text:
\section{Introduction}

The NASA {\em Kepler Mission} has left an indelible imprint on
exoplanet and stellar properties research through its unmatched
combination of photometric precision for a large number of stars
($\sim$150,000), over a long period of time (4 years) with a
standard observing cadence of 30 minutes \citep{koch10}.
The exquisite time series returned from \emph{Kepler\ } have provided the
first results for Earth-sized planets potentially in or near
the habitable zones of their host stars \citep[e.g.,][]{boru13,torr15}.
The standard cadence data have revolutionized
our ability to probe the properties of red giants with
asteroseismology \citep{bedd11}, while the limited short-cadence,
1 minute observations have similarly revolutionized asteroseismology
of dwarf stars \citep{chap11}.

While \emph{Kepler\ } photometric time series are excellent compared to anything
previously available, they are not perfect and one of the early
surprises in the {\em Kepler Mission} was a higher than expected
noise level, CDPP -- Combined Differential Photometric Precision \citep{chri12a},
a roll-up of all factors of relevance for detection of exoplanet
transits with widths of 3 -- 12 hours.  The {\em Kepler Mission} had
been designed \citep{koch10} to have roughly comparable noise levels
for fiducial 12th magnitude solar-type stars arising from irreducible
Poisson fluctuations, and intrinsic noise from the stars, with smaller
contributions expected from imperfections in the instrument, and the
software used to provide extracted and calibrated time series.
\emph{Kepler\ } provided the first opportunity to observe stars other than the
Sun at precision levels allowing well informed inferences about the
intrinsic variations of solar-type stars.  The total noise (CDPP
at the nominal 6.5 hours) was found to most commonly be 30 parts per
million (ppm) for 12th magnitude solar-type stars, compared to an
expected 20 ppm \citep{jenk02}.  This higher than expected noise level resulted in
the need for twice the data extent to reach the original mission goals,
and was a prime motivation in seeking to extend the original 3.5 year
mission.  An extended mission was approved to double the original
extent, however the loss of two (of four) reaction wheels brought
the prime mission to an end after rather precisely 4 years of
observing.  Analyses by \citet{gill11} showed that the primary factor
in increased CDPP was the contribution from stars, with
a smaller addition from imperfections of instrument and software.

Several studies have addressed general stellar variability with
\emph{Kepler\ } data.  \citet{ciar11} presented an overview of variability from
the first month of data over most stellar types.  \citet{mcqu12}
and \citet{robe13}
also analyzed the first month.  \citet{basr13} and \citet{walk13}
used one quarter of data to focus on
a multi-time scale consideration of solar-type stellar variability
concluding that the \emph{Kepler\ } stellar sample tended to be quieter than
the average Sun, a result at mild variance with \citet{gill11}
(hereinafter Paper 1)
and \citet{mcqu12} conclusions.  A primary critique by \citet{basr13}
(hereinafter BWR13)
with undeniable validity, was that CDPP at 6.5 hours of prime relevance
for exoplanet transit detection is not an optimal choice for study of
stellar variability where longer time scales of several days would better
elucidate behavior following from magnetic activity and rotation of
solar-type stars.

In this paper we revisit the Paper 1 analyses with two
primary considerations.  First, how do the original conclusions
regarding noise sources relevant to the detection of exoplanet
transits change with the consideration of data over 4 years, rather
than the 1.25 years originally used, and with use of data from
a more mature data processing pipeline providing the time series.
That will be the topic of Section 3.  Second, the topic of Section 4,
will be a consideration of noise for solar-type stars following
adoption of metrics on a longer time scale of greater relevance to
primary evidence of magnetic activity induced changes.
Section 5 provides results on simulating the expected
distribution of this longer timescale variability metric.

\section{{\em KEPLER} OBSERVATIONS, DATA RELEASES, AND PRIMARY NOISE METRIC}

Paper 1 provided an extensive discussion of how the
\emph{Kepler\ } photometer operated, the selection of targets relevant
to exoplanet detection, and hence the focus of the noise source study.
Also considered in detail were the primary noise (Poisson from stars,
Poisson from sky background, readout noise, instrument and/or software
imperfections, and intrinsic variability of the stars scaled from
solar observations) terms expected to be important for CDPP.
Rather than attempting to condense an original three page
discussion setting the stage for our primary study of noise
contributions we refer the interested reader to Section 2 of
Paper 1.

The data considered in this paper follow from three epochs:
1) As in Paper 1 the original release of Quarters 2 through 6
in 2009 to 2010.
Quarter 2 was re-released in the middle of this epoch bringing
the treatment of all five quarters to a roughly consistent level.
2) Quarters Q0 -- Q14 as uniformly reprocessed in early 2013.
Quarters 15 - 17 were released at a similar level of software
shortly after this.
3) All quarters as uniformly reprocessed in late 2014.

The early releases of data within three months of
having been telemetered to the ground used the
Science Operations Center (SOC) Pipeline 6, with 6.3
being representative of Quarters 2 -- 6 data as
analyzed in Paper 1.  Removal of instrumental systematics,
the key step in producing calibrated \emph{Kepler\ } time series
was handled via a least squares regression with basis
vectors associated with pointing records, temperature
records, and inferred telescope focus values.
For the early data releases the calibrated data
generated by the Presearch Data Conditioning (PDC) module
for which systematics have been removed was referred
to as {\tt ap\_corr\_flux} in the {\tt fits} files.
Details of processing may be found in the Data Release
Notes applicable to Quarter 5 as a representative
case \citep{mach10}, the Kepler Data Characteristics
Handbook \citep{chri12b}, and \citet{jenk10}.
While this early software did a good job of removing
instrumental systematics, inspection of light curves
(as discussed in the Data Release Notes) would
sometimes show clear evidence of spurious signals
being introduced, as well as frequent removal of likely real
stellar variability.

To address the common suppression of stellar signals a
Baysian approach to PDC was introduced in \emph{Kepler\ } SOC version 8.0.
This Baysian maximum a posteriori (MAP) approach to
cotrending \citep{stum12,smit12} more effectively removed
common mode instrumental systematics while preserving
stellar signals.  This earliest version of PDC-MAP was
first applied to Quarter 9 data, as then used in BWR13.

The 2013 data releases were the first time that a uniform reprocessing for the
bulk of \emph{Kepler\ } mission data was performed.  This used the
SOC Pipeline 8.3.
The primary change for this data release is that PDC
uses wavelet decomposition and multiple temporal
scales in performing the MAP processing.  It decomposes
each light curve into three characteristic bands, thus
improving the ability to deal with instrumental systematics,
while still preserving intrinsic stellar signals at
short to moderate ($\sim$20 days) timescales.  The
longest band ($\geq$21 days) performs a simple robust
fit to cotrending basis vectors evaluated for this
temporal band.  Stellar signals at timescales significantly
longer than this may be severely suppressed.
The middle band of 2 hours to 21 days performs a MAP fit.
The shortest band preserves all signals, i.e. no detrending
is performed.  The software evaluates on a star-by-star
basis whether to invoke the multi-scale MAP (msMAP), or
if on the basis of a goodness metric calculated by PDC
regular MAP performs better this is used to provide the
calibrated time series ({\tt PDCSAP\_FLUX}) in the fits file.
About 90\% of the time msMAP is adopted.  Details of
this processing may be found in \citet{smit12} and
\citet{stum12}, with an update in \citet{stum14}.
Quarters 15 -- 17 were processed by
slightly later versions of the SOC Pipeline, but the changes
were generally not such as to fundamentally affect
noise characteristics.

The third epoch of data releases in late 2014 considered here has been
the only time that all \emph{Kepler\ } data were processed
consistently with the same version of the \emph{Kepler\ } pipeline.
The large change introduced for SOC 8.3 of msMAP was retained.
The primary advance for this newest data release were
improvements to the lower-level treatment of data at the
pixel level, e.g. a more advanced consideration of overscan
in order to better deal with some of the more serious
sources of instrumental systematics at a root level.
For details see \citet{thom15}.
This processing used SOC Pipeline 9.2.

\section{STELLAR AND INSTRUMENTAL NOISE DECOMPOSITION}

\subsection{Summary of Original (and Current) Approach}

To facilitate determining the relative importance and quantitative
values of several terms contributing to CDPP we focused on a study
of a subset of the full \emph{Kepler\ } sample expected to have comparable
contributions from the primary terms of simple Poisson fluctuations,
intrinsic stellar variability, and instrument/software imperfections.
By design of the mission \citep{koch10} this led us to focus on
stars of roughly solar-type, and \emph{Kepler\ } magnitude,
Kp \citep{brow11}, of 12.0 $\pm$ 0.5.

We directly modelled contributions of noise from Poisson terms
on the stellar and sky fluxes, as well as the known CCD readout
noise \citep{chri12b} for each \emph{Kepler\ } CCD and removed these before attempting to
separate out stellar variability and instrument/software terms.

\emph{Kepler\ } observations were conducted on the same stellar field, with
primarily the same targets throughout the prime mission.  Four times
during each \emph{Kepler\ } orbit of the Sun, the spacecraft was reoriented by
90 degrees \citep{vanc09} in order to keep the solar panels illuminated, and
spacecraft radiator in shade.  The progressive reorientation results
in sets of stars cycling through four (of 84 total) CCD channels,
thus providing the primary leverage used to disentangle instrument
and intrinsic stellar contributions.  Considered as an ensemble, if
the noise of one set of stars changes as they cycle through 4 CCD
channels, then this demonstrates that the electronics associated
with those channels contribute different levels of noise.
Through adoption of a Singular Value Decomposition (SVD) formalism we
obtained noise terms in time (global value associated with each
quarter as might follow from unique operation of the instrument,
or external factors such as solar particle fluence), space (the
individual CCD channels), and for the stars.
The SVD formalism follows the discussion in, and uses subroutines
from \citet{pre92} for the solution of a highly over-determined
(more observables than unknowns) set of general linear least-squares
equations with degeneracies present.  A key assumption
was that ensembles of stars nearby on the sky should have the same
intrinsic variability, thus allowing us to put the independently
determined relation of quartets of channels on a common scale.

The original study considered a number of factors such as
dependence of stellar noise on galactic latitude, crowding of
sources, and the influence of fainter, superposed background stars.
These proved to be of second order and will not be considered here.
We refer interested readers to Sections 3.1 through
3.8 of Paper 1 for a full discussion of our approach.
In the remainder of this section we focus on results applying
the SVD formalism as before to updated data products, and the
use of all 17 quarters of data instead of the original 2 -- 6.

Since four years have passed since the original analysis was
performed, we started by locating the original codes, recompiling,
and attempting to replicate the sequential analyses of the original
study, using as well the data products used for the 2011 study.
This was successful in that new analyses of the original data
resulted in exactly the results quoted in Tables 1, 2 and 4,
and shown in Figure 8 of Paper 1 giving primary noise separation values.

\subsection{Repeat of Original Updated to New Data Products}

The 2011 study used time series produced within three months
of the end of each quarter, the last one analyzed (Q6) having
been written in December 2010.  There have since been two
primary releases in which most, or all of the prime mission data were
reanalyzed with more mature software at the Science Operations
Center for \emph{Kepler}.  We will provide results separately for the
processing version 8.3 data released over April through December
2013 for all 17 quarters, and processing version 9.2 released
over November through December 2014 for all the data.

Minor software adjustments needed to be made to accommodate the
newer fits formats of the 8.3 and 9.2 data sets, as well as
minor modifications in a few cases for date ranges provided in individual
quarters.  With the exception of such details, we have performed
analyses in exactly the same way as in Paper 1.

Adoption of the new data products led to rather dramatic shifts
in the noise levels attributed to individual CCD channels (or
imperfections in the pipeline software used to analyze them),
as well as dramatic shifts in the noise levels attributed to
each individual quarter in a global sense.  This was initially a
cause for concern, that perhaps the analyses were either inherently
unstable, or inadequately executed.  The linear correlation of
variances inferred per channel
between the original study (see Paper 1, Table 2)
and the new one using updated data
products was only $\sim$0.5.  However, examination of the inferred
intrinsic stellar variations between the original data products
for Q2--6, and the newer versions of the same data came in at
greater than 0.97.  The stars of course had intrinsically the same
behavior independent of how the data were analyzed to remove
various systematic effects from the time series.  The SVD procedure
successfully returned nearly identical behavior for the stars,
while showing different and generally smaller noise levels in time
and across the detector channels for the more recently
processed data.

Table 1 shows the assigned quarter-to-quarter excess variance
for the first five full quarters, with the first line being from
Paper 1.  In successive full data releases 8.3 and 9.2
the variance (square of noise) drops dramatically for quarters
2 and 3 which had been most affected by systematics.  This behavior
was expected since most pipeline development after the mission start
was devoted to dealing with and suppressing systematics arising from
imperfection in detector electronics and operational
and environmental variations.

\begin{table}
\begin{center}
\caption{Quarter-to-quarter excess variance.\label{tbl-1}}
\begin{tabular}{cccccc}
\tableline\tableline
Version & Q2 & Q3 & Q4 & Q5 & Q6 \\
\tableline
6.3 & 210.46 & 105.82 & 44.52 & 0.00 & 29.89 \\
8.3 & 62.35  &   0.00 & 15.02 & 23.15 & 8.64 \\
9.2 & 18.23 & 0.00 & 3.01 & 28.46 & 0.04 \\
\tableline
\end{tabular}
\tablecomments{Variances in ppm$^2$ over
the five quarters of {\em Kepler} data analyzed.
Variance of quietest quarter is forced to zero.
Version refers to SOC Pipeline version number used.}
\end{center}
\end{table}

Over the three data release versions shown in Table 1 the global
variance attributed to intrinsic variations of the stars was held
fixed, and as noted above the star-to-star variances were reproduced
at a very high level of fidelity across these.  The mean excess
variance over Quarters 2--6 is 78.1, 21.8 and 9.9 ppm$^2$ over
data processing releases using SOC 6.3, 8.3 and 9.2 respectively.

We defer showing the individual contributions per channel as in
Table 2, or Figure 12 of the original study until the next section
when data from the full mission are used to set this.  With the
mean stellar and Poisson contributions held fixed, it is worth noting
that the mean excess variance from both Table 1, plus the per-channel
excesses drops from 181 ppm$^2$ in the data releases made within
three months of each quarter end, to 137 ppm$^2$ for release 8.3,
and finally to 98 ppm$^2$ for release 9.2.  Since the sum of stellar
and Poisson terms is 664 ppm$^2$ the component of noise attributable
to imperfections in the detector electronics and the inability of
detrending software to perfectly compensate has become an increasingly
minor contributor to the overall noise budget, reflecting positive
changes in the pipeline software producing corrected time series.

\subsection{Extension from Quarters 2--6 to full Quarters 1--17}

We have performed full mission analyses for data releases 8.3 and 9.2.
The SVD analysis procedures remain unchanged, but now rather than
having a nearly minimal solution basis in which each quartet of stars
visited most detector channels only once (with redundancy of quarters
2 and 6), there is now a four-fold redundancy with each set of stars
cycling through the same detector multiple times.  We carry the same
assumption as before, namely that the ensemble properties of the stars
remain fixed in time, and to first order in space as well.  Over a
four year time span some individual stars are likely to have shown
significant evolution of intrinsic noise within the three-month
quarterly intervals, certainly in going from minimum to maximum
conditions the Sun shows significant variations.  The SVD solution
relies on having an average of 116 stars per quartet, i.e. the
individual sets cycling through the detector channels, and it is
a reasonable assumption that stellar cycle variations are not
synchronized and the ensemble of 116 stars remains sensibly fixed.
The intrinsic stellar variance star-to-star derived from Quarters 2--6
has a linear correlation of 0.959 with the same as derived from
Quarters 1--17, thus the evolution of intrinsic noise level for
individual stars is shown to be modest (for the two sets of 9.2
data).

The quarter-to-quarter global excesses are shown in Table 2.
The quietest quarter over 2-16 (the full length quarters) is forced
to zero within the SVD solution, and this happens to be Quarter 9
for both the SOC 8.3 and 9.2 data releases.  The small value shown for
Quarter 17 is likely an artifact of this being only about one month
long.  The SOC pipeline detrending removes signal on shorter time scales
for this shorter than normal quarter.  The mean of changes over time in the
two independent pipeline processing cases are modest:  50.5 ppm$^2$
on average at 8.3, and 46.6 ppm$^2$ for data release 9.2.  The changes
across time are generally well understood.  High values for quarters
1 and 2 result from a break-in period of less than optimal management
of \emph{Kepler\ }, e.g. the presence of variable guide stars removed for later
cycles, and multiple safings and repointings in Quarter 2.
Higher values later in the mission, Quarter 12 in particular phase
well with measures of solar activity indicative of increased
particle fluxes encountered by \emph{Kepler\ } as the Sun transitioned to
solar maximum activity.
A proxy for what \emph{Kepler\ } will have experienced is given by the
Planetary Ap index \citep{sieb71} in Table 2.  This is an average
over measurements of disturbance levels in two horizontal
field components observed at 13 selected, subauroral stations.
Since \emph{Kepler\ } was offset by as much as 0.4 AU from the Earth at
the end of mission, an Earth-based metric is only a rough
indication of the environment at \emph{Kepler}.  These results were
taken from http://www.solen.info/solar.
Other solar activity indicators such as sunspot number, 10.7 cm
flux, or flare counts also show rising trends with time and
a good correspondence with the rise in \emph{Kepler\ } noise in later quarters.

\begin{table}
\begin{center}
\caption{Quarter-to-quarter excess variance.\label{tbl-2}}
\begin{tabular}{rrrr}
\tableline\tableline
Quarter & 8.3 & 9.2 & Ap \\
\tableline
1 & 219.06 & 232.03 & 4.53 \\
2 & 95.07 & 58.55 & 5.43 \\
3 & 3.25 & 9.67 & 2.79 \\
4 & 31.65 & 23.85 & 3.48 \\
5 & 36.36 & 49.47 & 8.32 \\
6 & 20.44 & 16.14 & 7.04 \\
7 & 23.84 & 32.46 & 5.00 \\
8 & 72.68 & 72.19 & 6.65 \\
9 & 0.0 & 0.0 & 8.87 \\
10 & 23.56 & 20.41 & 9.68 \\
11 & 55.49 & 68.50 & 5.23 \\
12 & 125.57 & 127.96 & 11.17 \\
13 & 57.53 & 63.78 & 8.90 \\
14 & 78.85 & 62.93 & 10.07 \\
15 & 78.33 & 44.11 & 6.12 \\
16 & 52.92 & 48.44 & 7.53 \\
17 & -39.65 & -42.62 & 6.85 \\
\tableline
\end{tabular}
\tablecomments{Variances in ppm$^2$
over all 17 quarters of {\em Kepler} data analyzed.  The two columns
are for primary data processing release 8.3 (mid-2013), and 9.2 (late-2014).
Ap is the Planetary A index \citep{sieb71}.}
\end{center}
\end{table}

The next step in obtaining a separation of error terms between
the instrument (or residual inability of pipeline software to remove
the results of instrumental imperfections) and stars is to solve
for instrumental terms within each quartet of channels, while at
the same time solving for the intrinsic variance of each star.
The quartets are then placed on a common scale by requiring that
the ensemble average of stars within each quartet have a common value.
Figure~\ref{fig:focnoi} shows how the new by-channel variances
compare to those found in Paper 1.  For 52 of 84 channels
(62\%) the variance ascribed to the instrument has dropped.
In the original study the channels having poorer focus correlated
strongly with a linear correlation coefficient of -0.63 between
variance and focus.  In the new set of by-channel variances using
the 9.2 data release and all data as input, this correlation drops
to -0.34.  The correlation of excess noise with poor focus is still
noticeable, however this has been reduced significantly in amplitude.

\begin{figure}
\begin{center}
\includegraphics[width=80mm]{f1.eps}
\end{center}
\caption{By-channel intrinsic variance levels are plotted against
by-channel focus as represented by the fraction of total energy
in the central pixel for a star centered on a pixel.
Channels overplotted with a small circle represent nine cases
independently identified to have moderate Moir\'{e} pattern noise,
and the ten cases with strong Moir\'{e} noise have doubled circles
added.  Values from the original study, and the new 9.2 data release
based on all quarters are both plotted, connected with a green
line when the new solution has smaller variance, red when larger.}
\label{fig:focnoi}
\end{figure}

\subsection{Summary of Changes Using Newer and More Data}

In repeating the original noise study (Paper 1) using the current
data release (following four years of software development for the
pipeline), and all four years of data we have found generally expected results.
The noise levels attributed to the individual solar-type stars have
changed very little with adoption of the newer data release; a gratifying
result since pipeline updates cannot have affected the stars.
Figure~\ref{fig:starint} shows the updated version of Figure 8 from Paper 1,
the inferred intrinsic stellar noise, now based on all quarters with
use of up-to-date pipeline processing inputs.  Only differences of
minor detail can be noted with respect to the original.
The noise levels inferred for individual channels on the instrument
have dropped with the inclusion of more, and most significantly more
recently processed data.  The software developments within the pipeline
were of course motivated in large part to reduce the excess noise
attributed to the instrument.  The fraction of variance attributed to
factors potentially under the control of software development has
dropped from 22\% four years ago, to 13\% now.  This is of course an
over-simplified view.  The importance of changes for various applications
depends not only on a gross measure of noise level, but also on
detailed characteristics of residual noise.  Similarly the intrinsic
stellar noise may be amenable to suppression for some applications.
Nonetheless a consistent picture has developed of considerable
improvement in the pipeline-calibrated data products over time.

\begin{figure}
\begin{center}
\includegraphics[width=80mm]{f2.eps}
\end{center}
\caption{Upper panel shows the intrinsic stellar noise in
ppm for the {\em Kp} = 11.5 to 12.5 sample as a function of galactic
latitude.  Medians evaluated up to 100 ppm are shown as `o',
while means from up to 3$\times$ the median at each degree of
galactic latitude are shown as `+' symbols.  Standard errors
for the means are shown.  The lower panel shows a histogram
of number of stars per ppm bin.  The mean and rms distribution
for solar noise levels over Quarter-long intervals spanning a
solar Cycle are shown by the `+' and heavy horizontal line,
with the full extent of solar noise per Quarter the thin line.
\label{fig:starint}}
\end{figure}

\section{USE OF LONGER TIMESCALE NOISE METRIC}

CDPP was designed to capture those components of noise and intrinsic
stellar variability of greatest relevance to the detection of low-amplitude
exoplanet transits having characteristic time scales of 3 to 12 hours.
Such a metric need not be, and indeed is not an optimal one for other studies
such as determining the intrinsic variability of solar-type stars.
The CDPP metric depends on the low frequency tail of variability resulting
from stellar granulation, and only the high frequency tail of variability
resulting from magnetic activity induced variations.  If interested in the
stars it would be better to consider multiple metrics that individually
capture the primary sources of variability.
The ``flicker", or root mean square variation of stars on
timescales shorter than 8 hours \citep{bast13} has been useful
for characterizing variability at high frequencies, with resulting
ability to measure stellar gravities, as improved by \citet{kall14}.
At longer timescales the measure of intrinsic stellar behavior is
more difficult given the likelihood of contamination from systematics
in the \emph{Kepler\ } data.  Once a month \emph{Kepler\ } suspended science operations
to re-point the fixed high-gain antenna toward the Earth to telemeter
accumulated data to the ground.  This resulted in thermal perturbations
to the telescope and photometer introducing photometric changes large
compared to the stellar variations of quiet solar-type stars.
To deal with these systematics detrending was introduced that very
successfully removed many common mode variations from the instrumental
drifts, but at an additional cost of suppressing true stellar signals
in some regimes and introducing uncertainty in the final product.
Given the roughly month-long rotation period for quiet solar-type
stars, and the monthly cadence of \emph{Kepler\ } pointings, recovery
of intrinsic stellar signals on timescales of several days most useful
for characterization of activity variations was thus made challenging.

BWR13 have used two primary metrics to encapsulate stellar
variations on activity timescales.  These are physically well
motivated, and useful for characterizing stellar activity variations.
Caution, however, is due in application to \emph{Kepler\ } data where the
pipeline calibration of data may well suppress some variations of relevance
to forming these statistics.  The first diagnostic, $R_{var}$ was
introduced by \citet{basr11}, this ``range" parameter is found by
sorting all the photometric data points in a given interval (30 days
generally adopted), then taking the difference between the 5\% and 95\%
points (to avoid anomalous excursions) in this distribution.
The resulting statistic will be sensitive to both short timescale
excursions (if lasting more than 5\% of the time during 30 days)
as might follow from star spots, and more generally to longer
timescale variations approaching the monthly intervals adopted.
Since the \emph{Kepler\ } systematic noise removal process
(especially the latest msMAP version) strongly suppresses
noise on timescales as short as 30 days, we consider the $R_{var}$
parameter problematic for interpretation of \emph{Kepler\ } data, especially
when trying to obtain an absolute comparison of placing the Sun
within the distribution of stars assessed with \emph{Kepler}.
The second primary measure of variability introduced in BWR13
is the median differential variability MDV($t_{bin})$.  The MDV measures
the variability by forming bins of length $t_{bin}$, then taking
the absolute difference between adjacent bins.  The MDV follows as
the median value of the time series of absolute differences.
We have chosen to focus on MDV with $t_{bin}$ = 8 days.  Figure 8 of
BWR13 compares 8-day MDV for the Sun from SOHO data
\citep{froh97}
considering 30 day blocks over a full solar cycle, to an ensemble
of bright \emph{Kepler\ } stars for one quarter (Q9) of data.  In the comparison
given in their Figure 8 a significant fraction of the solar points
have MDV values well below the overall minimum value reached
for about 1,000 \emph{Kepler\ } stars -- an implausible result suggesting
that either something went wrong with evaluating the solar or
\emph{Kepler\ } values, or that the \emph{Kepler\ } time series have significant
residuals on time scales longer than 8-days creating this offset.

In order to further pursue a comparison of \emph{Kepler\ } stars with the Sun
in the 8-day MDV statistic, given the surprising result of many
solar values disjoint from the extrema of 1,000 \emph{Kepler\ } stars we have
formed our own metrics.  Figure~\ref{fig:mdvsun} shows the
distribution of solar values for 30 3-month long intervals of SOHO
\citep{froh97}
data, the same set as discussed in Paper 1, and the 4,529
solar-type \emph{Kepler\ } stars with Kp $<$ 12.5.  Differences with respect
to the BWR13 study include:  (1) We compute 8-day blocks
over 90 days (or length of \emph{Kepler\ } quarter of data), rather than within
30 day intervals to form MDV.  (2) We take the median over all 17
quarters of \emph{Kepler\ } data, rather than adopting Quarter 9.  (3) We use
the latest available (9.2) data release.  None of these differences
are significant.  Our distribution of 8-day MDV values for the Sun
compared to \emph{Kepler\ } stars is radically different than that in Figure 8
of BWR13.  In particular the distribution of MDV values
for the Sun falls within the extent of stellar values from a large
ensemble of stars.  The radically different distribution follows
primarily from our values for the solar MDV.  Our minimum solar
MDV is about 0.04 ppt, while the BWR13 value is about
0.0015 ppt.  We differ by over an order of magnitude in scale for
the solar MDV at 8 days.

\begin{figure}
\begin{center}
\includegraphics[width=80mm]{f3.eps}
\end{center}
\caption{Black dots show the median values over all 17 quarters for
the 4,529 \emph{Kepler\ } solar-type dwarfs for the BWR13 8-day MDV
and Range metrics as computed in this study.  The red crosses show
the same statistics for 30 90-day intervals of solar time series
spanning a full solar cycle.}
\label{fig:mdvsun}
\end{figure}

In order to pursue the latter discrepancy we have compared records
of solar variations used, with the time series used in BWR13
kindly provided by G. Basri.  The latter authors used the mean of
``green" and ``red" VIRGO (SPM) data from SOHO, starting with
hourly cadence data linearly interpolated to half-hour to roughly
match the \emph{Kepler\ } cadence.  Paper 1 also used VIRGO/SOHO data,
but started with a compilation at 60 second intervals and binned
this to 29.4 minutes.  We also adopted just the ``green" channel
and scaled this by 0.79 to adjust amplitudes to the longer average
wavelength of \emph{Kepler}.  Figure~\ref{fig:solboth} shows a representative
0.5 year interval between the adopted solar records of the two
studies.  A detailed comparison shows numerous differences, but
these are at the level of influencing 8-day metrics at the 10\%
level, not the nearly factor of 20 found in our two sets of 8-day
MDV metrics for the Sun.
Our difference from BWR13 for the solar 8-day MDV does not follow
from minor differences in color or sampling for adopted solar records.

\begin{figure}
\begin{center}
\includegraphics[width=80mm]{f4.eps}
\end{center}
\caption{Upper panel shows a 0.5 year interval of the solar
activity record adopted by BWR13.  The lower panel
shows the solar record for the same interval as used both in
Paper 1 and this study.}
\label{fig:solboth}
\end{figure}

Ironically our comparison of solar and \emph{Kepler\ } 8-day MDV better
support a primary contention of BWR13 that the
relative noise levels intrinsic to the stars compared to the
Sun are lower than concluded in earlier studies of Paper 1
and \citet{mcqu12} than does their own result for this metric.
We defer further discussion of typical
activity levels of solar-type \emph{Kepler\ } stars relative to the Sun
until after discussion of results from our favored long timescale
noise metric.

\subsection{Adoption of a CDPP-style Metric with Longer Timescale}

The CDPP metric of Paper 1 and in Section 3 above starts
with the calibrated (instrumental signatures removed to the extent
possible) pipeline data, removes a running 2-day quadratic polynomial
fit to the time series, block averages into 6.5 hour intervals, then
evaluates the standard deviation for quarter-long segments.
We choose here to adopt exactly the same procedure, but now use
timescales longer by $\times$12.  We start with a 24-day quadratic
polynomial fit that will preserve signal at much longer intervals
than the standard 6.5 hour CDPP, then follow this with binning into
3.25 day intervals before evaluating the standard deviation.
Figure~\ref{fig:filtresp} shows the response function of our filtering
and binning operations for this long timescale CDPP.
The 50\% transfer points are at about 8 and 15 days.
\citet{aigr04} found a timescale of 9.8 days best characterized
solar activity variations, rather than the rotation period
of $\approx$26 days.  Our metric nicely spans the timescale of 9.8 days.
We have also directly verified that the 8 -- 15 day bandpass
represents solar variations at high fidelity by evaluating
it for 30 ``quarters" of SOHO/Virgo \citep{froh97} data spanning
a full solar cycle.  The 8 -- 15 day bandpass correlates at the
85\% level with a 8 -- 30 day bandpass measure.  The upper range
of metric being set at 15 days avoids most of the damping inherent
with the calibrated data -- which at 30 days would be largely
removed, and at 20 days would be uniquely perturbed
star-to-star and quarter-to-quarter.  By design this
timescale was chosen to be as long as possible without the long
timescale end already having been significantly suppressed with
instrumental systematics removal in the \emph{Kepler\ } pipeline processing.
Figure~\ref{fig:mapdamp} illustrates the damping introduced
by the pipeline version (``regular MAP", data release 8.0) used
in BWR13, as well as the data products more recently available
(``msMAP", data release 9.2).
This figure supports the selection of an 8 -- 15 day bandpass
filter for our primary long-timescale metric.
This longer timescale CDPP would no longer be relevant to the
detection of 3 -- 12 hour transits, but is well suited to
attempting to characterize activity induced variations in a sample
of solar-like stars.

\begin{figure}
\begin{center}
\includegraphics[width=80mm]{f5.eps}
\end{center}
\caption{Light curves show filter components from the 24 day
quadratic polynomial filtering (zero response at zero frequency),
and sinc function representation of the 3.25 day binning adopted
for the long timescale CDPP.  The bold curve shows the adopted
net response function plotted against frequency with 50\%
transfer periods of about 8 and 15 days flagged.}
\label{fig:filtresp}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=80mm]{f6.eps}
\end{center}
\caption{This plot shows what fraction of existing signals
in the form of injected test sinusoids at amplitudes corresponding
to one standard deviation of the underlying time series are
preserved by data release 8.0 (`regularMAP' -- upper panel),
and data releases 8.3 and 9.2 (`msMAP' -- lower panel).
Signal corruption of unity corresponds to total loss of signal,
while small values indicate high fidelity retention of input
signals through the systematics removal step.
The corruption metric is qualitatively the same as fractional damping
of the signal amplitude.
Clearly, for the current `msMAP' signals with periods $\geq$20 days
are severely damped.
Not shown are more subtle, and less well characterized dependencies
on signal amplitude.  Larger input signals show relatively better
preservation at long periods, while smaller amplitudes show more
damping.}
\label{fig:mapdamp}
\end{figure}

With this much longer timescale metric, and consideration of the
same stellar sample used in Paper 1 some previously relevant
noise terms are now unimportant.  At 6.5 hours for CDPP the Poisson
noise was roughly comparable to the intrinsic stellar term.
At the $\times${12} longer metric the intrinsic stellar term rises
due to better sampling primary timescales of stellar activity,
while the Poisson term drops by $\sqrt {12}$.  Factors from readout noise
on the CCDs, Poisson fluctuations on the counts, and sky background
are now unimportant.

We have attempted to pursue the same type of Singular Value Decomposition
to isolate noise terms associated with individual quarters, the stars
themselves and contributions from the instrument.  This has been
relatively unsuccessful.  The original CDPP noise separation leveraged
off isolating nearly comparable terms, and benefited from a relatively
narrow range of intrinsic stellar noise.  The longer timescale CDPP
encounters much more discrepant components in which the instrumental
(or software inadequacy in dealing this these) terms are small compared to
intrinsic stellar, and more importantly the stars show a broader
distribution of intrinsic noise.  We therefore concentrate on showing
direct evaluations of the longer timescale CDPP for the \emph{Kepler\ } stars,
recognizing that if anything these will be over-estimates of the
intrinsic stellar noise.  We compute the solar metric using the same
algorithms and codes used for the stars.  The somewhat surprising
results are shown in Figure~\ref{fig:cdpplong}.  Panels are included
for analysis of both the simple aperture photometry (raw), and
the calibrated data (release 9.2, version for release 8.3 is identical
for all intents) for which the distribution of stellar values (median
for the 17 quarters is adopted for each star) is shown in relation to
statistics on the corresponding solar values.  Note that there is a
strong cluster in the calibrated data to the low range of solar
variability.  Even with consideration of the raw-data time series
for which no instrumental systematics have been removed the mode
for the stars is well below the mean for the Sun.

\begin{figure}
\begin{center}
\includegraphics[width=80mm]{f7.eps}
\end{center}
\caption{The upper panel shows a histogram of number of stars
(of 4,529 total) at different levels of the long timescale CDPP,
labelled as stellar noise in parts per million based on the
calibrated time series.  The lower panel shows the same based on
use of the direct, or raw data uncorrected for systematics.
The mean and rms distribution for solar noise levels over
quarter-long intervals spanning a solar Cycle are shown by the
``+" and heavy horizontal line, with the full extent of solar
noise per quarter the thin line.}
\label{fig:cdpplong}
\end{figure}

Great effort has been expended in an attempt to make the primary
feature (cluster of stellar values to low range of solar) go away.
While one can never be certain of any result, we have been unable
to resolve this finding.  We have verified that our solar record
in use is reasonable by comparing as in Figure~\ref{fig:solboth}
to an independent compilation.  We have verified that the same
code is used for the Sun and stars to form the CDPP.
We have verified that the stellar and solar time series are
normalized in the same way.  Something that could explain the
upper panel of Figure~\ref{fig:cdpplong} would be significant
suppression of stellar signal within our 8 -- 15 day passband already
by the \emph{Kepler\ } pipeline processing.  To pursue this we selected a
subset of stars having CDPP near the mode of 100 ppm in the bottom panel,
that also fell near the much smaller mode near 20 ppm in the upper panel.
We then visually inspected this subset looking for signals
of intermediate frequency (8 -- 15 days) in the raw data that
might have been improperly removed in creating the calibrated data.
While some intermediate frequencies could be seen in the raw
data cases, these invariably seemed to be common mode variations
across the several cases examined, and almost certainly not
inherent stellar signals.  Although expecting that this
(pipeline suppression of real stellar signals) was the most
logical explanation for the distribution in the calibrated
data of Figure~\ref{fig:cdpplong}, we have been unable
to find evidence in support of this contention.  Indeed, having
eliminated all potential contenders considered for an explanation we are
left with accepting the seemingly improbable one that for these
long timescales there is a large subset of the stars having
activity levels near the minimum recently experienced by the Sun.
However, the comparison shown in the upper panel of
Figure~\ref{fig:cdpplong} is also misleading in over-emphasizing
a quiet distribution for the stars.  To higher CDPP values there
is a very long tail not shown in the figure.  Indeed the number
of stars with CDPP greater than the highest encountered by the
Sun is 802, while the number quieter than the lowest solar value
is only 322.
The mean over all stars is 352 ppm$^2$, while the solar mean is 151 ppm$^2$.
The medians switch to 99 ppm$^2$ for the stars and 163 ppm$^2$ for the Sun.
Recall, though, that we have not made SVD-based
adjustments for other non-stellar contributions to the CDPP.
Although we believe such corrections would be minor at this
long timescale, doing so would not change large values, but
could shift some of the smaller (at $\lesssim$ 30 ppm)
stellar values to yet lower values.

Table 3 shows the first five lines for the electronically
available table documenting primary results in this paper.
A total of 4529 stars brighter than Kp = 12.5 met the selection
criteria for solar-type dwarfs as detailed in Paper 1.
For each of these stars Table 3 provides the Kp value,
the standard 6.5 hour CDPP analog, and the inferred intrinsic stellar
noise for this based on analysis of all \emph{Kepler\ } quarters and the
latest data release as discussed in Section 3.
Also provided are the 3.25 day CDPP raw and calibrated data values
as summarized in Figure~\ref{fig:cdpplong} of this section.

\begin{table*}
\begin{center}
\caption{Standard and long timescale CDPP values.\label{etable}}
\begin{tabular*}{0.6\textwidth}{rrrrrr}
\tableline\tableline
KIC & Kp & CDPP & Stellar Noise & $\times$12 CDPP(raw) & $\times$12 CDPP(cal) \\
\tableline
 1025494 & 11.822 &  24.85 &  15.40 &  154.13 &   22.45 \\
 1025986 & 10.150 & 119.56 & 118.46 & 5704.62 & 5722.35 \\
 1026669 & 12.304 &  28.21 &  17.19 &  244.60 &  105.55 \\
 1027030 & 12.344 &  30.40 &  20.27 &  184.49 &  22.97 \\
 1162051 & 12.475 &  53.24 &  52.81 &  982.96 &  947.07 \\
\tableline
\end{tabular*}
\tablecomments{All noise values are in ppm.  The standard timescale
values for our CDPP analog and inferred intrinsic stellar noise are discussed
in Section 3.  The longer timescale CDPP values corresponding to
analysis of both raw and calibrated time series are discussed in
Section 4.  Full version of table is available online.}
\end{center}
\end{table*}

\subsection{Is the Kepler Dwarf Sample More or Less Noisy than the Sun?}

The title of this subsection is a seemingly simple question.
The perhaps best simple answer would be:  It depends.

Previous, careful and reasonable studies that addressed this
question came up with conflicting answers.  BWR13 and
earlier studies sided with the \emph{Kepler\ } stars being at least as
quiet as the Sun, while Paper 1 and \citet{mcqu12}
sided with the \emph{Kepler\ } stars on average being a bit more active
than solar.

For the long timescale CDPP detailed in this section, one measure
is that more solar-type stars (giants have been excluded) have
variations at a level higher than the most active Sun,
than those having variations
at a level lower than the least active Sun.  However, the mode for
the stellar distribution of activity levels is distinctly toward the
quiet end of the solar range of variability.  This latter feature
persists were we to adopt our version of the 8-day MDV metric.
This feature would also persist were we to adopt a long timescale
CDPP metric at half the timescale, i.e. with a primary response
function of 4 -- 7.5 days.

Robust removal of instrumental signatures without in some cases
suppressing real stellar signatures is undeniably a difficult problem.
Careful inspection of many raw and calibrated time series suggests
that there is no obvious issue with the pipeline being too
aggressive and suppressing solar-type star intrinsic variations,
although we cannot fully rule this out as a factor contributing
to the stellar distribution.

Therefore, perhaps the best answer to settle on is:  We don't know,
it depends.  There probably is not a good, robust answer to the
simple question posed in this subsection.  What does seem quite
clear, though, in adopting an answer is that the distribution of
solar variability experienced over a recent solar cycle is well
within the range that a large number of \emph{Kepler\ } stars show on average.
The Sun is typical in the \emph{Kepler\ } distribution, which is quite wide
with a non-simple structure.

\section{SIMULATIONS OF STELLAR NOISE}

In Paper 1 we included extensive discussion of using the
galactic population synthesis package TRILEGAL \citep{gira00}
to provide a simulated set of stars appropriate for the
\emph{Kepler\ } field of view.  This was followed by detailed discussion
of granulation and stellar activity contributions, the two of
which were modelled as a function of the TRILEGAL generated
stellar parameters (mass and age).  Normalization was accomplished
for the activity contribution through consideration of both
ground based studies as presented in \citet{radi98}, \citet{lock07},
and \citet{hall09}, as well as reference to solar variations
as measured by SOHO \citep{froh97}.

With both the simulated stellar parameters and codes available
from the Paper 1 study, we have made only one change:  adoption
of the transfer function shown in Figure~\ref{fig:filtresp} for our $\times$12
longer timescale CDPP metric.  Since for this much longer timescale
metric we expect the stellar contributions at 12th magnitude to
generally dominate over Poisson, readout noise and instrumental
terms we have provided only the stellar terms from the simulation.

Figure~\ref{fig:simslong} shows the resulting distribution of
simulated stellar noise at the 3.25 day CDPP timescale considered
in the previous section.  The agreement with observations
as shown in Figure~\ref{fig:cdpplong} is generally quite good.
Stars with parameters close to solar map into mid-range of the
solar variation as measured directly from the SOHO data.
Most importantly the strong peak at low noise levels -- essentially
a pile-up near the lower range of solar variability levels
experienced over a solar cycle, is reproduced in the simulations.
Since the simulation codes were not tuned to reproduce the
distribution seen in the real data of Figure~\ref{fig:cdpplong}, we take the
general agreement as confirmation that the distribution of noise
seen in the real \emph{Kepler\ } data is a reasonable representation of reality.
The consistency between real and simulated data further
demonstrates that the pipeline is not significantly
suppressing stellar signals in our bandpass.

The population of stars in Figure~\ref{fig:cdpplong} at CDPP
values less than 70 ppm presumably arises from two factors.
The fraction of all stars sampled falling below 70 ppm is 38\%.
The first factor is that $\sim$20\% of the time the Sun
is this quiet.  The second factor is that 20\% of the stars in
the simulations of Figure~\ref{fig:simslong} have ages greater
than 5 Gyr.  Thus the very quiet stars sampled by \emph{Kepler\ } may
arise equally from stars similar to the Sun, and in quiet
phases of activity cycles, and from stars inherently older
than solar.

\begin{figure}
\begin{center}
\includegraphics[width=80mm]{f8.eps}
\end{center}
\caption{Simulated distribution of stellar noise arising from
activity and granulation for stars with Kp = 11.5 -- 12.5 as
modelled in Paper 1, with adoption of the transfer function of
Figure~\ref{fig:filtresp} appropriate to the 3.25 day CDPP metric defined in Section 4.}
\label{fig:simslong}
\end{figure}

\section{Summary}

We have repeated an earlier analysis studying noise in \emph{Kepler\ } data
at timescales relevant to the detection of exoplanet transits
using much longer time intervals, and making use of more
recent data products.  The inferred intrinsic stellar noise stayed
fixed with adoption of more, and newer data, thus providing
confidence in the analyses.  The inferred residual noise arising
from the instrument dropped with the consideration of newer data
products, this of course would be expected since the software
updates had been intended to do this.  Residual noise as a function
of time during the \emph{Kepler\ } mission correlates well with solar activity.
The earlier study by us (Paper 1) had shown a strong correlation
between excess noise by-channel with the mean focus offset of the
channels (in the sense that fuzzier images had poorer photometry).
That correlation is still present considering all of the data,
and the most recent data release, but is now relatively weak, consistent
with most possible gains in suppressing instrumental noise now being
in hand.

We have explored a longer timescale metric better suited to elucidating
levels of stellar magnetic activity induced variations.
This has shown mixed results.  We find that the spread of solar
variations over a recent cycle are well within the spread of mean
noise levels for a large sample of solar-type stars.  We also
find that there is a strong concentration of \emph{Kepler\ } noise levels
near the minimum values reached by the Sun.  We have not been
able to find evidence in support of any conclusion for this,
except the one directly presented:  there seem to be many \emph{Kepler\ }
solar-type stars that are as quiet as the quiet Sun, more than
we expected based on either the earlier (Paper 1) study
using a metric less well suited to characterizing stars, or to
modelling of expected noise levels using galactic population
synthesis models \citep[][TRILEGAL]{robi03} suggesting an age distribution averaging younger
than the Sun.  A direct simulation for the long timescale does,
however, show results consistent with the observations.
A significant fraction of stars are older, and hence quieter
than the Sun even though as argued in Paper 1 the overall age
distribution is younger than solar.
As such this study shows that the Sun may be
considered typical of the \emph{Kepler\ } distribution of solar-type star
activity levels.
The significant fraction of stars with activity levels at
or below the quiet Sun is a generally positive result for
habitability \citep{see14} of potential Earth-analogs in
the \emph{Kepler\ } field.
A simple answer to the question of whether the
Sun is quieter or noisier than the \emph{Kepler\ } sample has not been reached.

\acknowledgements

Funding for \emph{Kepler}, the tenth Discovery mission, was provided by NASA's Science Mission Directorate.
The many people contributing to the development of the {\em Kepler Mission}
are gratefully acknowledged.
We thank Joseph Twicken for shepherding evolution of the
\emph{Kepler\ } data processing pipeline.
R.L.G. and L.W.R. have been partially supported through grant
NNX14AK65G-S01 of the NASA ADAP program.
The Center for Exoplanets and Habitable Worlds is supported by the
Pennsylvania State University, the Eberly College of Science, and
the Pennsylvania Space Grant Consortium.
Data presented in this paper were obtained from the
Mikulski Archive for Space Telescopes.
Support to MAST for non-{\em HST} data is provided by the NASA
Office of Space Science via grant NNX13AC07G and by other
grants and contracts.

{\it Facilities:} \facility{{\em Kepler}}.


text:
\section{Introduction}

The spectral energy distributions (SED) of energetic particles outside their galactic sources are
important for determining basic properties of the populations and for assessing the impact of
the particle interactions in the magnetized plasma of galactic halos and galaxy clusters.
Knowledge of these distributions is generally quite limited and is based largely on just radio
observations. Measurements of non-thermal (NT) X-ray and more recently also $\gamma$-ray emission
from the extended lobes of several nearby radio galaxies provide, for the first time, a tangible
basis for detailed modeling of the spectral distributions of energetic electrons and protons in
the lobes. Sampling the SED over these regions by itself, with only limited spatial information,
yields important insight on the emitting electrons and possibly also on energetic protons whose
interactions with the ambient plasma may dominate any observed $\sim 100$ MeV emission (from the
decay neutral pions).

The very luminous nearby radio galaxy Fornax\,A best exemplifies the level of spectral modeling
currently feasible with supplementary X-and-$\gamma$-ray measurements. This system, located at a
luminosity distance $D_L = 18.6$\,Mpc (Madore et al. 1999) in the outer region of the Fornax
cluster, consists of the elliptical galaxy NGC\,1316 and two roughly circular (radius $R \simeq
11^\prime$) radio lobes centered (at a distance $d \simeq 17.5^\prime$) nearly symmetrically on
its east and west sides (e.g., Ekers et al. 1983; Isobe et al. 2006; McKinley et al. 2015). The
galaxy is at the center of a (sub-cluster) group of galaxies, thought to be falling towards the
cluster center (Drinkwater et al. 2001), which includes the star-forming galaxies NGC\,1310,
NGC\,1316C, and NGC\,1317.

Detailed modeling of emission from the lobes of Fornax A was carried out by McKinley et al. (2015)
and Ackermann et al. (2016); these included fitting radio, {\it WMAP}, {\it Planck}, NT emission at
1 keV, and {\it Fermi} measurements. Whereas the radio and X-ray data points can be fit with a
truncated electron spectrum, the indication from both analyses was that Compton scattering of the
(radio-emitting) electrons off the extragalactic background light (EBL) and some assumed level of
the local optical radiation fields, are too weak to account for the emission observed by {\it Fermi}.
This led to the conclusion that the
$\gamma$ emission could only be interpreted as $\pi^0$-decay from {\it pp} interactions; if so,
this would imply a very high proton energy density in the lobes.

Specifically, the deduced proton energy density was estimated to be $\sim 10^2$ higher than
the gas thermal energy density, $\sim 1$ eV\,cm$^{-3}$ (assuming $n_H = 3 \cdot 10^{-4}$
cm$^{-3}$, for $k_BT = 1$ keV; Seta et al. 2013). To ameliorate this clearly problematic result,
McKinley et al. (2015) suggested that {\it pp} interactions occur mostly in filaments
with very high gas over-density and enhanced magnetic field.
This could very well be unrealistic given the implied enhancement also in thermal X-ray emission
and the enhanced level of radio emission by secondary electrons and positrons produced in charged pion
decays. Such enhancements would have likely been detected in both radio maps and X-ray images of the
lobes (e.g., Tashiro et al. 2001, 2009; Isobe et al. 2006).

In an attempt to clarify and possibly remove some of the modeling uncertainty we re-assess
key aspects of conditions in the Fornax A lobes, and repeat detailed calculations of the
emission by energetic electrons and protons. We base our analysis on all the available radio,
X-ray, and $\gamma$-ray measurements, and on newly-published detailed surface-brightness
distribution of stellar emission from the central galaxy NGC\,1316. The latter emission is
sufficiently intense to constitute the most dominant optical radiation field in the lobes;
consequently, the predicted level of $\gamma$-ray emission is significantly higher than in
previous estimates. In Section 2 we briefly review the observations of NT emission in the
various bands and the optical radiation field in the lobe region. In Section 3 we describe
our calculations of the lobe SED model, which is fit to observations in Section 4. We conclude
with a short discussion in Section 5.

\section{Observations of NT Emission from Fornax\,A}

Fornax\,A lobes have been extensively observed over a wide range of radio and microwave
frequencies, in the (soft) X-ray band, and at high energies ($\geq 100$ MeV) with the
{\it Fermi}/LAT. We use the updated measurements of the lobes in all these spectral regions;
the dataset used in our analysis is specified in Table\,1 where the listed flux densities are the
combined emission from both the east and west lobes. On sufficiently large scales the radio
and X-ray emission appears smooth across the lobes, but with east lobe brightness $\sim1/2$
that of the west lobe.

We briefly review the most relevant observations used in our analysis, avoiding many of the
details that have already been presented in the two recent similar analyses by McKinley et al.
(2015) and Ackermann et al. (2016). A reasonably accurate description of the optical and IR
fields in the region of the lobes is important for a correct calculation of the predicted high
energy X-and-$\gamma$ emission from Compton scattering of energetic electrons (and positrons),
a new aspect of our analysis that leads to a different conclusion on the possible origin of the
emission observed by {\it Fermi}.

McKinley et al. (2015) compiled a comprehensive list of previously published radio measurements,
supplemented by {\it WMAP} and {\it Planck} data in the $22$-$143$\,GHz band, and at 154\,MHz
(observed with the Murchison Widefield Array). When available, single-lobe flux densities were
added up to obtain total lobe emission, implying a relatively low contribution from the central
galaxy. The flux drops sharply at $\nu \magcir 30$\,GHz.

X-ray observations and spectral analyses were reported by several groups. Early on Feigelson et
al. (1995), using spatially-unresolved {\it ROSAT} data, argued in favor of a diffuse NT origin
of the X-ray emission. Kaneda et al. (1995), based on a spectral decomposition of {\it ASCA}
$1.7$-$7$\,keV data, suggested that $\sim$$1/3$ and $\sim$$1/2$ of the west and east lobe 1\,keV
flux is NT, with a spectral index consistent with the radio index. This suggestion was confirmed
by Tashiro et al. (2001) who, based on follow-up {\it ASCA} data, concluded that $\sim$$1/2$ of
the west lobe flux is NT; Tashiro et al. (2009), analyzing west-lobe {\it Suzaku} data with a
multi-component spectral model, claimed the $0.5$-$20$\,keV spectrum to be fit by a power-law (PL)
with an energy index consistent with the radio index. Similarly, Isobe et al. (2006) found that
east-lobe {\it XMM-Newton} $0.3-6$\,keV data are well fit by a PL with an index consistent with
the radio index. Following McKinley et al. (2015), we assume that the 1\,keV flux densities
(Tashiro et al. 2009, Isobe et al. 2006) are indeed NT.

Based on 6.1\,yr of Pass\,8 {\it Fermi}/LAT data, Ackermann et al. (2016) reported extended
$>$$100$\,MeV lobe emission consistent with the radio lobes' morphology, and point-like emission
from the radio core. A similar level of emission, based on a smaller dataset and a point-source
spatial model, had been previously reported by McKinley et al. (2015).

\begin{table}
\caption[] {Emission from the lobes.}

\begin{tabular}{ l  l  l  l  l  l  l}
\hline
\hline

\noalign{\smallskip}
Frequency      &             Energy Flux     & Reference                       & & Frequency     &   Energy Flux            & Reference  \\
log($\nu$/Hz)  &$10^{-12}$erg\,cm$^{-2}$s$^{-1}$ &                             & & log($\nu$/Hz) &$10^{-12}$erg\,cm$^{-2}$s$^{-1}$ &     \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
7.476          &       $0.634 \pm 0.063$     & Finlay \& Jones 1973         & & 10.515        &  $3.08 \pm 0.185$     & McKinley et al. 2015 \\
7.933          &       $0.814 \pm 0.163$     & Mills et al. 1960            & & 10.607        &  $2.67 \pm 0.240$     & McKinley et al. 2015 \\
8.188          &        $1.16 \pm 0.220$     & McKinley et al. 2015         & & 10.644        &  $2.70 \pm 0.540$     & McKinley et al. 2015 \\
8.611          &        $1.06 \pm 0.106$     & Robertson 1973; Cameron 1971 & & 10.781        &  $2.84 \pm 0.426$     & McKinley et al. 2015 \\
8.778          &        $1.86 \pm 0.465$     & Piddington \& Trent 1956     & & 10.848        &  $2.11 \pm 0.359$     & McKinley et al. 2015 \\
8.926          &        $1.42 \pm 0.128$     & Jones \& McAdam 1972         & & 11.000        &  $1.20 \pm 0.300$     & McKinley et al. 2015 \\
9.151          &        $1.77 \pm 0.142$     & Ekers et al. 1983            & & 11.155        &  $0.415 \pm 0.212$    & McKinley et al. 2015 \\
9.179          &        $1.77 \pm 0.199$     & Fomalont et al. 1989         & & 17.38         &  $0.498 \pm 0.100$    & Tashiro et al. 2009; Isobe et al. 2006 \\
9.431          &        $2.65 \pm 0.265$     & Shimmins 1971                & & 22.60         &  $0.7 \pm 0.2$        & Ackermann et al. 2016 \\
9.699          &        $2.74 \pm 0.411$     & Gardner \& Whiteoak 1971     & & 23.15         &  $1.0 \pm 0.2$        & Ackermann et al. 2016 \\
10.353         &        $3.18 \pm 0.159$     & McKinley et al. 2015         & & 23.75         &  $1.1 \pm 0.2$        & Ackermann et al. 2016 \\
10.453         &        $3.10 \pm 0.217$     & McKinley et al. 2015         & & 24.31         &  $0.8 \pm 0.2$        & Ackermann et al. 2016 \\

\noalign{\smallskip}

\hline\end{tabular}
\end{table}

\subsection{Optical radiation fields in the lobes}

The superposed radiation field in the lobe region has cosmic and local components.
\medskip

\noindent
{\it a) Cosmic Fields}

\noindent
{\it i)} The cosmic microwave background (CMB) is described as an undiluted Planckian with $T_{\it CMB} = 2.735$\,K
and integrated intensity $\nu I_\nu = 960$\,nW\,m$^{-2}$\,sr$^{-1}$ and energy density $u_{\rm CMB}=0.25$\,eV\,cm$^{-3}$
(Cooray 2016);

\noindent
{\it ii)} The cosmic IR background (CIB), originating from dust-reprocessed starlight integrated over the star formation history
of galaxies, is described as a diluted Planckian with $T_{\rm CIB} = 29\,^o$K (from its 100\,$\mu$m peak) and integrated intensity
$\nu I_\nu = 30$\, nW\, m$^{-2}$\,sr$^{-1}$ and energy density $u_{\rm CIB}=7.85 \cdot 10^{-3}$\,eV\,cm$^{-3}$ (Cooray 2016);

\noindent
{\it iii)} The cosmic optical background (COB), originating from direct starlight integrated over all stars ever formed, is
described as a diluted Planckian with $T_{\rm COB} = 2900\,^o$K (from its 1\,$\mu$m peak) and integrated intensity $\nu I_\nu =
24$\,nW\, m$^{-2}$\,sr$^{-1}$, and $u_{\rm COB} = 6.28 \cdot 10^{-3}$\,eV\,cm$^{-3}$ (Cooray 2016).

The dilution factor, $C_{\rm dil}$, is the ratio of the actual energy density, $u$, to the energy density of an undiluted blackbody
at the same temperature $T$, namely $u = C_{\rm dil} a T^4$, where $a$ is the Stefan-Boltzmann constant. The dilution factors
of the cosmic fields are, $C_{\rm CMB} = 1$, $C_{\rm CIB} = 10^{-5.629}$, and $C_{\rm COB} = 10^{-13.726}$.
\medskip

\noindent
{\it b) Local Fields}

The local radiation fields are dominated by emission from the central galaxy, NGC\,1316, whose SED shows two thermal humps,
IR and optical. The IR hump peaks at 100\,$\mu$m and has a bolometric luminosity
\footnote{
The total-IR flux is computed from the {\it IRAS} flux densities at 12, 25, 60 and 100$\mu$m (Golombek et al. 1988)
using $f_{\rm IR}=1.8 \cdot 10^{-11} (13.48\,f_{12} + 5.16\,f_{25} + 2.58\,f_{60} + f_{100})$\, erg\,cm$^{-2}$s$^{-1}$
(Helou et al. 1988). }
$L_{\rm IR} \simeq 1.5 \cdot 10^{43}$\,erg\,s$^{-1}$; its effective temperature is $T_{\rm gal,\,IR} = 29$\,K. The optical hump
peaks at $1\,\mu$m, corresponding to an effective temperature $T_{\rm gal,\,OPT} = 2900$\,K, and has a bolometric luminosity
$L_{\rm opt} \sim 1.4 \cdot 10^{45}$\,erg\,s$^{-1}$ from the total $r$-band magnitude (Iodice et al. 2017), converted to bolometric
magnitude, as detailed below.

The optical stellar surface-brightness distribution of NGC\,1316, with a half-light radius of $135^{\prime \prime}$, can be
decomposed into a (nearly) de\,Vaucouleurs spheroid plus an exponential envelope (Iodice et al. 2017). Its deprojection, i.e.,
the emissivity distribution, will be used to model the distribution of the photon energy density.
\smallskip

\noindent
The de\,Vaucouleurs profile is given by $\sigma(R) = \sigma_0 \, {\rm exp}\left[-\delta\,(R/R_e)^{\alpha}\right]$, where
$\sigma_0$ is the central surface brightness, $\delta=7.67$, $\alpha = 1/4$, and $R_e$ is the radius encompassing half the
(integrated) luminosity. The corresponding deprojected profile can be well approximated by the analytical expression
$\rho(x)=\rho_0\, x^{-\beta} {\rm exp}\left(-x^{\alpha}\right)$ where $x=r\, \delta^4/R_e$,
$\rho_0 = \sigma_0 \, \delta^4/(2 R_e) \,\Gamma(8)/\Gamma\left((3-\beta) / \alpha)\right)$, and
$\beta=0.855$ (Mellier \& Mathez 1987). From $r$-band surface photometry measurements (Iodice et al. 2017) $R_e = 87^{\prime
\prime}$ ($\sim 7.8$\,kpc) and $\mu_{e,\,r} = 20.76$ mag\,arsec$^{-2}$. The latter can be transformed into $b$-band magnitudes
using $b = r + 1.33\,(g-r) + 0.20$ (Jester et al. 2005), with $g-r = 0.74$ inside the half-light radius (Iodice et al. 2017):
$\mu_{e,\,b} = 21.94$ mag\,arsec$^{-2}$. The $b$-band bolometric correction, needed to estimate the full-band magnitude, is
BC$_b = -0.85 - (b-v)$ mag (Buzzoni et al. 2006); with $b-v = 0.84$ (within $2\,R_e$; Cantiello et al. 2013), BC$_b = -1.69$.
Thus $\mu_{e,\, {\rm bol}} = 20.25$ mag\,arsec$^{-2}$, which corresponds to
\footnote{
$\mu($mag\,arsec$^{-2}) = M_{\rm bol \sun} + 21.572 -2.5\,$log\,$\sigma(L_{\rm bol \sun}$\,pc$^{-2})$,
with $M_{\rm bol \sun} = 4.74$ mag and $L_{\rm bol \sun} = 3.83 \cdot 10^{33}$ erg\,s$^{-1}$.
}
$\sigma_{e,\, {\rm bol}} = 264.9\, L_{{\rm bol}\, \sun}$\,pc$^{-2}$; from $\sigma_0 = e^{7.67} \sigma_e$ we finally derive
$\sigma_{0,\, {\rm bol}} = 228.3$\, erg\,cm$^{-2}$s$^{-1}$. The resulting volume brightness distribution of the NGC\,1316
stellar spheroid is
\begin{eqnarray}
\lefteqn{
\rho_{\rm sph}(r) ~=~ 4.7 \cdot 10^{-21} \, \left({r \over R_e}\right)^{-0.855} \, {\rm exp}\left[-7.67\,\left({r
\over R_e}\right)^{1/4}\right]  \hspace{0.2cm}  {\rm erg\, cm^{-3}\, s^{-1}} \,.  }
\label{eq:MelMa87}
\end{eqnarray}
\smallskip

\noindent
The deprojection of the exponential profile, $I(R) = I_0 \, {\rm exp}\left[-(R/h)\right]$, where
$I_0$ is the central surface brightness and $h$ is a characteristic scale, is expressed in terms of a
modified Bessel function of the second kind, $\rho(r) = I_0 /(\pi \,h) ~ K_0(r/h)$ (Baes \& Gentile
2011). From Iodice et al. 2017 we get $\mu_{0,\,r} = 22.79$ mag\,arsec$^{-2}$ and $h_r=317^
{\prime \prime}$ ($\sim 28.6$ kpc); however, the stellar envelope is measured to be considerably
bluer at larger radii (see their Figs.\,4,\,6-left), implying $h_g \sim 360^{\prime \prime}$ ($\sim
32.5$ kpc). Following the same procedure adopted for the spheroid, $I_{0,\,{\rm bol},} = 0.016$
erg\,cm$^{-2}$s$^{-1}$. The resulting emissivity distribution of the NGC\,1316 stellar envelope is
\begin{eqnarray}
\lefteqn{
\rho_{\rm env}(r) ~=~ 5.2 \cdot 10^{-26} \, \, K_0(r) \hspace{0.2cm}  {\rm erg\, cm^{-3}\, s^{-1}} \,.  }
\label{eq:BaesGent11}
\end{eqnarray}

We can now calculate the galaxy optical photon energy density, $u_{\rm gal\, OPT}$, in the lobes.
Given that the current X-ray and $\gamma$-ray data are not spatially resolved, we take a volume
average as a nominal value. The exact calculation accounting for the full light distribution and
averaging over the volume of the lobes requires a 6D integration, but given the approximate
nature of our treatment, we compute $u_{gal,\, OPT}$ as a line average along the
nearest-to-farthest--boundary lobe diameter of the stellar spheroid and envelope components
\begin{eqnarray}
\lefteqn{
u_{\rm j} = {\rho_0\, \lambda_{\rm j} \over 8 \pi c \rho_s} \,
\int_0^{2r_s \over \lambda_{\rm j}}
\int_0^\infty
\int_0^{2\pi}
\int_0^\pi
f_{\rm j} \left(
\sqrt{
\left[ \rho \sin(\theta) \sin(\phi) + d+ \xi \right]^2 +
\left[ \rho \cos(\theta) \right]^2 +
\left[ \rho \sin(\theta) \cos(\phi) \right]^2
}
\right) \, \sin(\theta) \, d\theta \, d\phi \, d\rho \, d\xi \hspace{0.2cm}  {\rm erg \over cm^{3}}   }
\label{eq:u_lobe}
\end{eqnarray}
where $j$ denotes either component, $\lambda_{\rm j}=R_e, \, h$, and $f_{\rm j}(R/\lambda)$ is given
by Eqs.\,(\ref{eq:MelMa87}) and (\ref{eq:BaesGent11}). The resulting $u_{\rm j}$ values are $7.500 \cdot
10^{-14}$ erg\,cm$^{-3}$ (disk) and $4.556 \cdot 10^{-14}$ erg\,cm$^{-3}$ (envelope).

Finally, $C_{\rm gal\, OPT} = 10^{-12.65}$. A similar calculation yields $C_{\rm gal\, IR} = 10^{-7.00}$.
Thus, $n(\epsilon)$ includes the CMB, cosmic and local (NGC\,1316) IR/optical radiation fields.

Further contributions to the local optical radiation field should also be assessed. The star-forming
galaxies NGC\,1310, NGC\,1316C, and NGC\,1317, members of the subcluster that contains Fornax\,A, lie
close to NGC\,1316. As their estimated projected separations from NGC\,1316 ($\sim 20^\prime$, $30^\prime$,
$6^\prime$) are comparable to the Fornax\,A size, they are likely located within the lobe region. Their
cumulative luminosity is relatively modest, $\mincir$15\% of the NGC\,1316 luminosity
(Iodice et al. 2017).

\section{NT emission in the lobes}

Radio emission in the lobes is by electron synchrotron in a disordered magnetic field whose mean value $B$
is taken to be spatially uniform, and X-$\gamma$ emission is by Compton scattering of the electrons by the
CMB and optical radiations fields. A significant energetic proton component could yield additional radio
emission by secondary electrons and positrons produced by $\pi^{\pm}$ decays, and $\gamma$-ray emission from
$\pi^{0}$ decay. The calculations of the emissivities from all these processes are well known and standard,
and need not be specified here. We will limit our brief treatment here to features in the calculations that
result from the observational need to explicitly assume (at the outset) that the particle spectral
distributions are truncated.

Given the lack of any information about temporal evolution of the lobe emission, it is appropriate
to represent the spectral electron and proton numbers by time-independent, truncated PL
distributions in the Lorentz factor $\gamma$ and energy $E$. The spectral (in $\gamma$) number
of electrons in a lobe is then
\begin{eqnarray}
\lefteqn{
N_e(\gamma) ~=~ N_{e,0} \, \gamma^{-q_e} \hspace{0.25cm}  ... \hspace{0.25cm}
\gamma_{min} < \gamma <  \gamma_{max}  \,,
}
\label{eq:electron_spectrum}
\end{eqnarray}
in the interval $[\gamma_{min}, \, \gamma_{max}$] with $\gamma_{min}>>1$ and finite $\gamma_{max}$.
Similarly, the spectral (in $E_p$) proton number in a lobe is
\begin{eqnarray}
\lefteqn{
N_p(E_p) ~=~ N_{p,0} \, E_p^{-q_p} \hspace{0.25cm}  ... \hspace{0.25cm}
E_p^{min} < E <  E_p^{max} \,,
}
\label{eq:proton_spectrum}
\end{eqnarray}
in the interval $[E_p^{min}, \, E_p^{max}$].

\subsection{Radiative yields of electrons and positrons}

\noindent
Synchrotron emissivity by the above (isotropically distributed) population of electrons is (Blumenthal \& Gould 1970)
\begin{eqnarray}
\lefteqn{
j_s(\nu) ~=~ { \sqrt{3} e^3 B N_{e,0} \over 4 \pi\,m_e c^2} \,
\int_\Omega \sin \theta \, d\Omega_\theta \,
\int_{\gamma_{min}}^{\gamma_{max}} \gamma^{-q_e} d\gamma \
\, {\nu \over \nu_c}
\int_{\nu/\nu_c}^\infty K_{5/3}(\xi) \, d\xi
\hspace{0.5cm}  {\rm erg}\, {\rm cm}^{-3}\, {\rm s}^{-1}\, {\rm Hz}^{-1} \,, }
\label{eq:synchro_emissivity1}
\end{eqnarray}
where $\nu_c=\nu_0 \gamma^2 \sin \theta$
($\nu_0 = {3eB \over 4 \pi m_ec}$ is the cyclotron frequency).
The modified Bessel function of the second kind, $K_{5/3}(\xi)$, has the integral representation
\begin{eqnarray}
\lefteqn{
K_{5/3}(\xi) ~=~ \int_0^\infty e^{-\xi\, \cosh(t)} \cosh\left({5 \over 3} t\right)\, dt \,.}
\label{eq:K53}
\end{eqnarray}
With $x = \nu/\nu_c = \nu/[\nu_0 \gamma^2 \sin \theta ]$ Eq.\,(\ref{eq:synchro_emissivity1}) transforms into
\begin{eqnarray}
\lefteqn{
j_s(\nu) = N_{e,0} {\sqrt{3} \over 4} {e^3 B \over m_e c^2} \left({\nu \over \nu_0}\right)^{-{q_e-1 \over 2}}
\int_0^\pi \sin \theta ^{q_e+3 \over 2}
\int_{\nu \over \nu_0 \gamma_{max}^2 \sin \theta }
^{\nu \over \nu_0 \gamma_{min}^2 \sin \theta }
x^{q_e-1 \over 2}
\int_x^\infty \int_0^\infty e^{-\xi \, \cosh(t)} \cosh\left({5 \over 3} t \right)\, dt \,d\xi\, dx \, d\theta
\hspace{0.2cm}  {{\rm erg} \over {\rm cm}^3 {\rm s \, Hz}} \,. }
\label{eq:synchro_emissivity_PL1}
\end{eqnarray}
This 4D integral can be readily reduced to a 3D integral
\begin{eqnarray}
\lefteqn{
j_s(\nu) ~=~ N_{e,0} {\sqrt{3} \over 4} {e^3 B \over m_e c^2} \left({\nu \over \nu_0}\right)^
{-{q_e-1 \over 2}}\int_0^\pi \sin \theta ^{q_e+3 \over 2}
\int_{\nu \over \nu_0 \gamma_{max}^2 \sin \theta }
^{\nu \over \nu_0 \gamma_{min}^2 \sin \theta } x^{q_e-1 \over 2}
\int_0^\infty e^{-x \, \cosh(t)} {\cosh\left({5 \over 3} t \right) \over \cosh\left(t \right)\,} dt \,dx \,
d\theta\hspace{0.5cm}  {{\rm erg} \over {\rm cm}^3 {\rm s \, Hz}} \,. }
\label{eq:synchro_emissivity_PL2}
\end{eqnarray}

The (differential) number of scattered photons in Compton scattering of electrons from the above distribution by
a diluted (dilution factor $C_{\rm dil}$) Planckian (temperature $T$) radiation field,
\begin{eqnarray}
\lefteqn{
n(\epsilon) ~=~ C_{\rm dil}~ {8 \pi \over h^3 c^3}~ {\epsilon^2 \over e^{\epsilon /  k_BT} -1}
\hspace{0.2cm}  {\rm cm^{-3}\, s^{-1}\, erg^{-1}} \,,}
\label{eq:phot_numb_dens1}
\end{eqnarray}
is (Blumenthal \& Gould 1970)
\begin{eqnarray}
\lefteqn{
{ d^2 N_{\gamma, \epsilon} \over dt\, d\epsilon_1 } ~=~
\int_{\epsilon_{min}}^{\epsilon_{max}}
\int_{ {\rm max} \{ {1 \over 2} \sqrt{\epsilon_1 \over \epsilon},\, \gamma_{min} \} }^
{\gamma_{max}}N_e(\gamma)\, {\pi r_0^2 c \over 2 \gamma^4} \,  {n(\epsilon) \over \epsilon^2} \,
\left( 2\epsilon_1 \, {\rm ln} {\epsilon_1 \over 4\gamma^2 \epsilon} + \epsilon_1 + 4\gamma^2
\epsilon - {\epsilon_1^2 \over 2\gamma^2 \epsilon} \right)
~ d\gamma \, d\epsilon
\hspace{0.5cm}  {\rm cm^{-3}\, s^{-1}\, GeV^{-1}} \,,}
\label{eq:IC_emissivity2}
\end{eqnarray}
where $\epsilon$ and $\epsilon_1$ are the incident and scattered photon energies, and $r_0 = (e^2/m_ec^2)^2$ is
the electron classical radius. In Eq.\,(\ref{eq:IC_emissivity2}) we set $\epsilon_{min}=0$ and $\epsilon_{max}/h
= 10^{12},~ 10^{13}$, and $10^{15}$\,Hz for the CMB, IR, and optical components, respectively.
\medskip

\subsection{Pion decay yields}

\noindent
Interactions between energetic and (thermal) gas protons lead to the production of $\pi^0$ and $\pi^\pm$, whose
decays yield $\gamma$\,rays and $e^\pm$, respectively.
\medskip

The $\pi^0$-decay $\gamma$-ray emissivity is (e.g., Stecker 1971)
\begin{eqnarray}
\lefteqn{
Q_\gamma(E_\gamma)|_{\pi^0}  ~=~
2 \, \int_{E^{min}_{\pi^0}}^{E^{max}_{\pi^0}} \,
{ Q_{\pi^0}(E_{\pi^0}) \over \sqrt{ E_{\pi^0}^2 - m_{\pi^0}^2 }} \,  dE_{\pi^0} \hspace{0.25cm}
{\rm cm^{-3}\, s^{-1}\, GeV^{-1}} }
\label{eq:hadr_emissivity1}
\end{eqnarray}
where the factor 2 accounts for the two photons emitted by the $\pi^0$ decay, $E^{min}_{\pi^0} = E_{\gamma} +
m_{\pi^0}^2 /(4 E_\gamma)$ GeV is the minimum $\pi^0$ energy required to produce a photon of energy $E_\gamma$,
$E^{max}_{\pi^0}$ is the maximum energy of a neutral pion specified in terms )of the above two masses and the
maximum proton energy $E^{max}_p$, and
\begin{eqnarray}
\lefteqn{
Q_{\pi^0}(E_{\pi^0})
 ~=~
{4 \over 3} \pi n_H \,\int_{E^{thr}_p(E_{\pi^0})}^{E^{max}_p} \, J_p(E_p) \,
{ d\sigma (E_{\pi^0}, E_p) \over dE_{\pi^0} } \, dE_p
\hspace{0.25cm}  {\rm cm^{-3}\,s^{-1}\,GeV^{-1}} }
\label{eq:pion_emissivity1}
\end{eqnarray}
is the spectral distribution of the neutral pions. Here, $n_H$ is the (thermal) gas proton density,
$J_p = {c \over 4 \pi} \, N_p$ is the (isotropic) energetic proton flux, $d\sigma (E_{\pi^0}, E_p) /
dE_{\pi^0}$ is the differential cross-section for production of neutral pions with energy $E_{\pi^0}$
from a collision of a proton with energy $E_p$ with a proton (essentially) at rest, and $E^{thr}_p
\simeq 1.22$ GeV is the threshold energy for production of a neutral pion with energy $E_{\pi^0}$.
We use a $\delta$-function approximation (Aharonian \& Atoyan 2000) to the differential cross-section
by assuming that a fixed average fraction, $\kappa_{\pi^0}$, of the proton kinetic energy, $E_p^{kin}
=E_p - m_p$, is transferred to the neutral pion, i.e., $d\sigma (E_{\pi^0}, E_p) / dE_{\pi^0} \sim
\delta(E_{\pi^0} - \kappa_{\pi^0} E_p^{kin}) \,\sigma_{pp}(E_p)$ with $\sigma_{pp}$ the total
cross-section for inelastic {\it p-p} collisions. Then
\begin{eqnarray}
\lefteqn{
Q_{\pi^0}(E_{\pi^0})
~=~
{c \, n_H \over 3\, \kappa_{\pi^0} } ~ N_p\Big\{m_p + {E_{\pi^0} \over \kappa_{\pi^0}}\Big\} ~
\sigma_{pp}\Big\{m_p + {E_{\pi^0} \over \kappa_{\pi^0}}\Big\} \,}
\label{eq:pion_emissivity2}
\end{eqnarray}
(in cm$^{-3}$s$^{-1}$GeV$^{-1}$) where $\kappa_{\pi^0} \simeq 0.17$ in the GeV-TeV region, and
$\sigma_{pp}(E_p)$ can be analytically approximated (see Eq.\,(79) of Kelner et al. 2006). Noting
that in the range $5 \mincir E_p/{\rm GeV} \mincir 50$ (relevant to our analysis), $\sigma_{pp}$
is roughly constant, $\sigma_{pp,0} \approx 30$\,mbarn, we set $\sigma_{pp} = \sigma_{pp,0}$. The
resulting $\gamma$-ray photon emissivity (in cm$^{-3}$ s$^{-1}$ GeV$^{-1}$; energies are in GeV) is
\begin{eqnarray}
\lefteqn{
Q_\gamma(E_\gamma)|_{\pi^0}  ~=~
{2 \over 3} \, {c \over \kappa_{\pi^0}} \, n_H \, N_{p,0} \, \sigma_{pp,0}
\int_{E^{min}_{\pi^0}}^{E^{max}_{\pi^0}} { (m_p+E_{\pi^0} / \kappa_{\pi^0})^{-q_p} \over \sqrt{ E_{\pi^0}^2 - m_{\pi^0}^2 }}\, dE_{\pi^0}
\hspace{0.25cm}  {\rm cm}^{-3}\, {\rm s}^{-1}\, {\rm GeV}^{-1} \,.}
\label{eq:hadr_emissivity2}
\end{eqnarray}
\medskip

The production rate of electrons and positrons is calculated from the $\pi^\pm$ decay rate density (e.g., Stecker 1971),
\begin{eqnarray}
\lefteqn{
Q_{\pi^\pm}(E_{\pi^\pm}) ~=~
{2 \over 3} c n_H \int_{E_{\rm thr}} N_p(E_p) ~ f_{{\pi^\pm}, p}(E_p, E_{\pi^\pm}) \, dE_p
\hspace{0.25cm}  {\rm cm}^{-3}\, {\rm s}^{-1}\, {\rm GeV}^{-1} \,,}
\label{eq:ch_pion_density1}
\end{eqnarray}
where $f_{{\pi^\pm}, p}(E_p, E_{\pi^\pm})$ is the $\pi^\pm$ energy distribution for an incident proton energy $E_p$.
The $\pi^\pm$ energy distribution can be well approximated by assuming that a constant fraction, $k_{\pi^\pm}$, of the
proton kinetic energy is transferred to charged pions, so that
\begin{eqnarray}
\lefteqn{
f_{{\pi^\pm}, p}(E_p, E_{\pi^\pm}) ~=~ \sigma_{pp}(E_p) ~\delta(E_{\pi^\pm} - k_{\pi^\pm} E_p^{\rm kin}) \, . }
\label{eq:ch_pion_distrib}
\end{eqnarray}
For the range of $q_p$ values of interest here, $k_{\pi^\pm}= 0.25$ (Kelner et al. 2006). Substituting
Eq.\,(\ref{eq:ch_pion_distrib}) in Eq.\,(\ref{eq:ch_pion_density1}) we obtain
\begin{eqnarray}
\lefteqn{
Q_{\pi^\pm}(E_{\pi^\pm}) ~=~ {2 \over 3} \,{c \over k_{\pi^\pm}}\, n_H \, N_{p,0}
~ \left( m_p+{E_{\pi^\pm} \over k_{\pi^\pm}} \right)^{-q_p} ~
\sigma_{pp}\Big\{m_p+{E_{\pi^\pm} \over k_{\pi^\pm}} \Big\}  	\hspace{0.25cm}
{\rm cm}^{-3} {\rm s}^{-1} {\rm GeV}^{-1} \,.}
\label{eq:ch_pion_density2}
\end{eqnarray}
The above equation for the $\pi^\pm$-decay rate density, Eq.\,(\ref{eq:ch_pion_density2}), allows calculation (e.g.,
Ramaty \& Lingenfelter 1966) of the source rate density of the secondary $e^{\pm}$ produced in charged-pion decays,
\begin{eqnarray}
\lefteqn{
Q_{se}(E_e) ~\approx~ {8 \over 3} \,{c \over k_{\pi^\pm}}\, n_H\, N_{p0}\, \sigma_{pp,0} \,
\left( {4 E_e + m_{\pi^{\pm}} \over \kappa_{\pi^{\pm}} } + m_p \right)^{-q_p}
\hspace{0.25cm}  {\rm cm}^{-3} {\rm s}^{-1} {\rm GeV}^{-1}  \,.}
\label{eq:secondary_source2}
\end{eqnarray}

The corresponding steady-state distribution, $N_{se}(\gamma)$, can be calculated via $d \left[b(\gamma) N_{se}(\gamma) \right]
/ d\gamma = Q_{se}(\gamma)$ -- where, assuming transport losses to be negligible, $b(\gamma)$ is the radiative energy loss term
(e.g., Rephaeli 1979). The latter is written as $b(\gamma) = b_0 + b_1(\gamma) + b_2(\gamma)$ (in s$^{-1}$), where $b_0(\gamma)
= 1.2 \cdot 10^{-12} n_e \left[ 1 + {\rm ln} \, (\gamma/n_e) / 84 \right]$ is the loss rate by electronic excitations in ionized
gas, $b_1(\gamma) = 10^{-15}\gamma \,n_e$ is the loss rate by electron bremsstrahlung in ionized gas, and $b_2(\gamma) = 1.3 \cdot
10^{-9} \gamma^2 \, (B^2 + 8\pi \rho_r)$ is the Compton-synchrotron loss rate. In these expressions $n_e$ is the thermal electron
density and $\rho_r$ is the ambient radiation field energy density. The secondary electron density is then
\begin{eqnarray}
\lefteqn{
N_{se}(\gamma) ~=~ { \int Q_{se}(\gamma) \, d\gamma  \over  b(\gamma) }   \hspace{0.25cm}
{\rm cm}^{-3} \,,}
\label{eq:ss_se_spectrum1}
\end{eqnarray}
where
\begin{eqnarray}
\lefteqn{
\int Q_{se}(\gamma) \, d\gamma
~\approx~
{2 \over 3} \,{c \over q_p -1}\, n_H\, N_{p0} \, \sigma_{pp,0} \, m_e^{1-q_p}
\left({4 \over \kappa_{\pi^{\pm}} } \gamma + { m_{\pi^{\pm}} + \kappa_{\pi^{\pm}} m_p \over \kappa_{\pi^{\pm}} m_e} \right)^{-q_p+1}
 	\hspace{0.25cm}  {\rm cm}^{-3} {\rm s}^{-1} \,. }
\label{eq:integrated_secondary_source}
\end{eqnarray}
The resulting $N_{se}(\gamma)$ can be approximated by
\begin{eqnarray}
\lefteqn{
N_{se}^{\rm fit}(\gamma) ~=~ N_{se,0} \left[1+\left({\gamma \over \gamma_f}\right)^\tau
\right]^{-(q_p+1)}   \,. }
\label{eq:fit_ss_se_spectrum2}
\end{eqnarray}
where $\tau$ is a (fit) parameter specified below (see section 5).

\begin{figure}
\vspace{9.0cm}
\special{psfile=lobe_emissions_color.eps  hoffset=-15   voffset=-105.  hscale=60. vscale=60. angle=0}
\special{psfile=lobe_gamma_color.eps  hoffset=225   voffset=-105.  hscale=60. vscale=60. angle=0}
\caption{
{\it Left:} broad-band SED of the Fornax\,A lobes. Data points (Table\,1) are shown by dots.
Emission components are:
synchrotron, black short/long-dashed curves;
Compton/CMB, black dotted--long-dashed curves;
Compton/IR, red long-dashed curves;
Compton/OPT, blue solid curves;
pionic, green dotted curve.
Thinner curves depict secondary-electron emissions.
{\it Right:} dashed curves bracket the $95$\% confidence level of the predicted Compton/OPT emission.
}
\label{fig:SED}
\end{figure}

\section{Modelling the lobe SED}

As has already been deduced in previous analyses (McKinley et al. 2015; Ackermann et al. 2016),
the energy range of electrons emitting the observed radio and-X-ray radiation is relatively narrow,
as is clear from the steep drop of the flux at high radio frequencies, and from the level and
spectral shape of the emission around $\sim 1$\,keV.
The exact emission level near this energy is pivotal to our analysis. X-ray observations were
reported by several groups, starting with spatially-unresolved {\it ROSAT} measurements by
Feigelson et al. (1995). From spectral analysis of {\it ASCA} $1.7 - 7$\,keV data Kaneda et al.
(1995) deduced that the emission had a NT component (with a PL index similar to the radio
index); specifically, they estimated that $\sim 1/3$ and $\sim 1/2$ of 1\,keV flux from the west
and east lobes, respectively, was NT; this was later confirmed by Tashiro et al. (2001). Similarly,
Isobe et al. (2006) found that east-lobe {\it XMM-Newton} $0.3-6$\,keV data are well fit by a
PL with an index consistent with the radio index. Additionally, from analysis of {\it Suzaku}
data Tashiro et al. (2009) estimated that $\sim 1/2$ of the west lobe flux to be NT.
Assuming all the measured flux at 1\,keV originates in Compton scattering off the CMB yields the
overall normalization factor, $N_{e,0} \simeq 8 \cdot 10^{64}$, and the lower energy limit,
$\gamma_{min}\simeq 200$.

Fitting the predicted synchrotron flux to measurements in the $0.03 - 143$ GHz yields the spectral
index of the electron distribution, $q_e = 2.43$, and the upper energy cutoff, $\gamma_{\max} = 7
\cdot 10^4$. The mean magnetic field strength in the lobes is computed using (e.g., Eq.\,4.53 in
Tucker 1975)
\begin{eqnarray}
\lefteqn{
B ~=~ \left[ 2.47 \cdot 10^{-19} ~(5.25 \cdot 10^3)^{(q_e-1)/2} ~ T_{\rm CMB}^{(q_e+5)/2}
~{b(q_e) \over a(q_e)} ~\left[ {F_c \over F_s} \left( {\nu_c \over \nu_s} \right)^{(q_e-1)/2}
\right]^{-1} \right]^{2/(q_e+1)} {\rm gauss} \,. }
\label{eq:B_synchro_IC}
\end{eqnarray}
Eq.\,(\ref{eq:B_synchro_IC}) applies when the synchrotron and Compton fluxes are emitted from the
same region by electrons with untruncated PL spectrum. Substituting the measured 5\,GHz and 1\,keV
flux densities for $F_s$ and $F_c$, respectively, Eq.\,(\ref{eq:B_synchro_IC}) returns $B = 2.64\,\mu$G.
The value we find is only slightly different, $B = 2.9\, \mu$G; the $\sim$$10$\% discrepancy mostly
stems from the scatter of radio data points about the synchrotron curve.

With the electron distribution fully determined, the Compton yields from scattering off the above
specified IR and OPT radiation fields can now be calculated; the results for the full SED are shown
in Fig.\,(\ref{fig:SED}) together with the radio, X-ray, and $\gamma$-ray measurements. The main
current interest in the predicted Compton yields is the emission in the range $0.1 - 100$ GeV probed
by {\it Fermi}-LAT; the fact that our predicted emission in this range matches well the measurements
is an important outcome of our analysis that is essentially a direct consequence of our full accounting
for the optical radiation field in the lobes which yielded a higher energy density than estimated by
Ackermann et al. (2016). As specified in the previous section, in our detailed estimate of the optical
radiation field in the lobes we determined that this field is dominated by the stellar emission of the
central galaxy, NGC\,1316. This conclusion is based on recent mapping of the extended surface brightness
distribution of this galaxy by Iodice et al. (2017).

The energetic proton density in the lobes is likely to be quite significant and the energy density
higher than that of the electrons; however, this does not necessarily imply that the associated
$\pi^{0}$-decay yield can exceed the above estimated level from Compton scattering of primary
electrons off the local optical radiation field. Assuming a proton energy distribution, with
$q_p=2.2$ and a theoretically-motivated nominal proton-to-electron energy-density ratio $\zeta
=\rho_{p}/\rho_{e} = 50$ (see below), we can calculate the secondary electron and $\pi^{0}$-decay
yields in all the above spectral bands. These emissions are all well below the corresponding levels
by primary electrons. Thus, we can only place a $\sim 30\%$ nominal upper limit on the pionic
contribution to the flux in the {\it Fermi}-LAT band, a limit that roughly reflects the modeling
uncertainties in our analysis and the level of precision of the current {\it Fermi}-LAT data.

\section{Discussion}

Detailed modeling of emission in radio lobes is clearly needed in order to determine key properties
of energetic particles, magnetic and radiation fields, and for assessing the impact of AGN jets on
their intergalactic or intracluster environment. In particular, fitting model predictions to
measurements of the lobe SED is critical for a robust determination of the relative significance of
energetic protons when gauged by their radiative yields from interactions with ambient gas. By virtue
of proximity, brightness of its lobes, and adequate multi-spectral observations, Fornax\,A is one of
the most suitable for such a study.

In our analysis of the broad-band SED of the Fornax\,A lobes we use the simplest truncated-PL spectral
distribution, the most constraining SED dataset currently available, and a recently published, sufficiently
precise NGC\,1316 light distribution. Assuming the X-ray flux originates from electron Compton scattering
off the CMB, our main result is that the related flux from Compton scattering off the optical radiation
field, which is dominated by the central galaxy, is consistent with the {\it Fermi}-LAT measurements. Thus,
there is no apparent need for an additional pionic component (at a level comparable to current observations).
This conclusion would only be strengthened if account is taken of the estimated $\mincir 15$\% contribution
to the observed emission by NGC\,1316 (Ackermann et al. 2016), and an additional $\mincir 15$\% enhancement
of the intra-lobe optical light by emission from nearby star-forming galaxies. In light of this, only an upper
limit can be set on a (likely) pionic component, in contrast with the conclusion reached by McKinley et al.
(2015) and Ackermann et al. (2016).

Limits on the ambient proton spectral parameters can be set by selecting the proton distribution's spectral
index and maximum energy to be close to the values that would be implied from fitting a pionic component to
the {\it Fermi}-LAT data; this yields $q_p=2.2$ and $E_p^{max}= 50$ GeV, respectively. The value of the
index is in the range of what is theoretically predicted. Assuming a nominal p/e energy density ratio of $50$,
as appropriate for an electrically neutral NT plasma for the deduced value of $q_e$ and assumed $q_p$ (Persic
\& Rephaeli 2014), then $N_{p0} = 1.7 \cdot 10^{62}$. This would imply a total NT proton energy of $7.2 \cdot
10^{58}$ erg, roughly two orders of magnitude lower than what would be required if the measured $\gamma$-ray
emission were of pionic origin (McKinley et al. 2015; Ackermann et al. 2016).

This low-level normalization implies also a correspondingly low secondary electron (and positron) contribution
to the observed emission; thus, the exact values of the proton and secondary-$e^\pm$ spectral indices are
of little practical interest. More relevant is the fact that the deduced spectral index, $q_e \sim 2.4$, is
significantly lower than the expected value for a population that had aged as result of efficient Compton-synchrotron
energy losses -- whose characteristic time, $t_{\rm CS} = \gamma / b_2(\gamma) \sim 0.08$\,Gyr (estimated using
the relevant photon and magnetic field energy densities and $\gamma = 1.35 \cdot 10^4$, i.e., the likely Lorentz
factor of an electron emitting at $\nu_c = 1.4$\,GHz in the deduced lobe magnetic field, derived from the expression
for $\nu_c$ with $<$$\sin \theta$$> = 2/\pi$), is shorter than the estimated age of the lobes, $t_{\rm lobe}
\sim 0.4$\,Gyr (Lanz et al. 2010). This is perhaps due to lack of information on the particle injection (by the
jet) and propagation mode and related spectro-spatial aspects, or perhaps a consequence of efficient re-acceleration
that can flatten pre-existing NT particle spectra (e.g., Bell 1978; Wandel et al. 1987; Seo \& Ptuskin 1994) within the lobes.
Given these uncertainties, taking $\tau = 3/4$ in Eq.\,(\ref{eq:fit_ss_se_spectrum2}) effectively implies a steady-state
secondary $e^\pm$ index of (asymptotically) 2.4, same as that of the primary electron spectrum. The secondary electron
maximum energy, $\sim {1 \over 4}E_p^{max}$
corresponds to $\gamma_{max} \sim 2.5 \cdot 10^4$.

To assess the accuracy of our quantitative results we focus on the impact of the main observational and modeling
uncertainties. Key parameters are the electron number normalization and endpoints of the energy range which were
determined from the radio and X-ray data. Specifically, the values of $N_{e,0}$ and $\gamma_{min}$ were deduced
from the measured 1\,keV flux, interpreted to be a consequence of Compton scattering off the CMB. While there is
appreciable uncertainty in the level of the NT emission at this energy, which was determined by Tashiro et al.
(2009) and Isobe et al. (2006), it largely impacts the modeled level of the radio spectrum, whose fit to the radio
data yields the value of the mean magnetic field. Therefore, the resulting uncertainty is essentially reflected in
the latter quantity whose exact value (in the range $B \sim 1-3\, \mu$G in all previous analyses) is clearly not
that important (also because it constitutes a nominal volume-averaged value across the lobes). The substantial
uncertainty in the spectral shape of the X-ray data results in some level of parameter degeneracy; e.g., the
combination $N_{e,0} = 1.1 \cdot 10^{65}$, $\gamma_{\min}=700$ is consistent with the 1\,keV flux density, but this
higher $\gamma_{min}$ causes a steeper rise of the predicted flux than suggested by the data. Also, the value of
$\gamma_{max}$, constrained by the high-frequency turnover of the radio spectrum, is affected by the observational
error in the value of $q_e$, whose relative level (at $1\,\sigma$ confidence level) is estimated to be $\sim 4$\%.

A source of (mostly) modeling uncertainty results from monochromatic to bolometric flux conversion required in order
to compute the dilution factor of the optical radiation field sourced by NGC\,1316. This involves converting $r$ to
$b$ and (then) to bolometric magnitudes based on the adopted stellar population synthesis model. The CIB and COB
intensities are known with a $\sim 30$\% uncertainty (see Cooray 2016).

Important are also the relatively large error intervals of the {\it Fermi}-LAT data (below $\sim 20$ GeV). With the
best-fit value of the spectral index determined from the radio data, $\alpha \simeq 0.71\pm 0.04$ ($1\,\sigma$
uncertainty), the propagated uncertainty in the predicted Compton flux in the {\it Fermi}-LAT range is $\sim 15$\%.

A similar joint analysis of the multi-spectral emission from the giant radio lobes of the nearby radiogalaxy Centaurus\,A,
the first extended extragalactic regions detected by {\it Fermi}-LAT (Abdo et al. 2010; Yang et al. 2012), is clearly of
much interest. The lack of unambiguous evidence for NT X-ray emission from the lobes, perhaps due to their large angular
size and complex X-ray morphology (e.g., Schreier et al. 1979; Hardcastle et al. 2009 and references therein), does not
allow a definite conclusion on the origin of the measured $\gamma$-ray emission. Assuming that the observed low energy
($\magcir 60$ MeV) emission originates mostly from Compton scattering off the CMB and EBL, allows calibration of the
electron spectrum in a purely leptonic model (Abdo et al. 2010
\footnote{The optical emission from the host galaxy, NGC\,5128,
was estimated to negligibly contribute to the Compton yield.}
). However, the improved spatial resolution attained in more recent radio measurements (Sun et al. 2016) indicates a
possible magnetic enhancement at the edge of the south lobe, and thus leads to a lower electron number normalization
that lowers the likelihood of the leptonic origin of the $\gamma$-ray emission. As previously suggested for
Fornax\,A, a combined lepto-hadronic model seems to require an unrealistically high proton energy density (Sun
et al. 2016).

Unequivocal observational evidence for energetic protons in the lobes of radio galaxies is of great interest also for
an improved understanding of the origin of extended radio halos in clusters. In a first detailed assessment of galactic
energetic proton and electrons sources in clusters (Rephaeli \& Sadeh 2016), it was assumed that electrons diffuse out
of radio and star-forming galaxies, and because of the lack of clear evidence for an appreciable proton component in the
lobes of radio galaxies, their (secondary electron) contribution to the radio halo emission was conservatively ignored.
The extended distribution of star-forming galaxies, whose relative fraction increases with distance from the cluster
center, is important in order to account for the large size of radio halos. This approach was applied to conditions in
the Coma cluster, where the number of star-forming galaxies was estimated from the total blue luminosity of the cluster,
and only the two central powerful radio galaxies were included as electron sources. It was found that for reasonable
models of the gas density and magnetic field spatial profiles, the predicted profile of the combined radio emission from
primary and secondary electrons is roughly consistent with that deduced from current measurements of the Coma halo.
However, the level of radio emission was predicted to be appreciably lower than the measured emission, suggesting that
there could be additional particle sources, such as AGN and (typically many) lower luminosity radio galaxies. Clearly,
therefore, quantitative evidence for energetic protons in radio lobes would have important implications also on our
understanding of the origin of cluster radio halos.
\medskip

\noindent
{\it Acknowledgement.}
We used the NASA/IPAC Extagalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute
of Technology, under contract with the National Aeronautics and Space Administration. This paper is dedicated to the memory
of our Bologna University colleague, Prof. Giorgio Palumbo.

redpajama_set_name: RedPajamaArXiv