (-1+m R)\right)}{32 \pi F m^3 R^3} \frac{e^{-m r}}{r}
& \text{if } r > R
\\
\end{cases} \,.
\ea
Note that outside the source, the field is proportional to $\frac{e^{-m r}}{r}$, i.e., it has the form of a Yukawa potential. Therefore, the parameter $m$ can be interpreted as the mass of the scalar field.

\subsection{Fourth order metric}
\label{ssec:solh4}

Since the only third order equations are trivially solved by \(\order{h}{3}^{\mathfrak{E}}_{0i} = 0\) in the case of a static, non-moving matter source, we continue directly with the metric field equation at the fourth velocity order. As it is rather lengthy, we give here only its generic form, while all appearing coefficients are stated explicitly in the appendix. This generic form reads
\ba
\label{BETA equ hE004 equation}
\triangle \order{h}{4}^{\mathfrak{E}}_{00}
= \begin{cases}
A_{h400}^{I1}
+\frac{A_{h400}^{I2}}{r^2}
+A_{h400}^{I3} r^2
+\frac{A_{h400}^{I4} e^{-m r}}{r}
+\frac{A_{h400}^{I5} e^{-2 m r}}{r^2}
+\frac{A_{h400}^{I6} e^{m r}}{r}
+\frac{A_{h400}^{I7} e^{2 m r}}{r^2}
& \text{if } r \leq R
\\
\frac{A_{h400}^{E1}}{r^4}
+\frac{A_{h400}^{E2} e^{-2 m r}}{r^2}
& \text{if } r > R
\\
\end{cases}\,.
\ea
Also its solution is lengthy, and so we proceed in the same fashion to display only its generic form here, which is given by
\bsub
\ba
\bs
\label{BETA equ hE004 solution int}
\order{h}{4}^{\mathfrak{E}}_{00} (r \leq R)
&= B_{h400}^{I1}
+B_{h400}^{I2} r^2
+B_{h400}^{I3} r^4
+\frac{B_{h400}^{I4} e^{-m r}}{r}
+\frac{B_{h400}^{I5} e^{-2 m r}}{r}
+\frac{B_{h400}^{I6} e^{m r}}{r}
\\
&\phantom{=}+\frac{B_{h400}^{I7} e^{2 m r}}{r}
+B_{h400}^{I8} \mathrm{Ei}(-2 m r)
+B_{h400}^{I9} \mathrm{Ei}(2 m r)
+B_{h400}^{I10} \ln\left(\frac{r}{R}\right) \,,
\es
\\
\label{BETA equ hE004 solution ext}
\order{h}{4}^{\mathfrak{E}}_{00} (r > R)
&=\frac{B_{h400}^{E1}}{r}
+\frac{B_{h400}^{E2}}{r^2}
+\frac{B_{h400}^{E3} e^{-2 m r}}{r}
+B_{h400}^{E4} \mathrm{Ei}(-2 m r) \,,
\ea
\esub
where $\mathrm{Ei}$ is the exponential integral defined as
\ba
\mathrm{Ei}(x) = -\fint_{-x}^{\infty}\frac{e^{-t}}{t}dt\,,
\ea
with $\fint$ denoting the Cauchy principal value of the integral. The values of the coefficients can be found in the appendix \ref{app coefficients fourth order metric}.
Note that $B_{h400}^{I8}=B_{h400}^{I9}$ and thus the exponential integral terms can be written more compactly as
\ba
B_{h400}^{I8} \mathrm{Ei}(-2 m r)+B_{h400}^{I9} \mathrm{Ei}(2 m r)
= 2 B_{h400}^{I8} \mathrm{Chi}(2 m r) \,,
\ea
where we used $\mathrm{Chi}$ for the hyperbolic cosine integral
\ba
\mathrm{Chi}(x) = \frac{\mathrm{Ei}(x) + \mathrm{Ei}(-x)}{2} = \upgamma + \ln x + \int_0^x\frac{\cosh t - 1}{t}dt\,,
\ea
and $\upgamma$ is Euler's constant.

\subsection{Fourth order scalar}
\label{ssec:solp4}

The final equation we must solve is the scalar field equation at the fourth velocity order, since the fourth order scalar field \(\order{\phi}{4}\) enters the Jordan frame metric perturbation \(\order{h}{4}^{\mathfrak{J}}_{00}\), from which the PPN parameter \(\beta\) is read off. This equation is similarly lengthy, and so also here we restrict ourselves to displaying only the generic form, which reads
\bsub
\ba
\bs
\label{BETA equ phi4 equation int}
\left( \triangle - m^2 \right) \order{\phi}{4}(r \leq R) &=
A_{\phi 4}^{I1}
+\frac{A_{\phi 4}^{I2}}{r^2}
+\frac{A_{\phi 4}^{I3}}{r^4}
+\frac{A_{\phi 4}^{I4} e^{-m r}}{r}
+\frac{A_{\phi 4}^{I5} e^{-2 m r}}{r^2}
+\frac{A_{\phi 4}^{I6} e^{-2 m r}}{r^3}\\
&\phantom{=}+\frac{A_{\phi 4}^{I7} e^{-2 m r}}{r^4}
+\frac{A_{\phi 4}^{I8} e^{m r}}{r}
+\frac{A_{\phi 4}^{I9} e^{2 m r}}{r^2}
+\frac{A_{\phi 4}^{I10} e^{2 m r}}{r^3}
+\frac{A_{\phi 4}^{I11} e^{2 m r}}{r^4}\\
&\phantom{=}+A_{\phi 4}^{I12} e^{-m r} r
+A_{\phi 4}^{I13} e^{m r} r \,,
\es
\\
\bs
\label{BETA equ phi4 equation ext}
\left( \triangle - m^2 \right) \order{\phi}{4}(r > R) &=
\frac{A_{\phi 4}^{E1} e^{-m r}}{r^2}
+\frac{A_{\phi 4}^{E2} e^{-2 m r}}{r^2}
+\frac{A_{\phi 4}^{E3} e^{-2 m r}}{r^3}
+\frac{A_{\phi 4}^{E4} e^{-2 m r}}{r^4}\,.
\es
\ea
\esub
The generic form of the solution then follows as
\bsub
\ba
\bs
\label{BETA equ phi4 solution int}
\phi_4(r \leq R)
&= B_{\phi 4}^{I1}
+\frac{B_{\phi 4}^{I2}}{r^2}
+\frac{B_{\phi 4}^{I3} e^{-m r}}{r}
+\frac{B_{\phi 4}^{I4} e^{-2 m r}}{r^2}
+\frac{B_{\phi 4}^{I5} e^{m r}}{r}
+\frac{B_{\phi 4}^{I6} e^{2 m r}}{r^2}
+B_{\phi 4}^{I7} e^{-m r}\\
&\phantom{=}+B_{\phi 4}^{I8} e^{-m r} r
+B_{\phi 4}^{I9} e^{-m r} r^2
+B_{\phi 4}^{I10} e^{m r}
+B_{\phi 4}^{I11} e^{m r} r
+B_{\phi 4}^{I12} e^{m r} r^2\\
&\phantom{=}+\frac{B_{\phi 4}^{I13} e^{-m r} \mathrm{Ei}(-m r)}{r}
+\frac{B_{\phi 4}^{I14} e^{m r} \mathrm{Ei}(-m r)}{r}
+\frac{B_{\phi 4}^{I15} e^{m r} \mathrm{Ei}(-3 m r)}{r}\\
&\phantom{=}+\frac{B_{\phi 4}^{I16} e^{m r} \mathrm{Ei}(m r)}{r}
+\frac{B_{\phi 4}^{I17} e^{-m r} \mathrm{Ei}(m r)}{r}
+\frac{B_{\phi 4}^{I18} e^{-m r} \mathrm{Ei}(3 m r)}{r} \,,
\es
\\
\bs
\label{BETA equ phi4 solution ext}
\phi_4(r > R) &=
\frac{B_{\phi 4}^{E1} e^{-m r}}{r}
+\frac{B_{\phi 4}^{E2} e^{-2 m r}}{r^2}
+\frac{B_{\phi 4}^{E3} e^{-m r} \mathrm{Ei}(-m r)}{r}
+\frac{B_{\phi 4}^{E4} e^{m r} \mathrm{Ei}(-2 m r)}{r}\\
&\phantom{=}+\frac{B_{\phi 4}^{E5} e^{m r} \mathrm{Ei}(-3 m r)}{r}
+\frac{B_{\phi 4}^{E6} e^{-m r} \ln\left(\frac{r}{R}\right)}{r} \,.
\es
\ea
\esub
The coefficients can be found in the appendix \ref{app coefficients fourth order scalar}.

\subsection{PPN parameters}
\label{sec PPN parameters}

We now have solved the field equations which determine all terms that enter the Jordan frame metric, and hence contribute to the PPN parameters \(\gamma\) and \(\beta\). The Jordan frame metric is then obtained by inserting the solutions obtained before into the relation~\eqref{BETA.equ:metric E to J frame} between the different invariant metrics.
Using the metric ansatz~\eqref{BETA.equ:PPN metric ansatz h200} we find the effective gravitational `constant'
\ba
G_\text{eff}(r)
= \frac{\order{h}{2}^{\mathfrak{J}}_{00}}{2 U}
= \begin{cases}
G \left[ 1 + 3
\frac{\sinh(mr)(1+m R)e^{-mR} - 2 m r }{(2\omega + 3) m^3 r (r^2 - 3 R^2)} \right]
& \text{if } r \leq R
\\
G \left[ 1 + 3
\frac{ mR\cosh(mR) - \sinh(mR) }{(2\omega + 3) m^3 R^3}e^{-mr} \right]
& \text{if } r > R
\\
\end{cases}\,.
\ea
Here we have introduced the abbreviation
\ba
\omega = 2F\frac{I_1^2}{I_1'^2} - \frac{3}{2}\,,
\ea
which is invariant under reparametrizations of the scalar field and chosen such that it agrees with the parameter $\omega$ in case of the Jordan-Brans-Dicke theory~\cite{BETA.Jordan:1959eg,BETA.Brans:1961sx}. In the next step, the PPN parameter $\gamma$ is obtained from the metric ansatz~\eqref{BETA.equ:PPN metric ansatz h2ij} giving
\ba
\gamma(r) = \frac{\order{h}{2}^{\mathfrak{J}}_{ii}}{2 G_\text{eff} U} =
\begin{cases}
1+\frac{12 \left[2 e^{m (r+R)} m r-\left(-1+e^{2 m r}\right) (1+m R)\right]}{6 \left(-1+e^{2 m r}\right) (1+m R)+2 e^{m (r+R)} m r \left[-6 + (2\omega + 3) m^2 \left(r^2-3 R^2\right)\right]}
& \text{if } r \leq R
\\
1-\left(\frac{1}{2}+\frac{(2\omega + 3) m^3 R^3 e^{m r}}{6 [m R \cosh (m R)-\sinh (m R)]}\right)^{-1}
& \text{if } r > R
\\
\end{cases} \,.
\ea
Finally, the PPN parameter $\beta$ is obtained from the ansatz~\eqref{BETA.equ:PPN metric ansatz h400}, and hence can be obtained from
\ba
\beta(r)
= -\frac{\order{h}{4}^{\mathfrak{J}}_{00}
- 2 G_\text{eff}[ (1+3\gamma)G_\text{eff}\Phi_2 + \Phi_3 + 3 \gamma \Phi_4 ]}{2 G_\text{eff}^2 (U^2 + 2 \Phi_2)} \,.
\ea
Due to the even more lengthy, and practically irrelevant solution for \(\beta\) inside the source, we omit this part of the solution. The solution outside the source, $r>R$, takes the generic form
\ba
\bs
\label{BETA.equ: beta}
\beta(r>R)
&= \left[ \left(\frac{1}{r^2} + \f1r \frac{12}{5R} \right) \left(1 + e^{-m r} \frac{C_{\beta}^{E4}}{2} \right)^2 \right]^{-1}
\Bigg[\frac{C_{\beta}^{E1}}{r}
+ \frac{C_{\beta}^{E2}}{r^2}
+ C_{\beta}^{E3} \frac{e^{-m r}}{r}
+ C_{\beta}^{E4} \frac{e^{-m r}}{r^2} \\
&\qquad\qquad+ C_{\beta}^{E5} \frac{e^{-2 m r}}{r}
+ C_{\beta}^{E6} \frac{e^{-2 m r}}{r^2}
+ C_{\beta}^{E7} \frac{e^{-  m r}}{r} \mathrm{Ei}{(-m r)}
+ C_{\beta}^{E8} \frac{e^{m r}}{r} \mathrm{Ei}{(-2 m r)}\\
&\qquad\qquad+ C_{\beta}^{E9} \frac{e^{m r}}{r} \mathrm{Ei}{(-3 m r)}
+ C_{\beta}^{E10} \mathrm{Ei}{(-2 m r)}
+ C_{\beta}^{E11} \frac{e^{-m r}}{r} \ln\left(\frac{r}{R}\right)
\Bigg] \,.
\es
\ea
The values of the coefficients can be found in the appendix \ref{app PPN Beta}, where we further introduce the abbreviations
\ba
\sigma = \frac{2F(I_1I_1'' + I_1'^2) - F'I_1I_1'}{2F^2I_1^2}\,, \quad \mu = \kappa^2I_1'\frac{2FI_2''' - 3F'I_2''}{4F^3I_1^2}\,.
\ea
Both $\gamma$ and $\beta$ depend only on the parameters $m, \omega, \mu, \sigma$ of the theory, which are invariant both under conformal transformations and redefinitions of the scalar field, and on the radius of the sphere $R$. As expected, it is independent of $M,\Pi_0,p_0$, which are absorbed in the metric potentials, and characterize the source only.

\subsection{Limiting cases}
\label{ssec:limits}

We finally discuss a number of physically relevant limiting cases. We start this discussion with the massless limit, i.e., the case of a vanishing potential $\mathcal{U} \rightarrow 0$, corresponding to $\mathcal{I}_2 \rightarrow 0$. This limit is achieved by successively applying to our result the limits $\mu \rightarrow 0$ and $m \rightarrow 0$. For $\gamma$, which does not depend on $\mu$, we obtain the limit
\ba
\gamma(m \rightarrow 0) = \frac{\omega + 1}{\omega + 2} = \frac{4FI_1^2 - I_1'^2}{4FI_1^2 + I_1'^2}\,.
\ea
For $\beta$ we find the limit
\ba
\bs
\beta(\mu \rightarrow 0, m \rightarrow 0) &= \frac{(2\omega + 3)\sigma - 8}{16(\omega + 2)^2}\\
&= 1 - \left( 1 + \frac{1}{4 F} \frac{I_1'^2}{I_1^2} \right)^{-2}
\left( \frac{F'}{32 F^3} \frac{I_1'^3}{I_1^3} + \frac{1}{16 F^2} \frac{I_1'^4}{I_1^4}
- \frac{1}{16 F^2} \frac{I_1'^2 I_1''}{I_1^3} \right) \,.
\es
\ea
These limits correspond to the result found in \cite{BETA.KuuskInvariantsMSTG2016}, if reduced to the case of a single scalar field.

Another interesting case is given by the large interaction distance limit. In the limit $r \rightarrow \infty$ we obtain
\ba
\gamma(r \rightarrow \infty) = 1 \,,
\ea
and
\ba
\bs
\beta(r \rightarrow \infty)
&= \frac{5 C_{\beta}^{E1} R}{12 I_1^2} \\
&= 1 + 5\frac{\left[39+m^2 R^2 (20 m R - 33)\right] -3 (1+m R) [13+m R (13+2 m R)]e^{-2 m R} }{16 (2\omega + 3) m^5 R^5} \,.\label{eqn:betainf}
\es
\ea
Note that it does not take the GR value $1$ as one might expect. This is due to the fact that the finite self-energy of the extended mass source influences $\beta$. If in addition we take the limit $R\rightarrow \infty$, we find that indeed $\beta$ goes to the GR value $1$,
\ba
\beta(r \rightarrow \infty, R \rightarrow \infty) = 1 \,.
\ea
Note that first we have to take the limit $r \rightarrow \infty$, since the solution we used here is valid only in the exterior region $r > R$, and the limit \(R \to \infty\) would otherwise be invalid. We finally remark that the same limit \(\beta \to 1\) is also obtained for \(m \to \infty\), which becomes clear from the fact that \(m\) always appears multiplied by either \(r\) or \(R\).

This concludes our section on the static spherically symmetric solution we discussed here. We can now use our results for \(\beta\) and \(\gamma\) and compare them to Solar System measurements of these PPN parameters. We will do so in the next section.

\section{Comparison to observations}
\label{sec Comparison to observations}
In the preceding sections we have derived expressions for the PPN parameters \(\beta\) and \(\gamma\). We have seen that they depend on the radius \(R\) of the gravitating source mass, the interaction distance \(r\) and constant parameters \(m, \omega, \mu, \sigma\), which characterize the particular scalar-tensor theory under consideration and are invariant under both conformal transformations of the metric and redefinitions of the scalar field. We now compare our results to observations of the PPN parameters in the Solar System, in order to obtain bounds on the theory parameters.

In the following we will not consider the parameters \(\mu\) and \(\sigma\), and set them to \(0\) in our calculations, as they correspond to higher derivatives of the invariant functions \(\mathcal{I}_1\) and \(\mathcal{I}_2\). Restricting our discussion to the parameters \(m\) and \(\omega\) will further allow us to plot exclusion regions which we can compare to previous results~\cite{BETA.HohmannPPN2013,BETA.HohmannPPN2013E,BETA.SchaererPPN2014}. To be compatible with the plots shown in these articles, we display the rescaled mass \(\tilde{m} = m/\sqrt{2\omega + 3}\) measured in inverse astronomical units \(m_{\mathrm{AU}} = 1\mathrm{AU}^{-1}\) on the horizontal axis. Regions that are excluded at a confidence level of \(2\sigma\) are shown in gray. In particular, we consider the following experiments:

\begin{itemize}
\item
The deflection of pulsar signals by the Sun has been measured using very long baseline interferometry (VLBI)~\cite{BETA.Fomalont:2009zg}. From this \(\gamma\) has been determined to satisfy \(\gamma - 1 = (-2 \pm 3) \cdot 10^{-4}\). The radar signals were passing by the Sun at an elongation angle of 3\textdegree, and so we will assume a gravitational interaction distance of \(r \approx 5.23 \cdot 10^{-2}\mathrm{AU}\). The region excluded from this measurement is shown in Fig.~\ref{fig:vlbi}.

\item
The most precise value for \(\gamma\) has been obtained from the time delay of radar signals sent between Earth and the Cassini spacecraft on its way to Saturn~\cite{BETA.Bertotti:2003rm}. The experiment yielded the value \(\gamma - 1 = (2.1 \pm 2.3) \cdot 10^{-5}\). The radio signals were passing by the Sun at a distance of \(1.6\) solar radii or \(r \approx 7.44 \cdot 10^{-3}\mathrm{AU}\). The excluded region, shown in Fig.~\ref{fig:cassini}, agrees with our previous findings~\cite{BETA.HohmannPPN2013,BETA.HohmannPPN2013E}.

\item
The classical test of the parameter \(\beta\) is the perihelion precession of Mercury~\cite{BETA.Will:2014kxa}. Its precision is limited by other contributions to the perihelion precession, most importantly the solar quadrupole moment \(J_2\). The current bound is \(\beta - 1 = (-4.1 \pm 7.8) \cdot 10^{-5}\). As the gravitational interaction distance we take the semi-major axis of Mercury, which is \(r \approx 0.387\mathrm{AU}\). We obtain the excluded region shown in Fig.~\ref{fig:mercury}. Note that for small values of \(\omega\) we obtain a tighter bound on the scalar field mass than from the Cassini tracking experiment, despite the larger interaction distance \(r\). This can be explained by the fact that the main contribution to \(\beta\) comes from a modification of the gravitational self-energy of the source mass, which is independent of the interaction distance, and depends only on the radius of the gravitating body.

\item
A combined bound on \(\beta\) and \(\gamma\) has been obtained from lunar laser ranging experiments searching for the Nordtvedt effect, which would cause a different acceleration of the Earth and the Moon in the solar gravitational field~\cite{BETA.Hofmann:2010}. In fully conservative theories with no preferred frame effects, such as scalar-tensor gravity, the Nordtvedt effect depends only on the PPN parameters \(\beta\) and \(\gamma\). The current bound is \(4\beta - \gamma - 3 = (0.6 \pm 5.2) \cdot 10^{-4}\). Since the effect is measured using the solar gravitational field, the interaction distance is \(r = 1\mathrm{AU}\). The excluded region is shown in Fig.~\ref{fig:llr}.

\item
A more recent measurement of both \(\beta\) and \(\gamma\) with higher precision has been obtained using combined ephemeris data and the Mercury flybys of the Messenger spacecraft in the INPOP13a data set~\cite{BETA.Verma:2013ata}. From these observations, combined bounds in the two-dimensional parameter space spanned by \(\beta\) and \(\gamma\) can be obtained, as well as bounds on the individual parameters by fixing one of them to its GR value. Since we have determined both parameters in our calculation, we do not perform such a fixing here, and use the full parameter space instead. From the 25\% residuals one finds a bounding region that can be approximated as
\begin{equation}
\left[(\beta - 1) - 0.2 \cdot 10^{-5}\right]^2 + \left[(\gamma - 1) + 0.3 \cdot 10^{-5}\right]^2 \leq \left(2.5 \cdot 10^{-5}\right)^2\,.
\end{equation}
Note that in this case one cannot easily define an interaction distance \(r\), since ephemeris from objects across the Solar System has been used. However, we may use the fact that for \(mr \gg 1\) the PPN parameters approach their limiting values \(\gamma \to 1\) and~\eqref{eqn:betainf}, so that the dominant effect is determined by the modified gravitational self-energy of the Sun. The excluded region under this assumption is shown in Fig.~\ref{fig:inpop}. One can see that for small values of \(\omega\) one obtains a bound on the scalar field mass which is approximately twice as large as the bound obtained from Cassini tracking and lunar laser ranging.
\end{itemize}

\begin{figure}[hbtp]
\centering
\includegraphics[width=100mm]{vlbi.png}
\caption{Region excluded by VLBI measurements.}
\label{fig:vlbi}
\end{figure}

\begin{figure}[hbtp]
\centering
\includegraphics[width=100mm]{cassini.png}
\caption{Region excluded by Cassini tracking.}
\label{fig:cassini}
\end{figure}

\begin{figure}[hbtp]
\centering
\includegraphics[width=100mm]{mercury.png}
\caption{Region excluded by the perihelion shift of Mercury.}
\label{fig:mercury}
\end{figure}

\begin{figure}[hbtp]
\centering
\includegraphics[width=100mm]{llr.png}
\caption{Region excluded by lunar laser ranging.}
\label{fig:llr}
\end{figure}

\begin{figure}[hbtp]
\centering
\includegraphics[width=100mm]{inpop.png}
\caption{Region excluded by the ephemeris data set INPOP13a.}
\label{fig:inpop}
\end{figure}

Our results must be taken with care, since they are based on a number of assumptions and simplifications. Most importantly, we have calculated the PPN parameters under the assumption of a homogeneous, non-rotating, spherical gravitational source. This is only a very crude approximation for the Sun, whose density decreases with growing distance from its center. A full treatment of the post-Newtonian limit of a non-homogeneous body would be required to improve on this assumption. However, since a larger amount of matter is located closer to the center of the Sun, hence increasing its gravitational self-energy and decreasing the effective radius \(R\), one might expect that the effect on \(\beta\) will be even larger in such a full treatment.

As another simplification we have assumed that experiments based on electromagnetic waves passing by the Sun can be described by a single effective interaction distance. A rigorous treatment would involve an explicit calculation of the wave trajectory~\cite{BETA.Devi:2011zz,BETA.Deng:2016moh}. However, this affects only the VLBI and Cassini measurements of \(\gamma\), while the measurements of \(\beta\), which are less dependent on the interaction distance, are unaffected.

\section{Conclusion}
\label{sec Conclusion}
We have calculated the PPN parameters \(\gamma\) and \(\beta\) of scalar-tensor gravity with a general potential for a homogeneous, spherical mass source. For our calculation we have used a formalism which is manifestly invariant under both conformal transformations of the metric and redefinitions of the scalar field. The result we have obtained depends on four constant parameters of the theory under consideration, which are derived from the invariant functions that characterize the theory. Further, the result also depends on the radius \(R\) of the gravitating mass source and the interaction distance \(r\) at which the PPN parameters are measured. We have finally compared our results to a number of measurements in the Solar System and derived bounds on two of the four constant theory parameters.

Our results improve on previous work in which we assumed a point-like mass source~\cite{BETA.HohmannPPN2013,BETA.HohmannPPN2013E,BETA.SchaererPPN2014}. We have seen that \(\gamma\) receives a correction which depends on the source mass radius, but retains the large distance limit \(\gamma \to 1\) for \(r \to \infty\). In contrast, \(\beta\) receives a modification also in the large distance limit. This is explained by a modified gravitational self-energy of the source mass, which influences its gravitational effects also at large distances, and which has been neglected for the point mass. As a result, measurements of \(\beta\) at an interaction distance which is large compared to the radius of the source mass, \(r \gg R\), are significantly more sensitive to modifications of GR by a massive scalar field than measurements of \(\gamma\) at the same interaction distance. We have shown this in particular for measurements of \(\beta\) using lunar laser ranging and planetary ephemeris, where the interaction distance is in the order of astronomical units, and which yield bounds on the scalar field mass comparable to or even better that the bound obtained from the Cassini tracking experiment, with an interaction distance in the order of the solar radius. Our work suggests that measurements of \(\beta\) in the gravitational field of smaller, more compact objects could yield even stricter bounds.

Of course also our assumption of a spherically symmetric and homogeneous source mass is still only an approximation. Further improvement of our results would require a weakening of this assumption, and considering the density profile of the gravitating source mass. Such a calculation would have to be done numerically. While we have provided all necessary equations in this article, we leave performing such a calculation for future work.

Finally, it is also possible to extend our results to more general or related theories. A straightforward generalization is given by considering multiple scalar fields, for which \(\gamma\) has been calculated for a point mass source~\cite{BETA.HohmannGamma2016}, or by allowing derivative coupling as in Horndeski gravity, where it would lead to a similar improvement on previous calculations of \(\gamma\) and \(\beta\)~\cite{BETA.Hohmann:2015kra}. Another possibility is to consider massive bimetric gravity, where GR is augmented by a massive tensor degree of freedom instead of a scalar field, and a similar result on \(\gamma\) for a point mass can be improved and extended to \(\beta\)~\cite{BETA.Hohmann:2017uxe}.

\section*{Acknowledgments}
The authors thank Sofya Labazova for pointing out an error in a previous calculation.
MH gratefully acknowledges the full financial support of the Estonian Research Council through the Startup Research Grant PUT790 and the European Regional Development Fund through the Center of Excellence TK133 ``The Dark Side of the Universe''.
MH and AS acknowledge support from the Swiss National Science Foundation.
This article is based upon work from COST Action CANTATA, supported by COST (European Cooperation in Science and Technology).

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