#  Changes in the Na D1 Absorption Components of $\eta$ Carinae Provide Clues
on the Location of the Dissipating Central Occulter

Connor S. Pickett Department of Physics and Astronomy, Embry-Riddle
Aeronautical University, 3700 Willow Creek Road, Prescott, AZ 86301, USA  Noel
D. Richardson Department of Physics and Astronomy, Embry-Riddle Aeronautical
University, 3700 Willow Creek Road, Prescott, AZ 86301, USA  Theodore Gull
Exoplanets and Stellar Astrophysics Laboratory, NASA/Goddard Space Flight
Center, Greenbelt, MD 20771  Space Telescope Science Institute, 3700 San
Martin Drive. Baltimore, MD 21218, USA  D. John Hillier Department of Physics
and Astronomy and the Pittsburgh Particle Physics, Astrophysics, and Cosmology
Center (PITT PACC), University of Pittsburgh, 3941 O’Hara Street, Pittsburgh,
PA 15260, USA  Henrik Hartman Department of Materials Science and Applied
Mathematics, Faculty of Technology and Society, Malmö University, SE-20506
Malmö, Sweden  Nour Ibrahim  Department of Physics and Astronomy, Embry-Riddle
Aeronautical University, 3700 Willow Creek Road, Prescott, AZ 86301, USA
Department of Astronomy, University of Michigan, 1085 S. University, Ann
Arbor, MI 48109, USA  Alexis M. Lane Department of Physics and Astronomy,
Embry-Riddle Aeronautical University, 3700 Willow Creek Road, Prescott, AZ
86301, USA  Emily Strawn  Department of Physics and Astronomy, Embry-Riddle
Aeronautical University, 3700 Willow Creek Road, Prescott, AZ 86301, USA
Augusto Damineli Universidade de São Paulo, Instituto de Astronomia, Geofísica
e Ciências Atmosféricas, Rua do Matão 1226, Cidade Universitária, São Paulo,
Brasil  Anthony F. J. Moffat Département de Physique and Centre de Recherche
en Astrophysique du Québec (CRAQ) Université de Montréal, C.P. 6128, Succ.
Centre-Ville, Montréal, Québec, H3C 3J7, Canada  Felipe Navarete SOAR
Telescope/NSF’s NOIRLab, Avda Juan Cisternas 1500, 1700000, La Serena, Chile
Gerd Weigelt Max Planck Institute for Radio Astronomy, Auf dem Hügel 69,
D-53121 Bonn, Germany

###### Abstract

The Na D absorption doublet in the spectrum of $\eta$ Carinae is complex, with
multiple absorption features associated with the Great Eruption (1840s), the
Lesser Eruption (1890s), and interstellar clouds. The velocity profile is
further complicated by the P Cygni profile originating in the system’s stellar
winds and blending with the He I $\lambda$5876 profile. The Na D profile
contains a multitude of absorption components, including those at velocities
of $-$145 km s-1, $-$168 km s-1, and $+$87 km s-1 that we concentrate on in
this analysis. Ground-based spectra recorded from 2008 to 2021 show
significant variability of the $-$145 km s-1 absorption throughout long-term
observations. In the high ionization phases of $\eta$ Carinae prior to the
2020 periastron passage, this feature disappeared completely but briefly
reappeared across the 2020 periastron, along with a second absorption at
$-$168 km s-1. Over the past few decades, $\eta$ Car has been gradually
brightening demonstrated to be caused by a dissipating occulter. The
decreasing absorption of the $-$145 km s-1 component, coupled with similar
trends seen in absorptions of ultraviolet resonant lines, indicate that this
central occulter was possibly a large clump associated with the Little
Homunculus or another clump between the Little Homunculus and the star. We
also report on a foreground absorption component at $+$87 km s-1. Comparison
of Na D absorption in the spectra of nearby systems demonstrates that this
red-shifted component likely originates in an extended foreground structure
consistent with a previous ultraviolet spectral survey in the Carina Nebula.

Massive stars (732), Binary stars (154), Luminous blue variable stars (944),
Ejecta (453), Circumstellar matter (241)

##  1 Introduction

Luminous Blue Variables (LBVs) are evolved massive stars that are near the
Eddington Limit, and are variable on time scales of days to centuries. The
variability of these stars is such that when the star is observed being
hotter, it is fainter in the optical and brighter in the ultraviolet. The
majority of active LBVs tend to fluctuate between spectral types of late-O or
early B- and F-type supergiants, while maintaining a nearly constant
bolometric luminosity, with only small changes (see for example the work on AG
Car; Groh et al., 2009, 2011). These stars have been observed to have stellar
eruptions where they eject up to several solar masses of material, creating
circumstellar ejecta around many of these objects. The population of LBVs was
recently documented in Richardson & Mehner (2018) and their properties have
been reviewed in the past by Humphreys & Davidson (1994) and van Genderen
(2001).

$\eta$ Carinae ($\eta$ Car; HD 93308; HR 4210; previously $\eta$ Argus) is a
massive binary star within the Carina Nebula and is sometimes considered a
prototype of the LBVs. The system is relatively close to us with a distance of
approximately only 2.3 kpc (Smith, 2006). $\eta$ Car has undergone two
observed eruptions, causing significant amounts of gas and other debris to
obscure the system itself, although there may have been additional eruptions
in the past (Kiminki et al., 2016). This material has resulted in a bipolar
shell of ejecta surrounding the system from the Great Eruption in the 1840s
and from the second Lesser Eruption in the 1890s, known as the Homunculus and
the Little Homunculus, respectively. The binary is in a long-period 5.54 year
orbit ($2022.7\pm 0.3$ d; Damineli, 1996; Damineli et al., 1997; Teodoro et
al., 2016) that is highly eccentric ($e=0.9$; e.g., Damineli et al., 2000;
Grant et al., 2020; Grant & Blundell, 2022). The primary star provides some
ionization to the surrounding gas, such as causing Fe II emission lines, while
the secondary star further ionizes the Homunculus and Little Homunculus.
During the system’s periastron passage, the secondary star passes into the far
side of the optically thick wind of the primary star, which causes the
ionization of gas in our line of sight to drop to normal levels for the
primary star. This variability, where the ionization of the gas temporarily
drops, has been referred to as spectroscopic events in the literature. As
noted in several studies (e.g., Melnick et al., 1982; Damineli et al., 1998;
Davidson et al., 2005; Johansson et al., 2005; Gull et al., 2006; Richardson
et al., 2010, 2015), the change in ionization causes increased absorption of
low-ionization species from the ejecta, especially at velocities associated
with the Little Homunculus (near $-$145 km s-1).

In recent years, the central source of the $\eta$ Carinae system has been
observed to be brightening. Unlike most normal LBV activity such as the S
Doradus cycles, the central source is brightening in both the optical and
ultraviolet in tandem. There are two competing hypotheses related to this
variability. The first is that the central star is quickly evolving as it
settles from the Great Eruption (Mehner et al., 2010, 2011, 2015, 2019;
Davidson et al., 2018; Martin et al., 2021). This suggestion relies on wind
lines, such as H$\alpha$ (Mehner et al., 2010), changing with a long-term
evolution, which could be interpreted as changes in the mass-loss properties.
The second hypothesis is that there was an clump in the line of sight that has
been dissipating and/or moving out of our line of sight (Hillier & Allen,
1992; Hillier et al., 2001; Damineli et al., 2019, 2021). This is supported by
the large amount of added extinction that is needed in the models of the
stellar spectra by both Hillier et al. (2001, 2006) and Groh et al. (2012). As
this extinction was fairly grey, a change in the extinction should allow most
wavelengths to change at similar rates.

We aimed to use the variability of the Na D absorption component associated
with the Lesser Eruption to understand how it relates to the documented
changes in the spectrum of $\eta$ Carinae. The ionization potential of neutral
Na is $\approx 5.14$ eV which corresponds to a wavelength of $\sim$2412Å. Thus
the ionization of neutral sodium is achieved by radiation in the near
ultraviolet. However the ground state photoionization cross-section of Na I
has a Cooper minimum111 A Cooper minimum is a region where the photoionization
cross-section approaches zero, and is typically exhibited by alkali elements
(Cooper, 1962; Pradhan & Nahar, 2011) near 2000 Å222The Na I photoionization
data were computed by K T Taylor for the Opacity project (Seaton, 1987) and
were obtained from http://cdsweb.u-strasbg.fr/topbase/home.html, and
consequently a large fraction of the ionizations will occur by radiation
shortward of 1800 Å. Given the large ultraviolet flux of $\eta$ Car, we expect
Na to mostly singly ionized. As a consequence the time variation of the Na I
equivalent widths will be primarily governed by the time scale for the flux
variations, and thus the orbital time scale around periastron. Thus we would
expect the Na equivalent width ($\propto$column density) variations, and the
ultraviolet flux variations to track each other. In Section 2, we describe 12
years of observations of the system from the Cerro Tololo Inter-American
Observatory (CTIO) 1.5 m telescope and its echelle spectrographs, along with
some complementary observations from the Hubble Space Telescope and the Space
Telescope Imaging Spectrograph (HST/STIS). Section 3 details the measurements
we made of three absorption components in the Na D spectrum. Section 4
discusses these findings, and we conclude this study in Section 5.

##  2 Observations

Our observations come from a time-series of high-resolution spectroscopy of
$\eta$ Car taken with the CTIO 1.5 m SMARTS (Small and Moderate Aperture
Research Telescope System) telescope. All data were taken with a fiber
centered on the object, with a projected angular size of 2.7″. The first
observations were taken in 2008–2010 with the fiber-fed spectrograph having a
resolving power of $R\sim 40,000$, and were described by Richardson et al.
(2010, 2015). Since the 2009 periastron passage, a time-series of higher
signal-to-noise and higher resolution spectra were obtained with the SMARTS
telescope and the new CHIRON spectrograph with modes having a resolving power
of $80,000-90,000$ for these observations (Tokovinin et al., 2013; Paredes et
al., 2021). For the CHIRON time-series, the data were described by Richardson
et al. (2016), with more observations having the same characteristics being
taken in the intervening time, especially around the 2020 periastron passage.
In total, we analyze 418 spectra taken with the CTIO 1.5 m telescope in this
study.

Data were previously reduced for all data previously published using
techniques described in Richardson et al. (2010, 2015) for the fiber echelle.
This was done with standard techniques using IRAF333IRAF was distributed by
the National Optical Astronomy Observatories, which is operated by the
Association of Universities for Research in Astronomy, Inc., under contract to
the National Science Foundation.. For the CHIRON data, the data were processed
with the standard pipeline described recently by Paredes et al. (2021). The
observations also included a standard observation of HR 4468 (B9.5Vn) or $\mu$
Col (O9.5V), which was used to empirically derive the blaze function and then
remove this through division with custom scripts in either IDL or with python.
This was done in orders without spectral lines, with the remaining orders
(primarily around H$\alpha$ or H$\beta$) were interpolated to match adjacent
orders.

The observations of the most recent periastron passage began with CTIO/CHIRON
data collected on 15 October 2019 and concluded with data collected on 18
March 2020 when the telescope was closed due to the COVID-19 pandemic. These
new observations did provide higher signal-to-noise Na D profiles than ever
before, owing to the ongoing increase in the visual brightness of $\eta$ Car
(e.g., Damineli et al., 2019, 2021).

Figure 1: A subset of the data taken for $\eta$ Carinae, from phase
$\phi=11.975$ to $\phi=14.256$ (indicated as numbers below the spectra on the
right), which highlights some of the variations seen in the data. The dotted
lines represent the multiple absorption features discussed in this paper
relative to the Na D1 line, with the $+87$ km s-1 component also shown as a
dashed line for the Na D2 line. We indicate the velocity of the absorptions at
the bottom of the figure. The data before phase 12.5 were taken with the
lower-resolution fiber-fed echelle spectrograph, while the more recent data
were taken with the CHIRON spectrograph. The variable emission on the red side
of this plot is from He I $\lambda$5876, for which the variability changes the
apparent spacing on the left.

In order to study the Na D complex, we first had to remove contamination from
water in Earth’s atmosphere. We utilized the telluric measurements from
Wallace & Livingston (2003) to construct a telluric template at the same
resolution and sampling rate as the CHIRON data. This template could then be
artificially strengthened or weakened to match each spectrum, under the
assumption that the relative strengths of different lines remains fixed. This
method was used over the entire Na D region, including regions with no
intrinsic features from $\eta$ Carinae to ensure that the remaining spectrum
was intrinsic to the source and the intervening material from the ejecta and
the interstellar medium.

When dealing with spectra of $\eta$ Car, it is important to use a reliable
clock with which we can compare observations to the binary ephemeris. To that
end, we adopt the ephemeris from Teodoro et al. (2016), which should be close
to the timing of periastron in 2014, and is given by the equation relating the
phase $\phi$ to the heliocentric julian date (HJD) by

$\phi=\frac{(HJD-2,456,874.4)}{2022.7}+13.$

In this context, the addition of the 13 in the equation means that the 2014
periastron was the thirteenth observed spectroscopic minimum since the first
reported by Gaviola (1953).

We found an absorption component which we suspected to be interstellar in its
origin, and we collected spectra of neighboring stars with the same high-
resolution mode of the CHIRON spectrograph in July 2021. The objects observed
were HD 93204, HD 93205, HD 303308, and CPD 59∘ 2628\. Using the same
methodology as with $\eta$ Car, the spectra were reduced and telluric
corrected.

We also used archived spectra taken with the Hubble Space Telescope/Space
Telescope Imaging Spectrograph (HST/STIS) in the E140M echelle mode with a
nominal resolving power, R$=$45,800. The three spectra were recorded at phases
$\phi=$ 11.119, 13.194 and 14.170, which is in the early recovery to the high-
ionization state of the nebular structures. These data were pipeline-processed
and then downloaded from MAST.

##  3 Absorption Components at Various Velocities

In Fig. 1, we show selected spectra from our time-series of the system. There
are several features in this figure, which we will describe in detail in the
following section. The interstellar components near 0 km s-1 (and near $-$300
km s-1 for the Na D2 component in the shown Na D1 reference frame) are both
very close to black in our data, but some small uncorrectable amount of
scattered light in the spectrographs makes the profile appear to have a
minimum near 1% of the normalized continuum light.

There is a dramatic disappearance of a feature at $-$145 km s-1 over our full
data set. This and other velocities were fit for a radial velocity by means of
a Gaussian profile or by measuring the point of minimum flux on select data
sets with high signal-to-noise. We caution that these profiles are not always
simple Gaussians, so we adopt these velocities to better describe the
variations of the individual components. Richardson et al. (2010) described a
similar component in the H$\alpha$ profile that appeared and strengthened
across the 2009 periastron passage, and Richardson et al. (2015) noted changes
in the Na D complex similar to those of H$\alpha$ during the 2009 periastron
passage. In the most recent periastron passage, the feature’s center has moved
to bluer velocities, and the profile has bifurcated so that a second component
at $-$168 km s-1 is now visible near periastron. Furthermore, there is a
component at $+$87 km s-1 that is seen in all of the CHIRON spectra, without
much variability present. For our analysis, we concentrate on the Na D1
$\lambda 5896$ line, as the Na D2 $\lambda 5890$ line is strongly blended with
both the D1 components originating in the Homunculus and the He I $\lambda
5876$ line.

Figure 2: The measured equivalent width strength of the $-$145 km s-1 feature,
in reference to Na D1, through the 2009 and 2014 events, where the strength
increases drastically near the two periastron passages. In the interim time,
the feature is seen to have a slow decline, which accelerates to higher
velocities after the 2014 event.

For the features that we discuss in this paper, we measure the equivalent
width defined as

$W_{\lambda}=\int{\frac{F_{C}(\lambda)-F_{\lambda}(\lambda)}{F_{C}(\lambda)}}d\lambda,$

where $F_{C}$ is the continuum flux and $F_{\lambda}$ is the flux as a
function of wavelength ($\lambda$). The continuum around these features is
difficult to define as the normalized spectrum shows broad Na D emission from
the outer portions of the stellar wind. Therefore, we re-normalized each
observation around the feature to measure the strength at that time of the
observation. In order to estimate the errors in Wλ of the measurements, we
adopted the method of Vollmann & Eversberg (2006), where the error depends on
the dispersion of the spectrum ($\Delta\lambda$), the number of spectral
pixels integrated over ($n$), and the signal-to-noise ratio ($SNR$) of the
spectrum per resolution element by the equation

$\sigma=\frac{{n\Delta\lambda-W_{\lambda}}}{SNR}.$

The errors for the weak feature at $-168$ km s-1 appear to be a bit larger
than the scatter around the trends in the points even when we carefully
measured the quantities needed in the equation above. Such a trend could mean
that we over-estimated the errors, however since we had to locally normalize
the data around the feature, within the emission component of the Na D
complex, there are additional errors involved in that process. There are also
sources of error that are not accounted for, such as how the strength could
change on a changing slope of the emission profile. We have therefore kept
these error bars at this level, since a measurement to a precision of $<5$mÅ
is difficult to achieve with the star and this instrument.

###  3.1 The Absorption Component at -145 km s-1

Figure 3: The measured equivalent width strength of the $-$145 km s-1 feature
through the 2009, 2014, and 2020 events, phased to the binary clock. For the
2020 event, the equivalent width of the blended components at $-145$ and
$-168$ km s-1 were measured, but multiplied by a factor of 3 to be displayed
on a similar scale as the previous two periastron passages.

Richardson et al. (2015) found that the absorption component near $-$145 km
s-1 increases near the periastron passage, similar to the H$\alpha$ component
that strengthens at the same time (Richardson et al., 2010; Damineli et al.,
2021). We measured these spectra, along with the CTIO/CHIRON data, in the
region surrounding the variable $-$145 km s-1 feature. The exact range changed
with time as the feature moved to bluer velocities in more recent
observations, but the range of integration was typically about 25 km s-1 wide
and moved with the absorption with time. The equivalent width measurements are
tabulated in the online appendix (Table 1). Gull et al. (2006) reported that
ultraviolet features of this velocity seemed to appear stronger at periastron,
yet remain fairly consistent in equivalent width ($W_{\lambda}$) throughout
the remainder of the orbit.

From our long time-series of measurements in Fig. 2, the absorption at $-$145
km s-1 is observed to be weakening over this time period. In Fig. 1 we see
that this feature seems to move to bluer velocities over this same time
period, while the interstellar absorption components stay constant in velocity
as expected. By a time shortly after the 2014 periastron event, this feature
was indiscernible in the spectra despite the higher signal-to-noise
(S/N$\sim$150–200 per pixel). We see similarities in the two events for 2009
and 2014 when phased to the binary clock (Fig. 3). The change in the
ultraviolet flux during the periastron passage is about a factor of three
according to Gull et al. (2021), consistent with the changes seen in this
line. The 2020 event is discussed in the following subsection.

It was reported in Gull et al. (2006) that the $-$145 km s-1 features in the
ultraviolet would change intensity across the spectroscopic minimum. The
absorption feature in Fe II, as measured at $\phi=0.739$, appears during the
broad maximum. However, when at a minimum, the feature appeared stronger and
was also accompanied by two more features at velocities near $-$145 km s-1. As
noted by Gull et al. (2006), this indicates a brief drop in ionization of the
Little Homunculus in our line of sight caused by the shadowing of that region
by the primary’s wind intervening in front of the secondary star.

###  3.2 The Absorption Component at -168 km s-1

Figure 4: The measured equivalent width strength of the $-$168 km s-1 feature
across the 2020 periastron passage, where the strength increases after
periastron. This measurement is a blend of the bluer component of the $-$145
km s-1 with the $-$168 km s-1 component.

During the recent 2020 periastron passage, an absorption component at $-$168
km s-1 has appeared. This absorption may have been present in earlier
periastra, but would have been blended with a much stronger component at
$-145$ km s-1. It was also seen in the H$\alpha$ line as observed by Damineli
et al. (2021), although at the velocity of $-180$ km s-1. This change in the
H$\alpha$ and Na D absorptions associated with the periastron passages is
likely due to a changing ionization structure in the Little Homunculus. A
similar velocity component was reported in low-ionization ultraviolet lines by
Gull et al. (2006) during the low-state corresponding to the periastron
passages. This component, while independent of the $-$145 km s-1 feature,
seems to have become similar in strength to its changing-velocity counterpart
at $-145$ km s-1, and could have been undetected in previous cycles due to the
stronger component at $-145$ km s-1. We compare the blended component to that
of the $-145$ km s-1 component in Fig. 3, where we see that the beginning of
the increase in absorption shows a similar time from the pre-event minimum to
maximum as the previous events.

While not as prominent as the $-$145 km s-1 feature had been, the $-$168 km
s-1 feature became significantly strong enough to take equivalent width
measurements during the last periastron. Measurements for the $-$168 km s-1
feature were taken between the wavelengths of 5892.45Å and 5893.20Å,
corresponding to the radial velocities of $-$180.82 km s-1 and $-$142.62 km
s-1 respectively. This integration includes the smaller contribution of the
$-$145 km s-1 component remnants, which can both be seen in the spectrum at
phase 14.010 in Fig. 1. We attempted to measure the two features at $-145$ and
$-168$ km s-1 independently, but the features were blended and weak and non-
Gaussian in nature. The resulting measurements were deemed unreliable by both
their component errors and visual inspection of the spectra compared with the
measurements.

As seen in Fig. 4, there is variability in $W_{\lambda}$ of the $-$168 km s-1
feature. This originates in a decreasing level of ionization near periastron.
The increased activity near $\phi$ = 14.00 resulted in the steady climb in
magnitude of $W_{\lambda}$, which then begins to decrease near $\phi$ =
14.018. This process is similar to that of the $-$145 km s-1 feature. The
timing of the disappearance of the $-$168 km s-1 feature was interrupted by
the temporary closure of the facilities at CTIO.

###  3.3 The Absorption Component at +87 km s-1

Figure 5: The measured equivalent width strength of the $+$87 km s-1 feature
from the CHIRON spectrograph, where the strength changes slightly near the
2014 periastron, presumably because of the changing profile of the occulter.

Following telluric correction, a previously-unnoticed absorption component was
recognized at $\sim$+87 km s-1. This component, seen in Figure 1, indicates
that material is moving towards $\eta$ Car and would represent an inflow of
material if originating in the gas surrounding $\eta$ Car. A second, identical
component at $-217$ km s-1 in the Na D1 reference frame is also observed and
represents the Na D2 transition at the same velocity.

Equivalent width measurements for the +87 km s-1 feature were taken between
the wavelengths of 5897.55Å and 5897.74Å, corresponding to the radial
velocities of 78.87 km s-1 and 88.53 km s-1and are shown in Fig. 5. Each
measurement was made within the bounds of .19Å, or 9.66 km s-1. We notice a
small change around the 2014 periastron passage, which may be explained by the
change in the $V$-band brightness across periastron such as that observed by
Damineli et al. (2019). In this regard, we see an increase in continuum flux
leading up to the event, followed by a dip in the flux after periastron. With
the equivalent width depending on the continuum, as shown by the associated
equivalent width equation earlier in this section, we see that the small
differences at that time can be explained by the change in flux. Furthermore,
oscillations at times away from periastron may reflect changes in the
photometric brightness, such as those recently documented by Richardson et al.
(2018). However, as noted by Richardson et al. (2010), usually changes in the
equivalent widths due to the continuum are primarily a concern for large flux
lines such as the H$\alpha$ line. It is also possible that these small changes
are due to changes in the Na D emission in the region of this absorption
component, which could alter our measurements. The exact amount of variation
that could be intrinsic to this component should therefore be very small when
compared to the large changes in the underlying flux from the system along
with the variability of the broad Na D wind emission features present in this
region of the spectrum.

##  4 Discussion

Our measurements of Na D1 absorption components from $\eta$ Carinae can be
used to illuminate some recent changes in the spectrum and photometric record.
In particular, Damineli et al. (2019, 2021) found that there are long-term
changes related to the circumstellar extinction in the line of sight that
should conclude in $\sim 2032$. Our results show that the Na D absorption from
the Little Homunculus is much weaker and likely absent at times away from
periastron, in broad agreement with the results from Damineli et al. (2019).
The spectral modeling of the system, first accomplished with HST spectroscopy
and the CMFGEN non-LTE radiative transfer code by Hillier & Miller (1998);
Hillier et al. (2001), required 2.0 magnitudes of grey extinction in the line
of sight to model both the stellar (wind) spectrum and achieve the correct
line strengths for the narrow line emission from the surrounding Weigelt
knots, but also had 5 magnitudes of extinction in $V$-band in the line of
sight towards the binary. We show a cartoon diagram of the most likely
geometry that could accommodate the observations in Fig. 6.

Figure 6: A cartoon diagram showing the geometry of the gas that provides Na D
absorption. A detailed discussion of this geometry is in the text.

If we assume that the component near $-$145 km s-1 originates in the Little
Homunculus as noted in several studies (e.g., Melnick et al., 1982; Damineli
et al., 1998; Davidson et al., 2005; Johansson et al., 2005; Gull et al.,
2006; Richardson et al., 2010, 2015), then we can begin to interpret the
variability and character of the components at $-$145 and $-$168 km s-1.
First, we consider the variability of the component at $-$145 km s-1.
Richardson et al. (2010) describe the variability of the component at the same
velocity for the H$\alpha$ transition. For H$\alpha$, Richardson et al. (2010)
postulate that during the spectroscopic minimum near periastron, the secondary
star would pass behind the primary, and thus drop the ionizing radiation in
our line of sight. This drop in ionization would then relax the gas and allow
lower ionization transitions to be observed in absorption.

These same principles can explain the changes we see in both the $-$145 km s-1
component (Fig. 2) across the 2009 and 2014 periastron passages and the $-$168
km s-1 component (Fig. 4) during the recent 2020 periastron event. The $-$168
km s-1 absorption was only previously observed in six low-ionization lines (Mg
I, Mg II, Cr II, Mn II, Fe II, and Ni II) in the near ultraviolet by Gull et
al. (2006) and only during the low-ionization state of the binary. In
contrast, the $-$145 km s-1 was observed in the same lines, as well as in He
I, Ti II, V II, Co II, and Ni II, although Gull et al. (2006) reported it as
$-146$ km s-1.

Figure 7: Velocity profiles of Si II, Si IV, Al II and Al III resonant lines
as recorded by HST/STIS. The three spectra were recorded at similar phases
following periastron passages 11, 13 and 14 when $\eta$ Carinae has re-entered
the high-ionizaton state. Long-term changes in the Little Homunculus are
apparent between $-$100 and $-$400 km s-1. The absorption at $+$87 km s-1 is
also present in the low ionization Si II and Al II profiles.

In the far-ultraviolet, several transitions have been observed by HST/STIS,
allowing us to make comparisons to the Na D1 observations described here. For
this discussion, we limit ourselves to the Al II $\lambda$1671, Al III
$\lambda$1855, Si II $\lambda$1527, and Si IV $\lambda$1394 transitions which
were observed during the previous two periastron passages where we also have
CHIRON spectroscopy. All of these ultraviolet transitions and Na D are
resonance lines. In Fig. 7, we show these transitions at phases 11.119,
13.194, and 14.170 so that the binary-induced phase differences are minimal,
and we can examine the long-term evolution of the profiles. Each profile was
normalized to the ‘continuum’ flux level near 1483Å recorded at phase $\phi=$
14.170. Gull et al. (2022, in prep) demonstrate that normalization of the
‘continuum’ is valid since virtually all continuum originates from deep within
$\eta$ Car-A.

These profiles show that these species have evolved to show less absorption at
$-$145 km s-1, although these changes are amplified because the normalization
of these data. This shows a similar trend as the Na D observations, where
recent observations since phase $\sim$13.5 have had no absorption present at
that velocity except for a brief appearance at the cycle 14 periastron event.
During the recent periastron, this absorption was seen at higher velocity and
with a companion absorption component at $-$168 km s-1. In the ultraviolet
profiles, the lower ionization Al II and Si II lines show that the absorption
components at these velocities are also seen at higher velocities than in
earlier cycles, such as just following the cycle 11 periastron passage.

We postulate that the absorption component was formed in a spatially
unresolved dense clump within the Little Homunculus. This clump would respond
to the ionization changes induced by the companion’s orbit around the primary
star. In this situation, the clump would be either moving out of our line of
sight or dissipating with time. Both scenarios allow for a weaker absorption
profile now than in the past, consistent with the observations of the
ultraviolet resonance lines and the Na D profiles.

We used a curve of growth method to find the column density of the absorbing
Na atoms during the times of just prior to the 2009 event (phases near 11.95),
near phase 12.5, and after the 2014 event at phase 13.1. We used these times
to avoid times when the ionization changes, which would therefore change the
column density of neutral sodium in our line of sight. At these three epochs,
the equivalent widths were roughly 70$\pm 12$ mÅ, 52$\pm$2.5 mÅ, and
14.7$\pm$2.2 mÅ respectively. With the known oscillator strength and
wavelength of Na D1, we can then calculate the values of $W_{\lambda}/\lambda$
to obtain the values of $F(\tau)$ using the curve of growth from Spitzer
(1978) with an assumed line width of 10 km s-1, which is complimentary to
techniques used for the local interstellar medium (Welsh et al., 1990). Then,
we derive a column density of $4.9\times 10^{11}$ cm-2, $3.5\times 10^{11}$
cm-2, and $0.8\times 10^{11}$ cm-2 for these three epochs respectively,
confirming the lower column density in recent times. This is in agreement with
the changes in the ultraviolet flux which has increased by about a factor of
10 in the same time period (Gull et al., 2021).

Figure 8: A comparison of the Na D profiles of $\eta$ Carinae at
$\phi=13.964$, CPD -59∘ 2628, HD 93204, HD 93205, and HD 303308 at 1 arcmin to
north. $\eta$ Carinae, CPD -59∘ 2628, and HD 303308 clearly have features
present at positive velocities similar to that of $\eta$ Car. Similar
interstellar features are denoted with arrows.

With a clump in the line of sight, we can also imagine that the angular size
was relatively small to us. The neighboring Weigelt knots, separated from the
central source by only about 0.2″(Weigelt & Ebersberger, 1986), did not
undergo the same amount of extinction as the central source according to the
models of Hillier et al. (2001). As a result, an angularly small clump similar
to the Weigelt knots, either within the Little Homunculus or in our line of
sight could provide the necessary extinction to provide the absorption lines
and the added extinction toward the binary. As Damineli et al. (2019, 2021)
have noted, the central occulter must be dissipating to explain the object’s
brightening and the lack of observed changes in the object’s spectrum. As a
result, the simplicity of a dissipating or moving clump in our line of sight
would help explain many of the unusual observations of this system, although
neither explanation can be fully refuted with the current data.

The component at $-$145 km s-1 is also seen to shift in its velocity during
the last several cycles as it was dissipating. The movement of the clump out
of the line-of-sight or its thicker parts dissipating on the interior of the
Little Homunculus shell, as illustrated in our Fig. 6, would allow for an
apparent shift in the radial velocity of the component by having the densest
part of the clump being at different points in the outflow of the Little
Homunculus. Furthermore, as the thickest part of the clump at the central
velocity of the clump dissipated or moved out of the line of sight, the
changing ionization front would penetrate into the Little Homunculus allowing
for the weaker $-$168 km s-1 to be visible during the most recent periastron
passage while not observed in Na D in past events.

Moreover, Gull et al. (2016) recently reported on spatially resolved HST/STIS
spectroscopy of the binary across an entire orbit of the system. The results
indicate that the Weigelt knot B (A4 in the analysis of Weigelt & Ebersberger,
1986) has fully dissipated or disappeared in the surrounding region of $\eta$
Carinae. If the clump is similar ejecta as the Weigelt knots that were ejected
during the Lesser Eruption, then the clump is traversing the region rather
than dissipating, then it could now be providing additional extinction toward
this clump. However, Weigelt B could also be dissipating with time, meaning
that the clumps left over from the Lesser Eruption of $\eta$ Car could all be
beginning to dissipate around this time, as the Weigelt knots have been shown
to be associated with the Lesser Eruption of $\eta$ Carinae (Weigelt & Kraus,
2012).

The origin of the component at $+$87 km s-1 is simpler to explain. With an
apparent lack of variability (Fig. 5), it is hard to envision a situation in
which that component would actually represent inflowing material from the
surrounding winds and ejecta. We anticipated this being interstellar in origin
so we tested our hypothesis in two ways. First, we see in the low-ionization
lines of Al II and Si II that there is a component at roughly the same
velocity as the $+$87 km s-1 component of Na D1 and Na D2. This component is
not seen to vary in its absolute absorption with the HST observations recorded
over an interval of 16.5 years (Fig. 7). The $+$87 km s-1 component is not
present in Si IV $\lambda$1394, Al III $\lambda$1855 nor in Si II
$\lambda$1533\. The absorbing cloud contains only singly-ionized ions and is
sufficiently cool that no Si II ions populate the low-lying energy level at
287 cm-1.

These features were reminiscent of the ultraviolet lines of other stars in the
Carina Nebula complex observed by Walborn et al. (2002). Strong absorption has
been previously observed in the nearby CPD -59∘2603, as noted in Danks et al.
(2001). Small line widths of high-velocity features, as well as column density
variations, indicate structures with transverse velocities when compared to
radial velocities. Walborn et al. (2002) depicts these absorptions from both
1997 and 1999 observations. These data show, in total, nine strong, low-
ionization absorptions within our line of sight. The strongest features appear
in lower-velocity regions, but it should be noted that dominant components of
high-ionization states appear at intermediate and negative velocities. The
high-ionization absorption corresponds to the global expansion of the H II
region. This expansion is further investigated in Walborn et al. (2007). These
observations of neutral sodium at varying velocities for stars in the Carina
Nebula raise the possibility that they are formed in low-density outflowing
gas from the region, perhaps originating from a past supernova or the
collective winds of the cluster.

To further test the interstellar origin of this component, we observed three
of the Carina stars observed by Walborn et al. (2002) with the CHIRON
spectrograph at the same resolving power as our $\eta$ Carinae observations,
along with CPD -59∘ 2628 which is very close to $\eta$ Car in the sky444CPD
-59∘ 2628 was seen to interfere with $\eta$ Car’s photometric light curve
observed by BRITE-Constellation reported by Richardson et al. (2018) due to
its close proximity in the sky.. For these observations of Na D shown in Fig.
8, we see many similarities in the interstellar lines.

The primary interstellar lines near 0 km s-1 all show similar components for
these stars. As noted by Walborn et al. (2002), many of these stars have lines
at high velocities. The Na D components of the neighboring CPD -59∘2603 show a
positive velocity component at lower velocity than that of $\eta$ Carinae,
with a similar profile showing a double absorption. The observations of $\eta$
Car show a similar double component of absorption. These similarities with
other stars within the Carina Nebula show that they could be explained by a
complicated foreground absorption complex, perhaps from a past supernova or
cluster wind that shows complexity across the region, but the physical cause
for these complex interstellar absorptions is beyond the scope of this paper.

##  5 Conclusions

Our study of the Na D complex of $\eta$ Carinae has found several interesting
trends that relate to other observations of the system.

  * •

The absorption component near $-$145 km s-1 has been observed to move to
higher negative velocities over the past decade. Simultaneously, the strength
of this absorption has weakened overall, becoming indiscernible by 2018. This
happened in parallel to the brightening of the star while the nebula has
remained more consistent in its brightness. This points to the central
occulter being a spatially unresolved clump in the Little Homunculus.

  * •

We find a new absorption in the spectra that is near $+$87 km s-1 which seems
to be interstellar in origin. Similarly shaped structures are seen in
neighboring stars, but at different velocities. The variability of this
component can likely be attributed to changes in the continuum flux and Na D
wind emission during the periastron passages.

  * •

The interstellar feature at $+$87 km s-1 is also seen in some low-ionization
ultraviolet resonance lines.

  * •

To further disentangle if the occulter is/was in the Little Homunculus or
another clump interior to the Little Homunculus, future observations of the Na
profile at high resolution and in reflected light from various lines of sight
in the Homunculus could illuminate this discussion.

## Acknowledgements

Many measurements in this project were originally made by Lucas St-Jean, and
we thank him for these early contributions that led to these results. The CTIO
observations are the result of many allocations of telescope time. We thank
internal SMARTS allocations at Georgia State University (in 2008–2009), as
well as NOIR Lab (formerly NOAO) allocations of NOAO-09B-153, NOAO-12A-216,
NOAO-12B-194, NOAO-13B-328, NOAO-15A-0109, NOAO-18A-0295, NOAO-19B-204,
NOIRLab-20A-0054, and NOIRLab-21B-0334. C.S.P. and A.L. were partially
supported by the Embry-Riddle Aeronautical University Undergraduate Research
Institute. E.S. acknowledges support from the Arizona Space Grant program.
N.D.R., C.S.P., A.L., E.S., and T.R.G. acknowledge support from the HST GO
Programs #15611 and #15992. A.F.J.M. is grateful to NSERC (Canada) for
financial aid.

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##  Appendix A Online Tables

Table 1: $-$ 145 km s-1Measurements  HJD | $W_{\lambda}$ | $\sigma$
---|---|---
-2,450,000 |  |
(d) | (mÅ) | (mÅ)
4755.8605 | 74.4100 | 6.0000
4784.8577 | 56.6100 | 6.0000
4818.8060 | 64.8500 | 10.7695
4819.8108 | 72.8200 | 7.6075
4821.8520 | 73.1500 | 6.1266
4824.7536 | 70.4100 | 4.5138
4829.7838 | 72.7000 | 6.0701
4831.7530 | 90.4600 | 4.2639
4836.7382 | 127.0000 | 5.5035
4836.7510 | 92.0700 | 5.6323
4837.7318 | 106.9000 | 3.6945
4838.7569 | 125.0000 | 3.4637
4839.7867 | 100.1000 | 3.3596
4840.7789 | 107.0000 | 3.2727
4841.7684 | 130.0000 | 4.6655
4842.7738 | 126.2000 | 2.6296
4843.7735 | 115.9000 | 2.8100
4844.7762 | 155.2000 | 3.0682
4845.7651 | 158.0000 | 3.4639
4846.7704 | 160.0000 | 3.7042
4847.7971 | 121.9000 | 4.0158
4848.7409 | 176.1000 | 5.2701
4849.7256 | 186.4000 | 4.1598
4850.7334 | 142.9000 | 3.0279
4851.7190 | 216.0000 | 4.8571
4852.7420 | 205.5000 | 4.6979
4853.7283 | 221.2000 | 5.3942
4854.7404 | 187.9000 | 4.5506
4855.7358 | 185.4000 | 3.6463
4857.7856 | 194.1000 | 3.2556
4858.7676 | 186.0000 | 3.2942
4859.7749 | 177.8000 | 3.8427
4861.7267 | 182.4000 | 2.7894
4862.7393 | 176.2000 | 4.8369
4863.7592 | 189.5000 | 2.4151
4864.7284 | 193.9000 | 10.1812
4865.7026 | 180.9000 | 4.0439
4866.7166 | 183.4000 | 3.5817
4867.7075 | 186.9000 | 3.2210
4875.6516 | 179.8000 | 4.0514
4879.6606 | 193.0000 | 4.7157
4880.6719 | 165.2000 | 5.1300
4881.6680 | 161.3000 | 6.8569
5125.8682 | 68.2600 | 6.0000
5139.8436 | 70.3600 | 6.0000
5140.8246 | 82.1900 | 6.0000
5164.8118 | 68.4000 | 6.0000
5213.7590 | 79.0900 | 6.7142
5226.7402 | 57.2000 | 4.0890
5230.6255 | 70.7500 | 6.0000
5240.6474 | 64.5900 | 4.7343
5250.6127 | 64.4800 | 3.0098
5254.5753 | 63.3300 | 3.6512
5259.6044 | 65.3400 | 4.1624
5270.5564 | 64.5900 | 3.2782
5278.4984 | 74.3400 | 4.6157
5284.5346 | 66.7000 | 3.5030
5293.5099 | 69.6000 | 3.0000
5306.5008 | 71.9600 | 2.4000
5311.5186 | 84.3400 | 1.8000
5314.5970 | 65.2800 | 1.2000
5317.4860 | 64.8600 | 1.5000
5327.4623 | 65.0300 | 1.0000
5335.4752 | 65.0700 | 1.0000
5360.3720 | 67.4400 | 1.2147
5989.6915 | 53.8782 | 1.2147
5993.7546 | 49.2825 | 1.2244
6001.7288 | 53.9522 | 1.2145
6013.6892 | 50.7330 | 1.2213
6221.8876 | 45.9027 | 1.2315
6236.8436 | 45.9535 | 1.2314
6238.8674 | 47.0313 | 1.2291
6248.8123 | 41.3750 | 1.2410
6254.8795 | 41.8054 | 1.2401
6260.7808 | 45.2798 | 1.2328
6289.7971 | 42.6745 | 1.2383
6361.6472 | 42.3128 | 1.2391
6401.5905 | 42.9964 | 1.2376
6417.5924 | 39.5673 | 1.2449
6607.8431 | 37.8184 | 1.2485
6612.8653 | 45.5610 | 1.2322
6656.7032 | 47.6766 | 1.2278
6659.7093 | 45.8957 | 1.2315
6664.6745 | 49.2717 | 1.2244
6670.7826 | 48.1998 | 1.2267
6672.8378 | 50.6189 | 1.2216
7038.7462 | 27.4097 | 1.2705
6677.7595 | 43.8922 | 1.2357
6687.7085 | 45.7443 | 1.2318
6690.6917 | 44.6191 | 1.2342
6697.7271 | 45.8772 | 1.2316
6698.7748 | 46.1766 | 1.2309
6710.6697 | 44.4810 | 1.2345
6712.6945 | 47.4635 | 1.2282
6718.7120 | 47.7831 | 1.2275
6725.5713 | 49.0575 | 1.2249
6729.5623 | 44.4447 | 1.2346
6732.6062 | 44.0142 | 1.2355
6739.5490 | 44.0139 | 1.2355
6746.5203 | 42.7372 | 1.2382
6750.5418 | 45.3872 | 1.2326
6754.5540 | 44.7233 | 1.2340
6765.5903 | 44.2191 | 1.2351
6766.5463 | 44.7315 | 1.2340
6767.5116 | 44.4612 | 1.2345
6774.5470 | 40.5219 | 1.2428
6775.4705 | 42.4427 | 1.2388
6777.5390 | 43.6835 | 1.2362
6781.4742 | 40.9965 | 1.2418
6786.5036 | 43.9414 | 1.2356
6787.5061 | 42.9262 | 1.2378
6791.5527 | 48.3870 | 1.2263
6795.5095 | 47.9230 | 1.2272
6800.4644 | 44.7013 | 1.2340
6801.5000 | 44.1911 | 1.2351
6802.5067 | 45.2528 | 1.2329
6803.5294 | 48.5563 | 1.2259
6806.5318 | 44.8490 | 1.2337
6807.6147 | 45.1985 | 1.2330
6809.4604 | 48.2952 | 1.2265
6810.4839 | 48.4599 | 1.2261
6811.4890 | 50.8294 | 1.2211
6818.5059 | 48.9579 | 1.2251
6819.5294 | 52.8926 | 1.2168
6823.5395 | 48.1640 | 1.2267
6824.5183 | 50.8258 | 1.2211
6825.5069 | 49.0499 | 1.2249
6829.4942 | 52.4642 | 1.2177
6832.5015 | 52.2552 | 1.2181
6833.5015 | 51.0414 | 1.2207
6835.5347 | 51.0836 | 1.2206
6836.4977 | 51.9177 | 1.2188
6837.4984 | 49.9543 | 1.2230
6838.5142 | 51.2240 | 1.2203
6839.5007 | 50.5233 | 1.2218
6840.4992 | 50.1879 | 1.2225
6845.4618 | 53.6993 | 1.2151
6847.4567 | 54.8509 | 1.2126
6850.5099 | 54.4124 | 1.2136
6855.4628 | 57.0751 | 1.2079
6856.4542 | 59.2667 | 1.2033
6857.4572 | 62.5979 | 1.1963
6858.4560 | 64.9401 | 1.1914
6859.4699 | 67.0792 | 1.1869
6863.4696 | 85.0692 | 1.1489
6864.4670 | 93.0745 | 1.1320
6866.5170 | 105.4488 | 1.1060
6867.4637 | 118.6766 | 1.0781
6870.4592 | 133.8580 | 1.0461
6871.4663 | 137.0995 | 1.0392
6872.4634 | 150.3551 | 1.0113
6873.4627 | 148.4821 | 1.0152
6874.4542 | 155.9589 | 0.9995
6876.4581 | 151.2320 | 1.0094
6878.4816 | 155.9124 | 0.9996
6879.4665 | 163.0676 | 0.9845
6880.4665 | 166.2434 | 0.9778
6881.4646 | 162.9879 | 0.9846
6882.4610 | 163.5481 | 0.9835
6883.4684 | 163.5807 | 0.9834
6885.4659 | 156.7293 | 0.9978
6886.4726 | 155.3184 | 1.0008
6887.4599 | 148.2345 | 1.0157
6941.9068 | 83.2774 | 1.1527
6942.9073 | 78.9372 | 1.1619
6943.9038 | 79.2119 | 1.1613
6950.8754 | 65.4337 | 1.1903
6951.8733 | 61.1214 | 1.1994
6952.8645 | 57.7236 | 1.2066
6953.8796 | 56.6173 | 1.2089
6954.8312 | 55.2559 | 1.2118
6955.8613 | 54.3239 | 1.2137
6956.8738 | 51.8092 | 1.2190
6957.8902 | 49.4740 | 1.2240
6958.8639 | 47.6115 | 1.2279
6959.8471 | 47.2173 | 1.2287
6961.8481 | 47.7860 | 1.2275
6964.8650 | 46.2104 | 1.2309
6965.8623 | 43.0701 | 1.2375
6968.8670 | 42.6636 | 1.2383
6969.8629 | 46.9872 | 1.2292
6972.8807 | 44.0054 | 1.2355
6974.8472 | 39.0575 | 1.2459
6977.8456 | 42.3540 | 1.2390
6979.8393 | 40.8341 | 1.2422
6980.8055 | 44.9754 | 1.2335
6982.8125 | 40.4891 | 1.2429
6984.8198 | 37.2191 | 1.2498
6985.8277 | 39.8160 | 1.2443
6986.8716 | 36.3258 | 1.2517
6988.7943 | 35.3878 | 1.2537
6990.7638 | 36.4655 | 1.2514
6992.7814 | 38.7161 | 1.2467
6994.8169 | 37.5105 | 1.2492
6996.8516 | 35.9668 | 1.2525
6998.8053 | 37.3780 | 1.2495
7000.8371 | 39.6315 | 1.2447
7007.8533 | 36.3848 | 1.2516
7008.7387 | 33.2607 | 1.2582
7009.7487 | 36.4337 | 1.2515
7012.8169 | 32.8265 | 1.2591
7013.8113 | 30.2994 | 1.2644
7014.7384 | 28.1325 | 1.2690
7015.7661 | 29.0392 | 1.2671
7016.7963 | 33.6223 | 1.2574
7018.7787 | 30.2510 | 1.2645
7021.8358 | 28.3041 | 1.2686
7022.7519 | 28.4105 | 1.2684
7025.8280 | 26.3324 | 1.2728
7026.7004 | 26.4958 | 1.2724
7030.6911 | 29.6624 | 1.2657
7032.8183 | 27.5399 | 1.2702
7035.7243 | 27.2459 | 1.2708
7037.8164 | 27.6261 | 1.2700
7038.8521 | 29.9518 | 1.2651
7039.8336 | 26.5309 | 1.2723
7040.7121 | 26.6035 | 1.2722
7043.8463 | 27.2521 | 1.2708
7046.6847 | 28.0830 | 1.2691
7047.6855 | 27.1237 | 1.2711
7048.8103 | 25.4844 | 1.2746
7050.7318 | 23.4698 | 1.2788
7051.8465 | 25.7026 | 1.2741
7059.7902 | 20.4370 | 1.2852
7060.6626 | 26.0203 | 1.2734
7061.8510 | 24.8567 | 1.2759
7062.6967 | 22.0224 | 1.2819
7063.7354 | 21.3341 | 1.2833
7064.7755 | 22.7382 | 1.2803
7068.7710 | 23.0571 | 1.2797
7070.6468 | 21.0678 | 1.2839
7074.8003 | 19.7124 | 1.2867
7075.6999 | 17.5518 | 1.2913
7078.8993 | 17.4877 | 1.2914
7083.6618 | 18.0506 | 1.2902
7088.6417 | 13.9511 | 1.2989
7091.5928 | 18.4445 | 1.2894
7092.7233 | 12.8638 | 1.3012
7093.7108 | 13.2399 | 1.3004
7095.8005 | 14.4630 | 1.2978
7097.6962 | 11.1866 | 1.3047
7104.5614 | 13.8121 | 1.2992
7115.7489 | 15.5300 | 1.2955
7118.5573 | 15.6659 | 1.2953
7120.5391 | 15.9264 | 1.2947
7124.4992 | 11.0033 | 1.3051
7125.5235 | 13.5067 | 1.2998
7127.6192 | 16.6063 | 1.2933
7130.5443 | 13.2235 | 1.3004
7132.5158 | 15.0305 | 1.2966
7134.6264 | 14.2428 | 1.2983

Table 2: $+$ 87 km s-1Measurements  HJD | $W_{\lambda}$ | $\sigma$
---|---|---
-2,450,000 |  |
(d) | (mÅ) | (mÅ)
5989.6915 | 6.9575 | 0.5690
5993.7546 | 6.4877 | 0.4666
6001.7288 | 7.0189 | 0.6171
6013.6892 | 6.7837 | 0.7300
6221.8876 | 5.6955 | 0.7066
6236.8436 | 6.1888 | 0.7984
6238.8674 | 5.4392 | 1.0361
6248.8123 | 5.6955 | 0.5610
6254.8795 | 6.2337 | 0.6488
6260.7808 | 5.5529 | 0.6005
6289.7971 | 6.2622 | 0.5882
6361.6472 | 6.8988 | 0.5884
6401.5905 | 6.8054 | 0.7523
6417.5924 | 7.6743 | 0.5769
6607.8431 | 6.4830 | 0.9925
6612.8653 | 7.0374 | 0.9321
6656.7032 | 6.1734 | 0.4011
6659.7093 | 6.5938 | 0.6570
6664.6745 | 7.5062 | 0.5782
6670.7826 | 6.2313 | 0.8487
6672.8378 | 6.1406 | 0.7884
7038.7462 | 8.8609 | 0.4875
6677.7595 | 5.5892 | 0.6148
6687.7085 | 5.3510 | 0.8682
6690.6917 | 6.9078 | 0.5528
6697.7271 | 7.8226 | 0.6497
6698.7748 | 6.5161 | 0.5798
6710.6697 | 5.6846 | 0.7924
6712.6945 | 6.5423 | 0.7655
6718.7120 | 7.1689 | 0.7073
6725.5713 | 5.9835 | 0.5270
6729.5623 | 7.7273 | 0.4005
6732.6062 | 6.9829 | 0.5528
6739.5490 | 7.1471 | 0.4769
6746.5203 | 7.0010 | 0.5962
6750.5418 | 7.6828 | 0.7995
6754.5540 | 6.2975 | 0.6330
6765.5903 | 7.2603 | 0.5293
6766.5463 | 7.4462 | 0.5885
6767.5116 | 6.5413 | 0.5541
6774.5470 | 6.1398 | 0.5976
6775.4705 | 6.0161 | 0.3171
6777.5390 | 6.1015 | 0.3593
6781.4742 | 6.5604 | 0.5276
6786.5036 | 6.0340 | 0.4806
6787.5061 | 6.0957 | 0.5503
6791.5527 | 6.0166 | 0.6992
6795.5095 | 7.2116 | 0.7634
6800.4644 | 7.1856 | 0.6102
6801.5000 | 5.8983 | 0.4060
6802.5067 | 4.9370 | 0.5346
6803.5294 | 6.3181 | 0.4875
6806.5318 | 3.6377 | 1.1296
6807.6147 | 4.3881 | 0.8387
6809.4604 | 5.4800 | 0.4523
6810.4839 | 6.5743 | 0.6446
6811.4890 | 5.4027 | 0.3113
6818.5059 | 5.9112 | 0.7276
6819.5294 | 6.4559 | 0.7309
6823.5395 | 5.9154 | 0.6874
6824.5183 | 7.5002 | 0.4767
6825.5069 | 5.7093 | 0.4808
6829.4942 | 4.9555 | 0.7242
6832.5015 | 5.5865 | 0.6765
6833.5015 | 5.4094 | 0.5411
6835.5347 | 5.4131 | 0.7196
6836.4977 | 6.0169 | 0.4646
6837.4984 | 5.8978 | 0.2831
6838.5142 | 5.1887 | 0.4357
6839.5007 | 5.1460 | 0.4750
6840.4992 | 5.0470 | 0.6156
6845.4618 | 5.2723 | 0.5693
6847.4567 | 6.2708 | 0.5228
6850.5099 | 5.0739 | 0.5690
6855.4628 | 3.2880 | 0.8426
6856.4542 | 5.3101 | 0.5070
6857.4572 | 5.4737 | 0.6054
6858.4560 | 5.6069 | 0.5388
6859.4699 | 5.2633 | 0.6833
6863.4696 | 6.1416 | 0.5253
6864.4670 | 4.3139 | 0.4121
6866.5170 | 2.9306 | 1.0689
6867.4637 | 5.1125 | 0.6642
6870.4592 | 3.9702 | 0.7236
6871.4663 | 3.4837 | 0.7023
6872.4634 | 4.8141 | 0.5248
6873.4627 | 4.5870 | 0.5503
6874.4542 | 6.1275 | 0.5268
6876.4581 | 5.5267 | 1.2326
6878.4816 | 7.1632 | 0.5748
6879.4665 | 5.3268 | 0.4990
6880.4665 | 5.7424 | 0.4804
6881.4646 | 5.5631 | 0.6196
6882.4610 | 6.3667 | 0.4998
6883.4684 | 5.2834 | 0.5567
6885.4659 | 3.5044 | 0.8973
6886.4726 | 3.7979 | 0.7180
6887.4599 | 5.4324 | 0.5923
6941.9068 | 5.5135 | 0.6352
6942.9073 | 6.0722 | 0.6552
6943.9038 | 7.4447 | 0.5162
6950.8754 | 7.1946 | 0.4945
6951.8733 | 8.1639 | 0.5595
6952.8645 | 8.1985 | 0.5905
6953.8796 | 8.2420 | 0.6296
6954.8312 | 9.3900 | 0.8081
6955.8613 | 8.3603 | 0.5537
6956.8738 | 9.1790 | 0.5338
6957.8902 | 8.5548 | 0.3678
6958.8639 | 7.9437 | 0.5608
6959.8471 | 8.5141 | 0.5679
6961.8481 | 9.3317 | 0.5193
6964.8650 | 8.7717 | 0.5678
6965.8623 | 8.9091 | 0.4967
6968.8670 | 9.1311 | 0.5058
6969.8629 | 8.4862 | 0.5019
6972.8807 | 9.0138 | 0.5930
6974.8472 | 9.0937 | 0.5025
6977.8456 | 8.9742 | 0.5387
6979.8393 | 7.3828 | 0.4877
6980.8055 | 8.8080 | 0.6131
6982.8125 | 9.3305 | 0.3660
6984.8198 | 9.3482 | 0.5050
6985.8277 | 9.0623 | 0.5774
6986.8716 | 9.2466 | 0.5336
6988.7943 | 8.7045 | 0.6594
6990.7638 | 8.9411 | 0.6907
6992.7814 | 9.3522 | 0.5176
6994.8169 | 8.3602 | 0.4506
6996.8516 | 7.8459 | 0.5787
6998.8053 | 9.0233 | 0.5028
7000.8371 | 9.0422 | 0.3881
7007.8533 | 8.0849 | 0.5467
7008.7387 | 9.1964 | 0.5310
7009.7487 | 9.7889 | 0.4396
7012.8169 | 8.1188 | 0.3972
7013.8113 | 9.5207 | 0.4539
7014.7384 | 9.6725 | 0.4726
7015.7661 | 11.3078 | 0.5339
7016.7963 | 10.3123 | 0.5559
7018.7787 | 9.5129 | 0.4672
7021.8358 | 9.3003 | 0.6022
7022.7519 | 9.6649 | 0.7746
7025.8280 | 9.2541 | 0.4911
7026.7004 | 9.5918 | 0.4978
7030.6911 | 10.5962 | 0.7018
7032.8183 | 9.7752 | 0.4993
7035.7243 | 9.6177 | 0.3830
7037.8164 | 9.6682 | 0.6541
7038.8521 | 9.6077 | 0.6162
7039.8336 | 8.6459 | 0.4043
7040.7121 | 9.6737 | 0.4904
7043.8463 | 10.5003 | 0.5425
7046.6847 | 9.5991 | 0.6591
7047.6855 | 9.0888 | 0.7053
7048.8103 | 9.1553 | 0.7067
7050.7318 | 9.3499 | 0.7276
7051.8465 | 9.5500 | 0.6863
7059.7902 | 10.4699 | 0.4528
7060.6626 | 10.3002 | 0.5673
7061.8510 | 10.8566 | 0.6771
7062.6967 | 9.5440 | 0.4022
7063.7354 | 9.0539 | 0.5210
7064.7755 | 10.3209 | 0.6899
7068.7710 | 8.5098 | 0.4984
7070.6468 | 11.1138 | 0.4044
7074.8003 | 9.9561 | 0.5636
7075.6999 | 10.5591 | 0.6446
7078.8993 | 10.1916 | 0.4279
7083.6618 | 11.6915 | 0.3053
7088.6417 | 9.5766 | 0.5989
7091.5928 | 11.7059 | 0.6554
7092.7233 | 10.3560 | 0.4098
7093.7108 | 10.5341 | 0.4324
7095.8005 | 11.3241 | 0.6450
7097.6962 | 9.0192 | 0.5465
7104.5614 | 11.1620 | 0.4707
7115.7489 | 11.3934 | 0.5647
7118.5573 | 11.2683 | 0.4354
7120.5391 | 10.2184 | 0.5184
7124.4992 | 12.2814 | 0.5199
7125.5235 | 9.8804 | 0.3943
7127.6192 | 10.3576 | 0.3959
7130.5443 | 10.3833 | 0.4921
7132.5158 | 9.5261 | 0.6126
7134.6264 | 10.3966 | 0.3892
8116.8606 | 5.2999 | 0.4137
8118.8165 | 5.4894 | 0.4418
8121.8505 | 4.9621 | 0.4014
8130.7972 | 5.2279 | 0.5254
8132.7853 | 4.2357 | 0.5714
8133.8294 | 4.8694 | 0.4552
8135.7970 | 5.0718 | 0.4733
8145.8177 | 5.7343 | 0.4658
8146.8286 | 5.2924 | 0.4023
8147.8066 | 5.9192 | 0.4301
8148.7815 | 5.9017 | 0.4739
8150.7582 | 5.3690 | 0.4648
8158.6342 | 5.7879 | 0.4606
8160.7501 | 5.8058 | 0.4439
8162.7974 | 6.0305 | 0.4349
8164.7352 | 5.5477 | 0.3781
8172.7982 | 6.0642 | 0.4702
8174.6811 | 5.8303 | 0.4847
8176.6893 | 6.1190 | 0.4063
8178.6189 | 6.1358 | 0.4895
8186.7597 | 5.9310 | 0.4177
8188.6939 | 5.6569 | 0.4698
8190.6400 | 5.7723 | 0.4124
8192.7439 | 5.8068 | 0.4421
8200.6405 | 5.7340 | 0.4374
8202.5807 | 5.7793 | 0.4124
8204.6013 | 4.9772 | 0.4790
8206.6028 | 4.9764 | 0.4171
8214.5330 | 5.6293 | 0.5741
8216.5715 | 5.2566 | 0.4795
8218.5708 | 4.5351 | 0.5933
8220.5664 | 5.1248 | 0.4728
8228.6131 | 4.9383 | 0.6416
8230.5611 | 4.8883 | 0.4610
8232.5621 | 4.9927 | 0.4078
8242.5156 | 4.8756 | 0.5288
8244.5167 | 4.7232 | 0.6114
8246.5215 | 4.1816 | 0.5869
8248.6102 | 4.1594 | 0.6660
8258.5731 | 4.6028 | 0.4507
8260.4913 | 4.1079 | 0.6641
8262.4987 | 4.7661 | 0.4661
8270.5720 | 3.1171 | 1.0923
8272.4622 | 4.4305 | 0.4789
8274.5431 | 4.2690 | 0.6600
8276.5334 | 3.7131 | 0.5313
8284.5375 | 4.2254 | 0.6119
8287.4957 | 4.8761 | 0.4310
8289.5123 | 4.1456 | 0.6214
8290.5663 | 4.9216 | 0.4633
8298.5078 | 4.0474 | 0.5453
8301.5267 | 4.8849 | 0.6908
8303.4741 | 4.4906 | 0.5030
8316.4768 | 4.4153 | 0.7053
8329.4678 | 4.8609 | 0.6449
8330.4688 | 4.7149 | 0.6765
8452.8630 | 4.5234 | 0.4178
8466.8571 | 4.4358 | 0.4701
8480.8632 | 4.7315 | 0.5157
8495.7624 | 3.6414 | 0.4933
8513.6726 | 3.5280 | 0.5242
8777.8660 | 5.3919 | 0.9074
8782.8819 | 5.8229 | 0.6407
8785.8744 | 4.7468 | 0.8103
8800.8592 | 6.3867 | 0.4662
8803.8497 | 6.1182 | 0.6688
8805.8312 | 6.8611 | 0.6851
8809.8619 | 6.5885 | 0.8066
8812.8630 | 6.1588 | 0.5554
8814.8574 | 6.4803 | 0.6234
8817.8610 | 5.7883 | 0.5637
8820.8104 | 6.1536 | 0.4876
8822.8133 | 6.0344 | 0.5743
8824.7950 | 6.0404 | 0.6142
8826.8360 | 5.7774 | 0.5780
8827.8608 | 6.2554 | 0.4869
8830.8214 | 6.2087 | 0.8675
8832.8593 | 6.9087 | 0.6375
8834.8460 | 6.7220 | 0.6081
8836.8174 | 5.8818 | 0.6183
8852.8207 | 5.7680 | 0.7024
8853.8375 | 6.5221 | 0.9110
8854.8133 | 5.1232 | 0.9352
8855.8436 | 5.9815 | 0.7943
8856.8048 | 5.7298 | 0.6826
8857.8663 | 5.2833 | 0.6781
8860.8514 | 5.1155 | 0.5466
8861.8533 | 6.2699 | 0.4861
8862.8308 | 5.6169 | 0.5726
8863.8699 | 5.6570 | 0.5895
8864.8254 | 6.8291 | 0.6214
8866.8648 | 5.3074 | 0.9421
8867.8760 | 5.1028 | 1.2621
8868.8816 | 5.4226 | 0.8160
8871.8362 | 5.1718 | 0.7734
8872.8191 | 5.1878 | 1.0159
8875.8653 | 5.3220 | 0.6858
8879.8167 | 5.3580 | 0.6493
8880.8738 | 5.9770 | 0.7499
8881.8639 | 5.4094 | 0.6616
8882.8219 | 5.4810 | 0.6340
8883.8449 | 5.9745 | 0.7162
8884.7985 | 6.0315 | 0.6037
8886.8221 | 5.4267 | 0.5306
8887.8172 | 5.9484 | 0.4732
8888.8627 | 5.9402 | 0.5647
8889.8636 | 5.9201 | 0.5624
8890.8608 | 5.2729 | 0.5544
8891.8962 | 5.4412 | 0.5985
8892.8507 | 4.8640 | 0.7056
8893.8244 | 5.5802 | 0.5533
8894.8883 | 5.0969 | 0.6055
8895.7919 | 6.8546 | 0.7473
8896.8512 | 5.1043 | 0.7421
8897.7793 | 4.7744 | 0.5408
8898.7409 | 4.9412 | 0.5014
8899.7816 | 5.3626 | 0.5083
8900.7151 | 5.4527 | 0.5108
8901.7245 | 5.1093 | 0.5539
8902.7389 | 5.4294 | 0.6548
8903.7444 | 4.8602 | 0.5707
8904.7448 | 5.0492 | 0.5340
8905.7200 | 4.6773 | 0.5662
8906.7154 | 5.2632 | 0.5417
8907.7330 | 5.0922 | 0.5449
8908.7334 | 5.0118 | 0.4672
8909.7035 | 4.7002 | 0.5172
8910.7072 | 6.0793 | 0.5556
8911.6643 | 4.9627 | 0.5609
8912.6790 | 4.9955 | 0.5705
8914.7153 | 4.5219 | 0.6795
8915.7889 | 5.7322 | 0.6033
8916.7380 | 4.9506 | 0.6643
8917.7593 | 6.3266 | 0.5666
8918.6995 | 5.4386 | 0.5988
8919.6849 | 5.6215 | 0.6395
8920.7120 | 5.3495 | 0.5328
8921.7181 | 5.4387 | 0.5980
8922.6421 | 4.2238 | 0.8165
8923.6323 | 5.0022 | 0.6467
8924.6323 | 4.7297 | 0.5665
8925.6034 | 5.2500 | 0.4730

Table 3: $-$ 168 km s-1Measurements  HJD | $W_{\lambda}$ | $\sigma$
---|---|---
-2,450,000 |  |
(d) | (mÅ) | (mÅ)
8116.8606 | 1.9270 | 2.0744
8118.8165 | 2.7298 | 2.2157
8121.8505 | 2.8522 | 2.0058
8130.7972 | 3.1699 | 2.6286
8132.7853 | 5.9941 | 2.8296
8133.8294 | 3.0175 | 2.2726
8135.7970 | 4.1528 | 2.3626
8145.8177 | 2.9724 | 2.3386
8146.8286 | 3.7882 | 2.0119
8147.8066 | 3.0426 | 2.1619
8148.7815 | 3.7487 | 2.3795
8150.7582 | 5.8807 | 2.3191
8158.6342 | 4.8814 | 2.3076
8160.7501 | 4.5840 | 2.2253
8162.7974 | 6.5519 | 2.1777
8164.7352 | 6.0477 | 1.8886
8172.7982 | 4.8312 | 2.3603
8174.6811 | 4.9278 | 2.4292
8176.6893 | 4.6321 | 2.0407
8178.6189 | 6.9212 | 2.4517
8186.7597 | 6.5458 | 2.0901
8188.6939 | 5.0281 | 2.3515
8190.6400 | 4.6226 | 2.0666
8192.7439 | 6.2721 | 2.2111
8200.6405 | 5.5918 | 2.1889
8202.5807 | 5.0442 | 2.0657
8204.6013 | 3.7935 | 2.3908
8206.6028 | 4.1444 | 2.0806
8214.5330 | 4.9041 | 2.8736
8216.5715 | 4.5730 | 2.3948
8218.5708 | 1.7513 | 2.9603
8220.5664 | 3.9028 | 2.3617
8228.6131 | 1.8869 | 3.2091
8230.5611 | 1.9861 | 2.3046
8232.5621 | 1.5371 | 2.0417
8242.5156 | 3.1707 | 2.6394
8244.5167 | 1.8863 | 3.0541
8246.5215 | 1.5284 | 2.9227
8248.6102 | 4.5160 | 3.3030
8258.5731 | 3.2661 | 2.2456
8260.4913 | 1.5134 | 3.3055
8262.4987 | 5.1208 | 2.3189
8270.5720 | 2.3722 | 5.3956
8272.4622 | 4.1881 | 2.3804
8274.5431 | 6.3511 | 3.2676
8276.5334 | 2.2828 | 2.6348
8284.5375 | 3.0182 | 3.0421
8287.4957 | 3.3950 | 2.1505
8289.5123 | 3.1830 | 3.0871
8290.5663 | 3.3647 | 2.3125
8298.5078 | 1.8562 | 2.7121
8301.5267 | 1.5341 | 3.4561
8303.4741 | 1.9703 | 2.5084
8316.4768 | 0.6062 | 3.5221
8329.4678 | 1.2385 | 3.2268
8330.4688 | 1.4468 | 3.3811
8452.8630 | 0.6586 | 2.0873
8466.8571 | 0.8109 | 2.3471
8480.8632 | 4.0911 | 2.5686
8495.7624 | 3.6353 | 2.4410
8513.6726 | -2.7691 | 2.6141
8777.8660 | -5.9810 | 4.0175
8782.8819 | -3.0090 | 2.8326
8785.8744 | -2.9033 | 3.5581
8800.8592 | -1.7399 | 2.0647
8803.8497 | -1.1761 | 2.9545
8805.8312 | 0.6667 | 3.0328
8809.8619 | -0.0915 | 3.5684
8812.8630 | 0.1391 | 2.4498
8814.8574 | -2.3675 | 2.7647
8817.8610 | 0.8139 | 2.4784
8820.8104 | -1.0610 | 2.1543
8822.8133 | -2.1444 | 2.5394
8824.7950 | -1.8230 | 2.7145
8826.8360 | -2.8950 | 2.5544
8827.8608 | -3.0781 | 2.1587
8830.8214 | -2.6295 | 3.8425
8832.8593 | -4.6527 | 2.8437
8834.8460 | -2.8550 | 2.7029
8836.8174 | -2.4430 | 2.7322
8852.8207 | -4.9671 | 3.1129
8853.8375 | -6.8060 | 4.0663
8854.8133 | -4.2678 | 4.1242
8855.8436 | -4.4630 | 3.5223
8856.8048 | -3.0746 | 3.0165
8857.8663 | -5.2199 | 2.9973
8860.8514 | -6.9145 | 2.4193
8861.8533 | -5.2517 | 2.1616
8862.8308 | -4.4471 | 2.5336
8863.8699 | -4.7974 | 2.6102
8864.8254 | -8.5891 | 2.7861
8866.8648 | -3.8925 | 4.1570
8867.8760 | -0.3216 | 5.5342
8868.8816 | -3.8268 | 3.6030
8871.8362 | -2.9384 | 3.4052
8872.8191 | -0.2323 | 4.4564
8875.8653 | -4.8696 | 3.0303
8879.8167 | -2.7358 | 2.8614
8880.8738 | -5.1024 | 3.3282
8881.8639 | -5.5478 | 2.9277
8882.8219 | -5.6743 | 2.8073
8883.8449 | -0.6598 | 3.1591
8884.7985 | -0.0741 | 2.6614
8886.8221 | 3.9531 | 2.3174
8887.8172 | 5.0786 | 2.0700
8888.8627 | 6.3572 | 2.4656
8889.8636 | 9.5681 | 2.4444
8890.8608 | 10.0664 | 2.3980
8891.8962 | 13.8973 | 2.5777
8892.8507 | 13.7307 | 3.0286
8893.8244 | 14.5997 | 2.3825
8894.8883 | 18.3980 | 2.5852
8895.7919 | 18.3334 | 3.2259
8896.8512 | 17.4236 | 3.1734
8897.7793 | 19.9830 | 2.2992
8898.7409 | 21.6081 | 2.1290
8899.7816 | 21.2090 | 2.1652
8900.7151 | 20.9040 | 2.1779
8901.7245 | 23.4560 | 2.3480
8902.7389 | 21.8435 | 2.7880
8903.7444 | 24.7157 | 2.4112
8904.7448 | 21.7421 | 2.2686
8905.7200 | 21.6379 | 2.4001
8906.7154 | 23.8953 | 2.2970
8907.7330 | 23.4400 | 2.3098
8908.7334 | 21.2560 | 1.9855
8909.7035 | 24.2127 | 2.1845
8910.7072 | 22.5546 | 2.3726
8911.6643 | 21.9787 | 2.3805
8912.6790 | 21.3513 | 2.4242
8914.7153 | 20.7693 | 2.8814
8915.7889 | 22.1137 | 2.5725
8916.7380 | 23.4706 | 2.8134
8917.7593 | 22.4455 | 2.4234
8918.6995 | 18.3622 | 2.5623
8919.6849 | 19.5739 | 2.7347
8920.7120 | 20.1201 | 2.2729
8921.7181 | 18.0060 | 2.5603
8922.6421 | 18.7528 | 3.4659
8923.6323 | 18.1697 | 2.7605
8924.6323 | 19.4647 | 2.4099
8925.6034 | 17.4798 | 2.0244


$\tilde{m}_{j}(x_{j})\in\widehat{C}_{\alpha}^{(j)}(x_{j}),\;\;m_{jk}(x_{j},x_{k})\in\widehat{C}_{\alpha}^{(j,k)}(x_{j},x_{k}),\ldots,\;\;\forall\;j=1,\ldots,d,\;\;k<j,\;\;\ldots,$

with probability at least $1-\alpha$. Analogous to the Lipschitz case in
Example 2 above, we can then construct the lower and upper bounds using the
lower and upper bounds of the confidence intervals,

$\displaystyle\widehat{B}_{\alpha\ell}(a,X)$
$\displaystyle=\sum_{j}\sup_{x^{\prime}\in\tilde{\mathcal{X}}_{a}}\left\\{\widehat{C}_{\alpha\ell}^{(j)}(x_{j}^{\prime})-\lambda|X_{j}-x_{j}^{\prime}|\right\\}+\sum_{j<k}\sup_{x^{\prime}\in\tilde{\mathcal{X}}_{a}}\left\\{\widehat{C}_{\alpha\ell}^{(j,k)}(x_{j}^{\prime},x_{k}^{\prime})-\lambda\|X_{(j,k)}-x_{(j,k)}^{\prime}\|\right\\}+\ldots$
$\displaystyle B_{\alpha u}(a,X)$
$\displaystyle=\sum_{j}\inf_{x^{\prime}\in\tilde{\mathcal{X}}_{a}}\left\\{\widehat{C}_{\alpha
u}^{(j)}(x_{j}^{\prime})+\lambda|X_{j}-x_{j}^{\prime}|\right\\}+\sum_{j<k}\inf_{x^{\prime}\in\tilde{\mathcal{X}}_{a}}\left\\{\widehat{C}_{\alpha
u}^{(j,k)}(x_{j}^{\prime},x_{k}^{\prime})+\lambda\|X_{(j,k)}-x_{(j,k)}^{\prime}\|\right\\}+\ldots.$

###  A.3 Regret for policy classes with finite VC dimension

######  Corollary 3 (Statistical safety with finite VC dimension policy
class).

If the policy class $\Pi$ has finite VC dimension $\nu<\infty$, under the
conditions in Theorem 2 and for any $0<\delta\leq e^{-1}$, the regret of
$\hat{\pi}$ relative to the baseline $\tilde{\pi}$ is

$R(\hat{\pi},\tilde{\pi},m)\ \leq\
\frac{C}{\sqrt{n}}\left(4c\sqrt{\nu}+14\sqrt{\log\frac{1}{\delta}}\right),$

with probability at least $1-\alpha-\delta$, where
$C=\max_{y\in\\{0,1\\},a\in\\{0,1\\}}|u(y,a)|$, and $c$ is a universal
constant.

######  Corollary 4 (Empirical optimality grap for bounded model class and
finite VC dimension policy class).

If the policy class $\Pi$ has finite VC dimension $\nu<\infty$, under the
conditions in Theorem 3 and for any $0<\delta\leq e^{-1}$, the regret of
$\hat{\pi}$ relative to the optimal policy $\pi^{\ast}$ is

$R(\hat{\pi},\pi^{\ast})\ \leq\
\frac{2C}{n}\sum_{i=1}^{n}\max_{a\in\mathcal{A}}\\{\widehat{B}_{\alpha
u}(a,X_{i})-\widehat{B}_{\alpha\ell}(a,X_{i})\\}+\frac{C}{\sqrt{n}}\left(4c\sqrt{\nu}+14\sqrt{\log\frac{1}{\delta}}\right),$

with probability at least $1-\alpha-\delta$, where
$C=\max_{y\in\\{0,1\\},a\in\\{0,1\\}}|u(y,a)|$, and $c$ is a universal
constant.

##  Appendix B Proofs and Derivations

###### Proof of Proposition 1.

$V(\tilde{\pi})=V^{\inf}(\tilde{\pi})\leq V^{\inf}(\pi^{\inf})\leq
V(\pi^{\inf}).$

∎

###### Proof of Theorem 1.

Since $V^{\inf}(\pi)\leq V(\pi)$ for all policies $\pi$, the regret is bounded
by

$\displaystyle R(\pi^{\inf},\pi^{\ast})$
$\displaystyle=V(\pi^{\ast})-V(\pi^{\inf})$ $\displaystyle\leq
V(\pi^{\ast})-V^{\inf}(\pi^{\inf})$
$\displaystyle=V^{\inf}(\pi^{\ast})-V^{\inf}(\pi^{\inf})+\sum_{a\in\mathcal{A}}u(a)\mathbb{E}\left[\pi^{\ast}(a,X)\\{1-\tilde{\pi}(a\mid
X)\\}m(a,X)\right]-J_{\mathcal{M}}(\pi^{\ast}\|\tilde{\pi}).$

Now since $\pi^{\inf}$ is a minimizer of $V^{\inf}(\pi)$,
$V^{\inf}(\pi^{\ast})-V^{\inf}(\pi^{\inf})\leq 0$. This yields

$\displaystyle R(\pi^{\inf},\pi^{\ast})$
$\displaystyle\leq\sum_{a\in\mathcal{A}}u(a)\mathbb{E}\left[\pi^{\ast}(a\mid
X)\\{1-\tilde{\pi}(a\mid
X)\\}m(a,X)\right]-\inf_{f\in\mathcal{M}}\sum_{a\in\mathcal{A}}u(a)\mathbb{E}\left[\pi^{\ast}(a\mid
X)\\{1-\tilde{\pi}(a\mid X)\\}f(a,X)\right]$
$\displaystyle\leq\sup_{f\in\mathcal{M}}\left\\{\sum_{a\in\mathcal{A}}u(a)\mathbb{E}\left[\pi^{\ast}(a\mid
X)\\{1-\tilde{\pi}(a\mid X)\\}f(a,X)\right]\right\\}$
$\displaystyle\qquad-\inf_{f\in\mathcal{M}}\left\\{\sum_{a\in\mathcal{A}}u(a)\mathbb{E}\left[\pi^{\ast}(a\mid
X)\\{1-\tilde{\pi}(a\mid X)\\}f(a,X)\right]\right\\}$
$\displaystyle=|u|\mathcal{W}_{\mathcal{M}}\left(\pi^{\ast}(1-\tilde{\pi})\right)$
$\displaystyle\leq|u|\sup_{\pi\in\Pi}\mathcal{W}_{\mathcal{M}}\left(\pi(1-\tilde{\pi})\right).$

Now notice that $\sum_{a\in\mathcal{A}}\pi(a\mid x)\\{1-\tilde{\pi}(a\mid
x)\\}\leq 1$ and $\pi(a\mid x)\\{1-\tilde{\pi}(a\mid x)\\}\geq 0$ for any
$\pi\in\Pi$, so

$R(\pi^{\inf},\pi^{\ast})\leq|u|\mathcal{W}_{\mathcal{M}}.$

∎

###### Proof of Corollary 1.

The width of $\mathcal{M}=\\{B_{\ell}(a,x)\leq f(a,x)\leq B_{u}(a,X)\\}$ in
the direction of $g$ is

$\mathcal{W}_{\mathcal{M}}(g)=\mathbb{E}\left[\sum_{a\in\mathcal{A}}g(a,X)\\{B_{u}(a,X)-B_{\ell}(a,X)\\}\right].$

By Hölder’s inequality,

$\displaystyle\mathcal{W}_{\mathcal{M}}$
$\displaystyle=\sup_{g\in\mathcal{G}}\mathcal{W}_{\mathcal{M}}(g)$
$\displaystyle=\mathbb{E}\left[\sup_{\sum_{a\in\mathcal{A}}g(a,x)\leq
1}\sum_{a\in\mathcal{A}}g(a,x)\\{B_{u}(a,X)-B_{\ell}(a,X)\\}\right]$
$\displaystyle\leq\mathbb{E}\left[\max_{a\in\mathcal{A}}B_{u}(a,X)-B_{\ell}(a,X)\right]$

∎

######  Lemma 1.

Define the empirical value of a policy $\pi$ as

$\hat{V}(\pi)=\frac{1}{n}\sum_{a\in\mathcal{A}}\pi(a\mid
X_{i})\left\\{u(a)\left[\tilde{\pi}(a\mid X_{i})Y_{i}+\\{1-\tilde{\pi}(a\mid
X_{i})\\}m(a,X_{i})\right]+c(a)\right\\}.$

Then with a binary action set $\mathcal{A}=\\{0,1\\}$, for any $\delta>0$,

$\sup_{\pi\in\Pi}|\hat{V}(\pi)-V(\pi)|\leq
4C\mathcal{R}_{n}(\Pi)+\frac{4C}{\sqrt{n}}+\delta,$

with probability at least $1-\exp\left(-\frac{n\delta^{2}}{2C^{2}}\right)$,
where $C=\max_{y\in\\{0,1\\},a\in\\{0,1\\}}|u(y,a)|$.

###### Proof of Lemma 1.

First, with binary actions, the empirical value is

$\displaystyle\hat{V}(\pi)$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}u(0)\left\\{(1-\tilde{\pi}(X_{i}))Y_{i}+\tilde{\pi}(X_{i})m(0,X_{i})\right\\}+c(0)$
$\displaystyle\qquad+u(1)\pi(X_{i})\left\\{\tilde{\pi}(X_{i})(Y_{i}-m(0,X_{i}))+(1-\tilde{\pi}(X_{i}))(m(1,X_{i})-Y_{i})\right\\}$
$\displaystyle\qquad+\pi(X_{i})(c(1)-c(0)).$

Define the function class with functions $f(x,y)$ in

$\displaystyle\mathcal{F}$
$\displaystyle=\left\\{u(0)\left[(1-\tilde{\pi}(X_{i}))Y_{i}+\tilde{\pi}(X_{i})m(0,X_{i})\right]+c(0)\right.$
$\displaystyle\qquad\left.+u(1)\pi(X_{i})\left[(\tilde{\pi}(X_{i})(Y_{i}-m(0,X_{i}))+(1-\tilde{\pi}(X_{i}))(m(1,X_{i})-Y_{i})\right]+\pi(X_{i})(c(1)-c(0))\mid\pi\in\Pi\right\\}.$

Now notice that

$\sup_{\pi\in\Pi}|\hat{V}(\pi)-V(\pi)|=\sup_{f\in\mathcal{F}}\left|\frac{1}{n}\sum_{i=1}^{n}f(X_{i},Y_{i})-\mathbb{E}\left[f(X,Y)\right]\right|.$

The class $\mathcal{F}$ is uniformly bounded by the maximum absolute utility
$C=\max_{y\in\\{0,1\\},a\in\\{0,1\\}}|u(y,a)|$, so by Theorem 4.5 in
Wainwright (2019)

$\sup_{f\in\mathcal{F}}\left|\frac{1}{n}\sum_{i=1}^{n}f(X_{i},Y_{i})-\mathbb{E}\left[f(X,Y)\right]\right|\leq
2\mathcal{R}_{n}(\mathcal{F})+\delta,$

with probability at least $1-\exp\left(-\frac{n\delta^{2}}{2C^{2}}\right)$.

Finally, notice that the Rademacher complexity for $\mathcal{F}$ is bounded by

$\displaystyle\mathcal{R}_{n}(\mathcal{F})$
$\displaystyle\leq\mathbb{E}_{X,Y,\varepsilon}\left[\left|\frac{1}{n}\sum_{i=1}^{n}\left\\{u(0)((1-\tilde{\pi}(X_{i}))Y_{i}+\tilde{\pi}(X_{i})m(0,X_{i}))+c(0)\right\\}\varepsilon_{i}\right|\right]$
$\displaystyle\;+\sup_{\pi\in\Pi}\mathbb{E}_{X,Y,\varepsilon}\left[\left|\frac{1}{n}\sum_{i=1}^{n}\left[u(1)\left\\{\tilde{\pi}(X_{i})(Y_{i}-m(0,X_{i}))+(1-\tilde{\pi}(X_{i}))(m(1,X_{i})-Y_{i})\right\\}+c(1)-c(0)\right]\pi(X_{i})\varepsilon_{i}\right|\right]$
$\displaystyle\leq\mathbb{E}_{\varepsilon}\left[\left|\frac{u(0)+c(0)}{n}\sum_{i=1}^{n}\varepsilon_{i}\right|\right]+\sup_{\pi\in\Pi}\mathbb{E}_{X,Y,\varepsilon}\left[\left|\frac{1}{n}\sum_{i=1}^{n}(u(1)+c(1)-c(0))\pi(X_{i})\varepsilon_{i}\right|\right]$
$\displaystyle\leq\frac{2C}{n}\mathbb{E}_{\varepsilon}\left[\left|\sum_{i=1}^{n}\varepsilon_{i}\right|\right]+\sup_{\pi\in\Pi}2C\;\mathbb{E}_{X,Y,\varepsilon}\left[\frac{1}{n}\sum_{i=1}^{n}\pi(X_{i})\varepsilon_{i}\right]$
$\displaystyle\leq\frac{2C}{\sqrt{n}}+2C\mathcal{R}_{n}(\Pi).$

∎

###### Proof of Theorem 2.

The regret is

$\displaystyle R(\hat{\pi},\tilde{\pi})$
$\displaystyle=V(\tilde{\pi})-V(\hat{\pi})$
$\displaystyle=V(\tilde{\pi})-\hat{V}(\tilde{\pi})+\hat{V}(\tilde{\pi})-\hat{V}(\hat{\pi})+\hat{V}(\hat{\pi})-V(\hat{\pi})$
$\displaystyle\leq
2\sup_{\pi\in\Pi}|\hat{V}(\pi)-V(\pi)|+\hat{V}(\tilde{\pi})-\hat{V}(\hat{\pi})$

Now if $\mathcal{M}\subset\widehat{\mathcal{M}}_{n}(\alpha)$, then
$\hat{V}(\hat{\pi})\geq\hat{V}^{\inf}(\hat{\pi})$, and
$\hat{V}(\tilde{\pi})=\hat{V}^{\inf}(\tilde{\pi})$. Also, note that
$\hat{\pi}$ maximizes $\hat{V}^{\inf}(\pi)$. Combining this, we can see that
if $\mathcal{M}\subset\widehat{\mathcal{M}}_{n}(\alpha)$, then

$\hat{V}(\tilde{\pi})-\hat{V}(\hat{\pi})\leq\hat{V}^{\inf}(\tilde{\pi})-\hat{V}^{\inf}(\hat{\pi})\leq
0.$

So, with probability at least $1-\alpha$ (the probability that
$\mathcal{M}\subset\widehat{\mathcal{M}}_{n}(\alpha)$),

$R(\hat{\pi},\tilde{\pi})\leq 2\sup_{\pi\in\Pi}|\hat{V}(\pi)-V(\pi)|.$

Now, using Lemma 1 and the union bound, we have that

$R(\hat{\pi},\tilde{\pi})\leq 8C\mathcal{R}_{n}(\Pi)+\frac{8C}{\sqrt{n}}+2t,$

with probability at least $1-\alpha-\exp\left(-\frac{nt^{2}}{8C^{2}}\right)$.
Choosing $t=C\sqrt{\frac{8}{n}\log\frac{1}{\delta}}$ and noting that
$8+2\sqrt{8\log\frac{1}{\delta}}\leq(8+2\sqrt{8})\sqrt{\log\frac{1}{\delta}}\leq
14\sqrt{\log\frac{1}{\delta}}$ gives the result.

∎

###### Proof of Theorem 3.

The regret of $\hat{\pi}$ relative to $\pi^{\ast}$ is

$\displaystyle R(\hat{\pi},\pi^{\ast})$
$\displaystyle=V(\pi^{\ast})-V(\hat{\pi})$
$\displaystyle=V(\pi^{\ast})-\hat{V}(\pi^{\ast})+\hat{V}(\pi^{\ast})-\hat{V}(\hat{\pi})+\hat{V}(\hat{\pi})-V(\hat{\pi})$
$\displaystyle\leq\sup_{\pi\in\Pi}2|\hat{V}(\pi)-V(\pi)|+\hat{V}(\pi^{\ast})-\hat{V}(\hat{\pi}).$

We have bounded the first term in Lemma 1, we now turn to the second term.

$\displaystyle\hat{V}(\pi^{\ast})-\hat{V}(\hat{\pi})$
$\displaystyle=\hat{V}^{\inf}(\pi^{\ast})-\hat{J}_{\widehat{\mathcal{M}}_{n}(\alpha)}(\pi^{\ast}\|\tilde{\pi})+\frac{1}{n}\sum_{a\in\mathcal{A}}u(a)\pi^{\ast}(a,X_{i})\\{1-\tilde{\pi}(a\mid
X_{i})\\}m(a,X_{i})-\hat{V}(\hat{\pi})$

Since
$\hat{V}(\hat{\pi})\geq\hat{V}^{\inf}(\hat{\pi})\geq\hat{V}^{\inf}(\pi^{\ast})$,
conditioned on the event $\mathcal{M}\in\widehat{\mathcal{M}}_{n}(\alpha)$ and
with probability at least $1-\alpha$, we have,

$\displaystyle\frac{\hat{V}(\pi^{\ast})-\hat{V}(\hat{\pi})}{|u|}$
$\displaystyle\leq\sup_{f\in\widehat{\mathcal{M}}_{n}(\alpha)}\frac{1}{n}\sum_{a\in\mathcal{A}}\pi^{\ast}(a,X_{i})\\{1-\tilde{\pi}(a\mid
X_{i})\\}f(a,X_{i})$
$\displaystyle\qquad-\inf_{f\in\widehat{\mathcal{M}}_{n}(\alpha)}\frac{1}{n}\sum_{a\in\mathcal{A}}\pi^{\ast}(a,X_{i})\\{1-\tilde{\pi}(a\mid
X_{i})\\}f(a,X_{i})$
$\displaystyle=\widehat{\mathcal{W}}_{\widehat{\mathcal{M}}_{n}(\alpha)}\left(\pi^{\ast}(1-\tilde{\pi})\right)$
$\displaystyle\leq\max_{\pi\in\Pi}\widehat{\mathcal{W}}_{\widehat{\mathcal{M}}_{n}(\alpha)}\left(\pi(1-\tilde{\pi})\right).$

Now since $\max_{x\in\mathcal{X}}\sum_{a\in\mathcal{A}}\pi(a\mid
x)\\{1-\tilde{\pi}(a\mid x)\\}\leq 1$ for any $\pi\in\Pi$, we get that with
probability at least $1-\alpha$,

$\hat{V}(\pi^{\ast})-\hat{V}(\hat{\pi})\leq|u|\widehat{W}_{\widehat{\mathcal{M}}_{n}(\alpha)}.$

Combined with Lemma 1 and the union bound this gives that

$R(\hat{\pi},\pi^{\ast})\leq|u|\widehat{W}_{\widehat{\mathcal{M}}_{n}(\alpha)}+8C\mathcal{R}_{n}(\Pi)+\frac{8C}{\sqrt{n}}+2t,$

with probability at least $1-\alpha-\exp\left(-\frac{nt^{2}}{8C^{2}}\right)$.
Choosing $t=C\sqrt{\frac{8}{n}\log\frac{1}{\delta}}$ and noting that
$8+2\sqrt{8\log\frac{1}{\delta}}\leq(8+2\sqrt{8})\sqrt{\log\frac{1}{\delta}}\leq
14\sqrt{\log\frac{1}{\delta}}$ and $u\leq 2C$ gives the result.

∎

###### Proof of Corollary 2.

The empirical width of
$\widehat{\mathcal{M}}_{n}(\alpha)=\\{\widehat{B}_{\alpha\ell}(a,x)\leq
f(a,x)\leq\widehat{B}_{\alpha u}(a,X)\\}$ in the direction of $g$ is

$\widehat{\mathcal{W}}_{\widehat{\mathcal{M}}_{n}(\alpha)}(g)=\frac{1}{n}\sum_{i=1}^{n}\sum_{a\in\mathcal{A}}g(a,X)\\{\widehat{B}_{\alpha
u}(a,X)-\widehat{B}_{\alpha\ell}(a,X)\\}.$

By Hölder’s inequality,

$\displaystyle\widehat{\mathcal{W}}_{\widehat{\mathcal{M}}_{n}(\alpha)}$
$\displaystyle=\sup_{g\in\mathcal{G}}\widehat{\mathcal{W}}_{\widehat{\mathcal{M}}_{n}(\alpha)}(g)$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\sup_{\sum_{a\in\mathcal{A}}g(a,x)\leq
1}\sum_{a\in\mathcal{A}}g(a,x)\\{\widehat{B}_{\alpha
u}(a,X)-\widehat{B}_{\alpha\ell}(a,X)\\}$
$\displaystyle\leq\frac{1}{n}\sum_{i=1}^{n}\max_{a\in\mathcal{A}}\\{\widehat{B}_{\alpha
u}(a,X)-\widehat{B}_{\alpha\ell}(a,X)\\}.$

∎

##  Appendix C A Simulation Study

Figure C.1: Monte Carlo simulation results as the sample size $n$ increases,
varying the multiplicative factor on the empirical Lipschitz constant and the
significance level $1-\alpha$. The left panel shows the difference in the
expected utility between the empirical safe policy $\hat{\pi}$, and the
baseline policy $\tilde{\pi}$, normalized by the regret of the baseline
relative to the oracle, i.e.
$\frac{V(\hat{\pi})-V(\tilde{\pi})}{V(\pi^{\ast})-V(\tilde{\pi})}$. The right
panel shows the regret of the safe policy relative to the oracle, scaled by
the regret of the baseline relative to the oracle, i.e.
$\frac{V(\pi^{\ast})-V(\hat{\pi})}{V(\pi^{\ast})-V(\tilde{\pi})}$.

We have a single discrete covariate with 10 levels, $x\in\\{0,\ldots,9\\}$,
and a binary action so that the action set is $\mathcal{A}=\\{0,1\\}$. We
choose a baseline policy $\tilde{\pi}=\mathbbm{1}\\{x\geq 5\\}$, and set the
utility gain to be $u(0)=u(1)=10$ and the costs to be $c(0)=0,c(1)=-1$, so
that action 0 is costless and action 1 costs one tenth of the potential
utility gain. For each simulation we draw $n$ i.i.d. samples
$X_{1},\ldots,X_{n}$ uniformly on $\\{0,\ldots,9\\}$. Then we draw a smooth
model for the expected control potential outcome
$m(0,x)\equiv\mathbb{E}[Y(0)\mid X=x]$ via random Fourier features. We draw
three random vectors: $\omega\in\mathbb{R}^{100}$ with i.i.d. standard normal
elements; $b\in\mathbb{R}^{100}$ with i.i.d. components drawn uniformly on
$[0,2\pi]$; and $\beta\in\mathbb{R}^{100}$ with i.i.d. standard normal
elements. Then we set

$m(0,x)=\text{logit}^{-1}\left(\sqrt{\frac{2}{100}}\beta\cdot\cos\left(\omega\frac{x}{9}+b\right)\right),$

where the cosine operates element-wise. See Rahimi and Recht (2008) for more
discussion on random features. For the potential outcome under treatment,
$m(1,x)=\mathbb{E}[Y(1)\mid X=x]$, we add a linear treatment effect on the
logit scale:

$m(1,x)=\text{logit}^{-1}\left(\text{logit}\left(m(0,x)\right)+\frac{1}{2}\left(x-\frac{9}{2}\right)-\frac{8}{10}\right).$

We then generate the potential outcomes $Y_{i}(0),Y_{i}(1)$ as independent
Bernoulli draws with probabilities $m(0,X_{i})$ and $m(1,X_{i})$,
respectively.

With each simulation draw, we consider finding a safe empirical policy by
solving Equation (16) under a Lipschitz restriction on the model as in Example
2 and with the threshold policy class $\Pi_{\text{thresh}}$ in Equation (21).
Note that the true model is in fact much smoother than Lipschitz; here we
consider using the looser assumption. Following our empirical analysis in
Section 7.1, we take the average outcome at each value of $x$, and compute the
largest difference in consecutive averages as pilot estimates for the
Lipschitz constants $\lambda_{0}$ and $\lambda_{1}$. We then solve Equation
(16) using $\frac{1}{2}$, 1, and 2 times these pilot estimates as the
Lipschitz constants, and setting the significance level to 0, 80% and 95%. We
additionally compute the oracle threshold policy that uses the true model
values $m(0,x)$ and $m(1,x)$. We do this for sample sizes
$n\in(500,1000,1500,2000)$.

Figure C.1 shows how the empirical safe policy $\hat{\pi}$ compares to both
the baseline policy $\tilde{\pi}$ and the oracle policy $\pi^{\ast}$ in terms
of expected utility. First, we see that on average, the empirical safe policy
improves over the baseline, no matter the confidence level and the choice of
Lipschitz constant. This improvement is larger the less conservative we are,
e.g. by choosing a lower confidence level or a smaller Lipschitz constant.
Furthermore, as the sample size increases, the utility of the empirical safe
policy also increases due to a lower degree of statistical uncertainty. We
find similar behavior when comparing to the oracle policy. Less conservative
choices lead to lower regret, and the regret decreases with the sample size.
Importantly, the regret does not decrease to zero; even when removing all
statistical uncertainty the safe policy can still be suboptimal due to the
lack of identification.

##  Appendix D An Additional Figure

(a) Varying the confidence level $1-\alpha$

(b) Varying the Lipschitz constant

Figure D.2: New safe threshold values solving Equation (16) for the NVCA flag
threshold rule as the cost of an NVCA increases from 100% to 1,000% of the
cost of triggering the NVCA flag, and (a) the confidence level varies between
0% and 100% and (b) the multiplicative factor on the estimated Lipschitz
constant varies from 1 to 10.