# DMPP-4: Candidate sub-Neptune mass planets orbiting a naked-eye star

J.R. Barnes1, M.R. Standing1, C.A. Haswell1, D. Staab1,2, J.P.J. Doherty1,M.
Waller-Bridge1, L. Fossati3, M. Soto4, G.Anglada-Escudé5,6, J.Llama7, C.
McCune1, F.W. Lewis8
1 School of Physical Sciences, The Open University, Walton Hall, Milton
Keynes. MK7 6AA. UK
2 AVS, Rutherford Appleton Laboratory, Harwell, Oxford, OX11 0QX United
Kingdom
3 Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6,
8042 Graz, Austria
4 School of Physics and Astronomy, Queen Mary, University of London, 327 Mile
End Rd, London. UK
5 Institut de Ciencies de l’Espai (ICE, CSIC), Campus UAB, Can Magrans s/n,
08193 Bellaterra, Spain
6 Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain
7 Lowell Observatory, 1400 West Mars Hill Rd Flagstaff AZ, 86001, USA
8 Faulkes Telescope Project, School of Physics and Astronomy, Queen’s
Building, The Parade, Cardiff. CF24 3AA. Wales. UK

(Accepted for publication in MNRAS, July 2023.)

###### Abstract

We present radial velocity measurements of the very bright ($V\sim 5.7$)
nearby F star, DMPP-4 (HD 184960). The anomalously low Ca ii H&K emission
suggests mass loss from planets orbiting a low activity host star. Periodic
radial velocity variability with $\sim 10$ m s-1 amplitude is found to persist
over a $>4$ year timescale. Although the non-simultaneous photometric
variability in four tess sectors supports the view of an inactive star, we
identify periodic photometric signals and also find spectroscopic evidence for
stellar activity. We used a posterior sampling algorithm that includes the
number of Keplerian signals, $N_{\textrm{p}}$, as a free parameter to test and
compare (1) purely Keplerian models (2) a Keplerian model with linear activity
correlation and (3) Keplerian models with Gaussian processes. A preferred
model, with one Keplerian and quasi-periodic Gaussian process indicates a
planet with a period of $P_{\textrm{b}}=3.4982^{+0.0015}_{-0.0027}$ d and
corresponding minimum mass of
$m_{\textrm{b}}\,\textrm{sin}\,i=12.2^{+1.8}_{-1.9}$ M⊕. Without further high
time resolution observations over a longer timescale, we cannot definitively
rule out the purely Keplerian model with 2 candidates planets with
$P_{\textrm{b}}=2.4570^{+0.0026}_{-0.0462}$ d, minimum mass
$m_{\textrm{b}}\,\textrm{sin}\,i=8.0^{+1.1}_{-1.5}$ M⊕ and
$P_{\textrm{c}}=5.4196^{+0.6766}_{-0.0030}$ d and corresponding minimum mass
of $m_{\textrm{b}}\,\textrm{sin}\,i=12.2^{+1.4}_{-1.6}$ M⊕. The candidate
planets lie in the region below the lower-envelope of the Neptune Desert.
Continued mass loss may originate from the highly irradiated planets or from
an as yet undetected body in the system.

###### keywords:

exoplanets – stars: late-type – techniques: radial velocities

††pubyear: 2022

##  1 Introduction

The dearth of planets, now commonly dubbed the “Neptune Desert”, is a very
clear region in the planetary mass vs orbital period plane (Mazeh et al.,
2016). It is delineated by a sharp upper edge with a planet mass inversely
proportional to the orbital period, the trend first noted by Mazeh et al.
(2005), and a lower boundary with masses that are roughly linearly
proportional to the period. Several mechanisms to explain both boundaries were
discussed by Mazeh et al. (2016) and more recently by Vissapragada et al.
(2022). The upper boundary can be explained as a death line where inward
migrating planets lose much of their mass due to insolation from the host
star. They consequently move down below the lower boundary of the desert where
a large number of transiting Kepler planets were found in a ridge in the
planet radius vs orbital period diagram (Mazeh et al., 2016). The lack of
intermediate mass planets at short periods exists because the timescale for
this mass-loss is thought to be short. A number of planets have nevertheless
been found in the Neptune Desert (West et al., 2019; Jenkins et al., 2020;
Díaz et al., 2020; Armstrong et al., 2020; Burt et al., 2020; Jordán et al.,
2020; Dreizler et al., 2020; Smith et al., 2021; Murgas et al., 2021; Kanodia
et al., 2021; Mori et al., 2022). These planets orbit stars with
$11.1<V<17.0$, except for a single example with $V=9.8$ (Jenkins et al.,
2020). Identification of planetary systems with brighter stellar hosts that
are either in, or have potentially transitioned the Neptune Desert, would be
advantageous for follow-up characterisation.

Chromospheric emission has been found to be depressed in stars harbouring
mass-losing planets (Vidal-Madjar et al., 2003; Haswell et al., 2012). The Ca
ii H&K emission in these stars is below the basal level for inactive stars
owing to absorbing gas that is lost from close-orbiting planets (Haswell et
al., 2020). Hence, a priori identification of stars that harbour hot ablating
planets is possible. The Dispersed Matter Planet Project (DMPP) targets bright
stars within 100 pc, with typical apparent magnitudes of $m_{v}<10$, making
follow-up characterisation easier (Haswell et al., 2020). A total of 39 host
stars with anomalously low Ca II H&K emission have been carefully selected by
DMPP. Because this gas is concentrated in the orbital plane of a planet
(Debrecht et al., 2018), the method preferentially picks out edge-on systems.
Consequently, a significantly higher than average fraction of these planets
are expected to transit. Most of the DMPP planets reported to date lie just
below the lower Neptune desert boundary (Haswell et al., 2020; Barnes et al.,
2020; Staab et al., 2020). One explanation is that they are the stripped cores
of more massive planets that have crossed the Neptune Desert. It is possible
that the DMPP planets identified so far are part of the same population as the
dense concentration or “ridge” of Kepler planets in the period-radius diagram
(Mazeh et al., 2016). The DMPP planets thus offer valuable opportunities to
understand the mechanisms sculpting exoplanet demographics, and thus the
evolution of individual hot planets.

The compact multiplanet system, DMPP-1, comprises some of the most irradiated
rocky planets yet known (Staab et al., 2020). The DMPP planets are not
typically mass-losing hot Jupiters; rather, it is likely that the quenching of
Ca ii H&K emission comes from the low-mass, potentially rocky planets, or from
additional planets that are below the radial velocity (RV) detection
threshold. Jones et al. (2020) identified a candidate transiting signature in
tess observations of DMPP-1 that would correspond to such a low-mass planet
with an extended mass-losing atmosphere. This planet was not identified in the
RVs reported by Staab et al. (2020). DMPP target stars could thus host
analogues and precursors of catastrophically disintegrating planets (Rappaport
et al., 2012; Rappaport et al., 2014; Sanchis-Ojeda et al., 2015).

DMPP has proved very successful, with an effectively 100$\%$ success rate
(Haswell et al., 2020) on targets with sufficient observations. While most
targets are in the southern hemisphere, the DMPP sample comprises a number of
northern targets. Here we present SOPHIE and HARPS-N RV observations of our
brightest target, the late-F star, DMPP-4, which is visible to the naked eye.
We present system parameters for DMPP-4 in §2. In §3, we discuss ground-based
and tess photometric observations and search for evidence of the stellar
rotation period. Spectroscopic activity indicators are analysed in §4 before
searching for Keplerian signals in the RVs in §5. A summary and further
discussion in §6 is followed by a brief conclusion in §7.

##  2 System parameters of DMPP-4

At $V\sim 5.7$, DMPP-4 is visible to the naked eye. It is a northern
hemisphere star (J2000 coordinates 19:34:19.8, +51:14:11.8). Stellar observed
and derived parameters are listed in Table 1. The Gaia Data Release 3 parallax
is $\pi=39.323$ mas, from which a directly estimated distance ($d=1/\pi$) of
$25.43$ pc is implied111The widely reported systematics in Gaia parallaxes
(e.g. see Lindegren et al. 2021) would have a small effect for nearby
objects..

A range of spectral types between F5 (Roman, 1949) and F8V (Eggen, 1960) have
been estimated222CDS database https://simbad.u-strasbg.fr. Table 1 includes
estimates of $V$ and also $\textrm{B-V}=0.4344\pm 0.0076$ derived from
photometric conversions333Gaia DR3 documentation
https://www.cosmos.esa.int/web/gaia-users/archive/gdr3-documentation
documented in Gaia Data Release 3 (Gaia Collaboration et al., 2016, 2023),
which is more consistent with an earlier spectral type of F5V (Pecaut et al.,
2012; Pecaut & Mamajek, 2013). Our own estimates of $T_{\textrm{eff}}$,
$M_{*}$ and $T_{*}$ using our HARPS-N observations and species (Soto &
Jenkins, 2018) are also more consistent with an earlier spectral type
classification of F5V. We also find an age of $2.15$ Gyr using species. With
careful consideration of macroturbulence, $v_{\rm mac}$, the equatorial
rotation velocity, $v$ sin $i$ and $v_{\rm mac}$ have been obtained following
the method described in Murphy et al. (2016). Macroturbulence contributes
significantly to the broadening, with $v_{\rm mac}=7.0\pm 2.0$ kms-1. The
derived $v$ sin $i$ $=7.5^{+0.5}_{-2.0}$ kms-1 is consequently significantly
lower than would be measured without including $v_{\rm mac}$.

Table 1: DMPP-4 stellar parameters. (1) CDS database (2) Roman (1949) (3) Eggen (1960) (4) Gaia Data Release 3 (Gaia Collaboration et al., 2023) (5) Stassun et al. (2018) (6) Pace (2013) (7) Soto & Jenkins (2018) (8) Murphy et al. (2016). Parameter | value | Reference
---|---|---
Spectral Type | F5 - F8V | 1,2,3
Mean G [mag] | 5.5974 $\pm$ 0.0005 | 4
Mean BP [mag] | 5.8437 $\pm$ 0.0009 | 4
Mean RP [mag] | 5.1896 $\pm$ 0.0024 | 4
V [mag] | 5.7034 $\pm$ 0.0302 | this work, from 4
B-V [mag] | 0.4344 $\pm$ 0.0076 | this work, from 4
tess [mag] | 5.22 | 5
$\pi$ [mas] | 39.323 $\pm$ 0.028 | this work, from 4
$1/\pi$ [pc] | 25.43 $\pm$ 0.02 | this work, from 4
log($R^{\prime}_{\rm HK}$) | -5.24 | 6
$T_{\rm eff}$ [K] | 6400 $\pm$ 50 | this work, 7
Fe/H | -0.007 $\pm$ 0.056 | this work, 7
$\log{g}$ [cms-2] | 4.24 $\pm$ 0.10 | this work, 7
$v\sin{i}$ [kms-1] | 7.5${}^{+0.5}_{-2.0}$ | this work, 8
$v_{\rm mac}$ [kms-1] | 7.0 $\pm$ 2.0 | this work, 8
$M_{*}$ [M☉] | 1.25 $\pm$ 0.02 | this work, 7
$R_{*}$ [R☉] | 1.38 $\pm$ 0.01 | this work, 7
$L_{*}$ [L☉] | 2.88 $\pm$ 0.08 | this work, 7
Age [Gyr] | 2.15 $\pm$ 0.32 | this work, 7
Figure 1: LCOGT B band lightcurve and tess PDCSAP lightcurves for sectors 14,
15, 40 and 41. The data are binned with a 0.1 d interval and are mean-
subtracted. The change in flux, $\Delta F$, is plotted in parts per thousand
for the LCOGT data and parts per million for the tess data. Figure 2:
Generalised Lomb Scargle periodograms.

##  3 Photometric observations

###  3.1 LCOGT photometry

Photometric observations of DMPP-4 were collected using the Las Cumbres
Observatory global telescope network (LCOGT) robotic telescopes (Brown et al.,
2013). A total of 761 exposures of $\sim 1$ s duration each were collected
over 46 observing nights. Between 3 and 63 observations were made each night,
with 1 hour cadence between batches of $3-9$ observations. The upper panels of
Fig. 1 shows the data as mean subtracted flux in parts per thousand (ppt)
collected between 2016 June 17 and Aug 28. The most significant peak in the
corresponding periodogram (Fig. 2 upper panel) is found at $5.119$ d, with an
additional short period peak at $1.979$ d and a longer $10.95$ d peak. The
peak at $1.245$ d is the 1-day alias of the $5.119$ d peak. The peak at
$10.95$ d corresponds to a sinusoidal signal with amplitude $5640$ ppm. When
phase-folded on this periodicity, the data are found to be clustered into 11
distinct groups. The $5.119$ d peak corresponds to a sinusoidal signal with
semi-amplitude of 7.56 ppt (7560 ppm). We modelled a B-band lightcurve using
the image code DoTS (Collier Cameron, 2001) for a single spot with $T_{\rm
phot}-T_{\rm spot}=2000$ K (Berdyugina, 2005), finding that a $r_{\rm
spot}=5.04^{\rm o}$ is required to reproduce this semi-amplitude. A small spot
group with the same effective area as a single spot could also potentially
yield the same photometric amplitude, though a more widely distributed
collection of spots would likely need to cover a larger total effective area
to yield the same sinusoidal or near sinusoidal photometric amplitude. The
implied effective spot area appears to be at odds with the low
log($R^{\prime}_{\rm HK}$) activity of DMPP-4, especially given that the
largest sunspots are never this large when the solar maximum
log($R^{\prime}_{\rm HK}$) is still considerably greater than is seen on
DMPP-4. Moreover, we find that a $5.04^{\rm o}$ spot would induce a $K=82$
ms-1 RV signal, which is more than an order of magnitude greater than we find
in our RV data (§4). We therefore conclude that the periodicities recovered
from the LCOGT B band data are not genuine activity-related signals.

Figure 3: Continuous wavelet transform (CWT) for the tess photometry. The most
significant continuous wavelet transform (CWT) powers are shown in red. The
shaded area denotes the wavelet cone of influence region. The right hand panel
shows the global spectrum for Sectors 14 and 15 (dotted line), Sectors 40 and
41 (dashed line) and all sectors (solid line).

###  3.2 TESS photometry

The Transiting Exoplanet Survey Satellite (tess, Ricker et al. 2015) monitored
DMPP-4 in Sectors 14, 15, 40 and 41. The Mikulski Archive for Space
Telescopes444https://mast.stsci.edu provides detrended Pre-search Data
Conditioning Simple Aperture Photometry (PDCSAP) lightcurves. Release notes
for each tess
sector555https://heasarc.gsfc.nasa.gov/docs/tess/documentation.html (figure 6
of Fausnaugh et al. 2018) give the Combined Differential Photometric Precision
(CDPP) for 1hr timescales on $\sim 20000$ targets. The tess magnitude for
DMPP-4 is 5.22 (Table 1), indicating an expected CDPP of $\sim 15$ ppm. This
scales to $\sim 10$ ppm for the $2.4$ hr ($0.1$ d) bins we used, and is in
close agreement with the formal errors on our binned data of $11.4-11.6$ ppm.

Photometric semi-amplitude variability on a scale of $<100$ ppm (Fig. 1) can
be seen at times in the tess observations; considerably less than the 7560 ppm
seen in the LCOGT data. However, periodic variability is only easily discerned
during part of Sectors 15 and 41. Log-likelihood periodogram searches using
sinusoidal signals, are shown in Fig. 2 for Sectors 14 and 15, Sector 40 and
41 and all sectors combined. Highly significant peaks appear in the
periodograms with the most significant periods at respectively $6.33$ d,
$6.47$ d, $6.40$ d. The respective half width at half maxima (HWHM) of these
peaks of $0.33$ d, $0.44$ d and $0.33$ d give an indication of the spread of
power, but should not be treated as period uncertainties, as discussed by
VanderPlas (2018). We also note that the longer period at $11.6$ d (HWHM =
$1.4$ d) is also significant in the Sector 40 and 41 periodogram, along with a
shorter 4.770 d peak. As with the $6.40$ d strongest peak when using all
sectors, the longer peak at $12.1$ d is strongly aliased due to sampling
(VanderPlas, 2018). The envelope of the aliased peak possesses HWHM = $1.7$ d.
The $12.1$ d peak is marginally significant with $\Delta$log$L=13.0$ (FAP
$=0.003$).

To investigate the periodicities further, we performed a wavelet analysis
using the scaleogram package666https://github.com/alsauve/scaleogram, which is
based on the PyWavelets library (Lee et al., 2019). The resulting continuous
wavelet transform in Fig. 3 confirms the visual inspection of periodicities in
Fig. 1. The periodicities identified in the log likelihood periodograms are
not present throughout the extent of the timeseries and the periodicities are
often localised. The global spectrum reveals peak power in Sectors 14 and 15
at $5.8$ d and in Sectors 40 and 41 at $6.0$ d and $10.7$ d (peaking at 11.1 d
for the maximum power).

We again used DoTS to model the photometric variability expected from tess
observations and the absorption line profiles for a cool spot on DMPP-4
observed with $550$ nm central wavelength. The tess flux half-amplitude of
$\sim 29$ ppm found by phasing Sectors 15 and 41 is commensurate with an
equatorial cool starspot of radius $0.34^{\rm o}$. A spot of this size is
expected to induce RV variability of $K<$ $16$ cms-1 at 550 nm and is thus
below the level that could easily be detected by an RV instrument achieving
$\sim 1$ ms-1 precision. If a Solar-like facular/spot area ratio of 11.6 is
present on DMPP-4 (Shapiro et al. 2014; see also Barnes et al. 2023), a
$0.082^{\rm o}$ region of plage surrounding a $0.024^{\rm o}$ spot could
induce a flux half-amplitude of $\sim 29$ ppm. In this case, the expected RV
amplitude is $K\sim 38$ cms-1 at $550$ nm. The RV amplitude rises by an order
of magnitude for a cool spot with radius $0.2^{\rm o}$ and surrounding plage
of $0.68^{\rm o}$. Although the tess observations and our spot estimates
provide further evidence in support of the low activity of DMPP-4, we cannot
rule out the possibility of higher activity levels at other epochs when our RV
observations were made.

Figure 4: Monte Carlo simulations for stellar rotation period, $P$, and axial
inclination, $i$ Top: The distribution of stellar rotation periods based on
the parameters in Table 1, assuming uniformly distributed periods with the
indicated ranged. Bottom: Distribution of stellar axial inclination assuming
tess periodicities identified in Fig. 2.

###  3.3 Spot evolution and the orbital period of DMPP-4

A study of sunspot group data spanning over a century by Berdyugina (2005)
found that two active longitudes persist and are separated by $\sim
180^{\textrm{o}}$. One region usually dominates, periodically alternating
between the two. More recent work by Basri & Nguyen (2018) has looked at the
phenomenon of “double-dipping”, where a stellar photometric lightcurve spends
some of time exhibiting a single sinusoid behaviour at the rotation period and
some time in a double-dip mode with two dips or sinusoids within one rotation.
The phenomenon was found in the analysis of over 34,000 Kepler lightcurves by
McQuillan et al. (2014), where periodogram peaks at the rotation period and
half the rotation period were identified. While two distinct active groups may
be present, any spot distribution will typically yield a photometric
lightcurve with either one to two dips. For more active stars, spots may thus
be distributed over a range of longitudes rather than in two distinct
longitude regions. The exact lightcurve morphology is determined by either the
relative strength, the distribution of the spots, or by both effects. Basri &
Nguyen (2018) used the Kepler lightcurves to examine how much time stars spend
in single-dip mode and how much in double-dip mode. For their hottest sample
corresponding to stars with $T_{\rm eff}=6000-6200K$, the time spent in
double-dip mode is 1.8 times that spent in single-dip mode. This bimodal
behaviour is seen in the tess observations of DMPP-4. At the tess Sector 14
and 15 epochs, two spot groups of similar intensity could yield the periodic
signatures seen at $\sim 5.5-6.5$ d, while in Sectors 40 and 41, there appears
to be evidence for one spot group for some of the time and two spot groups for
some of the time. Evolution of spot groups appears to be relatively rapid and
on the timescale of a single rotation, resulting in a rapidly changing
lightcurve. The lightcurve in Sectors 40 and 41 can be explained by a single
weak/small spot that grows in strength with the simultaneous appearance of a
second spot group resulting in a switch from single-dip to double-dip mode.

In summary, we believe that DMPP-4 exhibits small spots, as evidenced by the
low log($R^{\prime}_{\rm HK}$), photometric amplitude and spot size estimate.
It possibly follows the solar paradigm of two persistent and distinct active
regions rather than that of potentially more distributed activity on an active
star. However, interpreting periodicities at $P/2$ from this scenario may be
simplistic since other effects such as differential rotation and spot
evolution can result in both double dipping and multiple periodicities. A
rotation period of $\sim 11$ d rather than $\sim 6$ d thus seems more likely
given the estimated stellar rotation velocity and radius.

###  3.4 Period and inclination distributions

We expect DMPP to find systems with high orbital inclinations where the
stellar Ca ii H&K emission would be more readily obscured (Haswell et al.,
2012, 2020). So, under the assumption that the stellar spin axis and planetary
orbital planes are aligned, we expect a stellar rotation axis which is highly
inclined to our line of sight. The upper plot in Fig. 4 shows the distribution
of expected periods from the $R_{*}=1.38\pm 0.01$ R⊙ (Gaussian distribution)
and $v$ sini $i$ = $7.5^{+0.5}_{-2.5}$ ms-1 (asymmetric Gaussian distribution)
estimates listed in Table 1. A uniform distribution of axial inclinations of
$0^{\rm o}<i<90^{\rm o}$ was simulated. We find a resulting modal period of
$P$ = $8.89$ d with a range between $2.74$ d to $11.10$ d at the 16 per cent
and 84 percent intervals. A simulation with $60^{\rm o}<i<90^{\rm o}$ only
affects the most likely period slightly ($P$ = $9.02$ d), but skews the
posterior distribution to higher values of $P$ ($8.69$ d to $13.50$ d). The
lower panel in Fig. 4 shows the axial inclination distributions, $i$, assuming
period distributions with $P=6.0$ d and $11.0$ d from the TESS periodicities
identified in §3.2 (assuming Gaussian distributions of $\sigma=\pm 1$ d and
$\pm 2$ d). For the $P=6.0$ d case, a low axial inclination with modal value,
$i=32.8^{\rm o}$ ($23.2^{\rm o}$ to $43.8^{\rm o}$ at 16 per cent and 84
percent intervals) is found. For $P=11.0$ d, inclinations $>49.7^{\rm o}$ (16
per cent) are found with a modal sin $i>1$. The large uncertainty in our $v$
sin $i$ estimate, particularly in the lower limit is a likely explanation for
the difference between the $11$ d photometric rotation period and Monte Carlo
modal period of $8.89$ d. If $i=90^{o}$, $v$ sin $i$ must be $\leq 6.35$ kms-1
when $P=11$ d.

Table 2: DMPP-4 correlations between RV and chromatic index, $\kappa$, full width at half maximum (FWHM), RV vs bisector span (BIS) and RV vs Ca H&K S-index. Pearson’s r, and student’s $p$ statistic are shown for each correlation for all SOPHIE observations combined and all HARPS-N observations combined (top rows). The remaining rows show the same statistics for individual observing runs. | Kappa | FWHM | BIS | S-index
---|---|---|---|---
|  $r$ ($p$) |  $r$ ($p$) |  $r$ ($p$) |  $r$ ($p$)
All SOPHIE | 0.019 ( 0.902 ) | 0.057 ( 0.712 ) | -0.366 ( 0.014 ) | -0.054 ( 0.726 )
All HARPS-N | 0.287 ( 0.146 ) | -0.204 ( 0.308 ) | -0.255 ( 0.199 ) | 0.236 ( 0.236 )
SO-15A | 0.178 ( 0.703 ) | 0.299 ( 0.514 ) | 0.544 ( 0.207 ) | -0.119 ( 0.799 )
SO-15B | 0.032 ( 0.925 ) | 0.253 ( 0.452 ) | -0.406 ( 0.215 ) | 0.028 ( 0.934 )
SO-16A | -0.208 ( 0.496 ) | -0.268 ( 0.376 ) | -0.601 ( 0.030 ) | -0.114 ( 0.712 )
SO-16B | -0.271 ( 0.350 ) | -0.613 ( 0.020 ) | -0.732 ( 0.003 ) | -0.242 ( 0.405 )
HN-16A | 0.405 ( 0.096 ) | -0.235 ( 0.348 ) | -0.317 ( 0.201 ) | 0.403 ( 0.097 )
HN-19A | -0.007 ( 0.986 ) | -0.287 ( 0.454 ) | 0.540 ( 0.134 ) | -0.018 ( 0.962 )
|
---|---
Figure 5: Left: Activity index correlations with RV for SOPHIE (panels a-c)
and HARPS-N (panels d-f) observations. The points are colour coded according
to the observation semester. Right: Activity index periodograms for line full
width at half maximum (FWHM), bisector inverse span (BIS) and Calcium S-index.
FAP at 0.1%, 1% and 10% are (top to bottom) are indicated by the dashed
horizontal lines.

Figure 6: Stacked periodograms of the RVs (top) and residual RVs after
accounting for the first signal (bottom). The dotted lines indicate the end of
each observing run. Figure 7: Radial velocity log likelihood periodograms
using (a) RVs only and (b) with a BIS linear correlation (RV+linBIScorr). The
periodograms for single ($N_{\textrm{p}}=1$) Keplerian signals are shown in
the top panels. The period searches with a second recursively added signal
($N_{\textrm{p}}=2$) are shown in the bottom panels. The vertical dashed line
indicates the BIS period of 2.54 d. Horizontal dashed lines are the 0.1%, 1%
and 10% FAPs as in Fig. 5.

##  4 Activity indicators

Spectroscopic observations of DMPP-4 were made with with SOPHIE (Perruchot et
al., 2008) at the 1.9 m Observatoire de Haute Provence Telescope during four
semesters in 2015 and 2016. Further data were obtained with HARPS-N (Cosentino
et al., 2014) at the Telescopio Nazionale Galileo telescope. Observations were
made in 2015 Apr-May on 3 nights over a time span of 4 nights; and again later
in 2015 Nov on 5 nights. Similarly, observing runs in 2016 May and 2016 Nov-
Dec each resulted in 4 nights of observations over a 5 night time span. We
refer respectively to the individual SOPHIE runs as SO-15A, SO-15B, SO-16A and
SO-16B. In 2016, HARPS-N observations on 4 nights (HN-16A) were made in 2016
Jul between the SO-16A and SO-16B runs. The last set of HARPS-N observations
were made in 2019 Aug on a single night (HN-19A). Exposures were monitored to
ensure relatively stable counts with SOPHIE. Because DMPP-4 is very bright,
observation times of $200$ s to $900$ s were used, depending on observing
conditions. When observations were cycled with other targets, exposures on
DMPP-4 were made for at least $\sim 900$ s in a single block to ensure that
possible 5 minute stellar oscillations were minimised.

The RVs were derived using the matched-template code, harps-terra (Anglada-
Escudé & Butler, 2012). We also used the activity indices provided by the
standard data reduction software pipelines for both instruments. A total of
111 observations were made with SOPHIE, while 87 observations were made with
HARPS-N. After optimally weighted binning into $900$ s observations, the
respective data sets comprise 45 and 27 observations. The binned data are
tabulated in Tables 5 & 6.

The low altitude of the Observatoire de Haute Provence, small telescope
aperture and instrumental effects such as reference/target fibre drift,
potential chromatic $\kappa$-correlation (see below) and charge transfer
inefficiency effects limit the precision that is achieved with SOPHIE. The
formal uncertainties for our 2015 and 2016 observations, before $900$ s
binning, are $2.25$ ms-1 and $2.26$ ms-1. The binned data uncertainties are
respectively $1.69$ ms-1 and $1.60$ ms-1. A measurement precision of $\sim
1-2$ ms-1 has been demonstrated with SOPHIE on RV standard stars on timescales
of a few tens of days (Bouchy et al., 2013). For HARPS-N, the formal
uncertainties for single $200$ s observations is $1.17$ ms-1, and $0.73$ ms-1
for $900$ s binned data.

###  4.1 Kappa correlation with RVs

We investigated correlations between possible diurnal chromatic systematics
and the RVs. Bourrier et al. (2014) and Berdinas et al. (2016) identified a
significant intra-night systematic effect on observations made with the
HARPS-N spectrograph. The primary cause appears to be an incomplete correction
of differential atmospheric refraction by the atmospheric dispersion corrector
(ADC), causing colour-dependent (chromatic) flux-losses. The effect is thus
potentially important for targets observed throughout the night at different
airmasses. We measured the ‘chromatic index’, $\kappa$, defined as the slope
of a linear portion of the pseudo spectral energy distribution about a
specified wavelength. In practice, we measured the slope in mean flux per
échelle order across several orders in a similar manner to Berdinas et al.
(2016); see Haswell et al. (2020) for full details. Fig. 5 (left panel) shows
the chromatic index, $\kappa$, plotted against the RVs for all SOPHIE and
HARPS-N observations. The degree of correlation is measured via Pearson’s $r$
coefficient and listed in Table 2; only weak or very weak correlations are
seen during most observing runs. There is a moderate correlation in the HN-16A
data set, but the significance is also moderate, as indicated by the student’s
t test p-value. This suggests that there is no clear evidence of a
correlation. The correlation is driven by the two outlying RVs (greatest
positive values); removing them yields $r=0.05$ and $p=0.86$.

###  4.2 RV correlations with activity measures

The correlation of RVs with the full width at half-maxiumum (FWHM), line
Bisector Inverse Span (BIS) and Ca ii H&K S-index are also plotted in Fig. 5.
Correlation measurements are listed in Table 2. We include the HN-19A data in
Table 2 for completeness; since this run comprised a single night, activity
correlations are likely to be less meaningful.

The BIS shows significant moderate anti-correlation in the SO-16A and SO-16B
data sets, but is less pronounced in SO-15B. We note that the SO-15A data is
of significantly lower precision than the subsequent observations. For BIS in
SO-15A, removing the data point with largest positive RV $=8.49$ ms-1 results
in $r=-0.64$ and $p=0.17$. For individual observing runs, the BIS anti-
correlation is largest and most significant in the SO-16A and SO-16B epochs.
In SO-16B, the RVs also show a significant (p=0.02) moderate anti-correlation
with FWHM.

HARPS-N shows moderate correlations, but the significance is low. The HN-16A
run shows a moderate S-index variability, although the probability of no
correlation is still nearly 10% (i.e. $p=0.097$). Of the three activity
indicators, only BIS shows a consistently moderate-good correlation.

Table 3: Candidate maximum likelihood periodogram search summary for (a) RV period search and (b) RV period search with a linear BIS correlation (RV+linBIScorr model). The corresponding Bayesian Information Criteria (BIC) are also given along with the $\Delta\textrm{BIC}$ values for $\textrm{BIC}(N_{\textrm{p}}-(N_{\textrm{p}}-1))$ in parentheses and directly derived estimates of the corresponding Bayes Factors (BF). Keplerians |  | (a) RV |  |  | (b) RV+linBIScorr
---|---|---|---|---|---
$N_{\textrm{p}}$ | P [d] | logL ($\Delta\mathrm{log}L$) | BIC ($\Delta\mathrm{BIC}$) | BF |  | P [d] | logL ($\Delta\mathrm{log}L$) | ($\Delta\mathrm{BIC}$) | BF
0 | - | -200.2 | 417.6 |  |  | - | -195.0 | 419.7 |
1 | 3.498 | -172.6 (27.7) | 375.1 (-42.5) | $1.7\times 10^{9}$ |  | 3.498 | -168.7 (26.3) | 382.1 (-37.8) | $1.6\times 10^{8}$
2 | 2.459 | -155.2 (17.3) | 353.2 (-21.9) | 56954 |  | 2.459 | -154.1 (14.6) | 367.8 (-14.3) | 1274

###  4.3 Periodicities in activity measures

The right panels in Fig. 5 show periodograms for the $\kappa$-correlation and
each activity index. Aliases and integer fractions of a day in the Kappa and
FWHM periodograms are seen. The $\kappa$-correlation and FWHM show no other
significant peaks. For BIS, we find a 2.54 d peak with FAP = $2.7\times
10^{-6}$ and $\Delta$ Log$L=27.2$ relative to the scenario with no signal. A
neighbouring 2.68 d peak with log$L_{2.54-2.68}<5$ ($\Delta$ Log$L=22.9$)
relative to the main peak is also present. This periodicity, taken with the
moderate anti-correlation between BIS and RV, suggests that the RVs may be
affected by stellar activity. It is nevertheless difficult to reconcile this
finding with our analysis that reveals very low photometric variability and
the low log($R^{\prime}_{\rm HK}$), suggesting that DMPP-4 is an inactive
star. The variable degree of correlation between BIS and RV at different
observing epochs suggests that the star shows variable activity. This should
however be treated with some caution since each observing epoch comprises only
a few nights of observations and so may not sample a complete stellar
rotation.

Further, the most significant BIS periodicity is likely to be a fraction of
the true period. Period analysis of apparent RV shifts due to line-shape
changes as described by BIS, are a proxy measure of the third central moment
or line skewness. Barnes et al. (2023) finds that for stars with realistic
spot and facular distributions and solar activity levels, RV periodicities may
appear with dominant power at harmonics of the rotation period, including
$P_{\textrm{rot}}/2$ and $P_{\textrm{rot}}/4$. This behaviour was first noted
by Boisse et al. (2011). For instance, a single high latitude spot can easily
induce RV and BIS variability at $P_{\textrm{rot}}/2$, while a low latitude
spot could simultaneously induce RV variability at $P_{\rm rot}/2$ and BIS
variability at $P_{\textrm{rot}}/4$. A pair of low-latitude spots located at
active longitudes 180${}^{\textrm{o}}$ apart, as has been observed for the sun
(Berdyugina & Usoskin, 2003), could then also be expected to lead to
predominant periodicities in RV and BIS at $P_{\textrm{rot}}/4$. The exact
periodicities are likely to vary from star to star, but depend on the exact
spot patterns, presence of absence of faculae and the combination of spot
latitude and stellar axial inclination and $v\,\textrm{sin}\,i$.

The BIS periodicities of $2.54$ d and $2.68$ d may thus be manifestations of
active regions that have true periodicities at twice or four times these
values. In other words, $P_{\rm rot}$ inferred from BIS may be either $5.08$ d
and $5.36$ d ($2\times$) or $10.2$ d and $10.7$ d (4$\times$). The longer
periods are close to photometric periodicities identified from the tess Sector
40 and 41 observations in §3.2. There is tentative evidence for low
significance peaks (1 - 10 per cent FAP) at $\geq 10$ d in the S-index
periodogram, which shows a most significant peak at $11.1$ d ($\Delta$
Log$L=15.9$), again in close agreement with the tess observations.

Parameter | Model fits to RV data only | RV + BIS
---|---|---
| N$\mathbf{{}_{p}=1}$ | N$\mathbf{{}_{p}=2}$ |  N$\mathbf{{}_{p}=1}$ \+ GP |  N$\mathbf{{}_{p}=1}$ \+ GP
$T_{\rm 0,b}$ [d] | $2457139.7^{+1.5}_{-1.2}$ | $2457140.26^{+0.81}_{-0.65}$ | $2457139.8^{+1.3}_{-1.3}$ | $2457139.9^{+1.2}_{-1.3}$
$P_{\rm b}$ [d] | $3.49791^{+0.00046}_{-0.00143}$ | $2.4570^{+0.0026}_{-0.0462}$ | $3.4982^{+0.0015}_{-0.0027}$ | $3.4982^{+0.0022}_{-0.2327}$
$K_{\rm b}$ [ms-1] | $4.24^{+0.55}_{-0.56}$ | $3.26^{+0.46}_{-0.57}$ | $4.58^{+0.59}_{-0.67}$ | $4.43^{+0.66}_{-0.69}$
$e_{\rm b}$ | <0.067 | <0.065 | <0.063 | <0.063
$\omega_{\rm b}$ [rad] | $3.9^{+1.4}_{-2.7}$ | $2.7^{+2.4}_{-1.6}$ | $3.6^{+1.8}_{-2.5}$ | $3.5^{+1.9}_{-2.3}$
$a_{\rm b}$ [$\rm AU$] | $0.04853^{+0.00028}_{-0.00047}$ | $0.03836^{+0.00039}_{-0.00044}$ | $0.04854^{+0.00033}_{-0.00054}$ | $\hskip 14.22636pt0.04853^{+0.00036}_{-0.00197}\hskip 14.22636pt$
$m_{p}$ sin $i$ b [M⊕] | $11.6^{+1.5}_{-1.6}$ | $8.0^{+1.1}_{-1.5}$ | $12.6^{+1.6}_{-1.8}$ | $12.2^{+1.8}_{-1.9}$
Planet candidate c
$T_{\rm 0,c}$ [d] |  | $2457138.9^{+1.6}_{-1.8}$ |  |
$P_{\rm c}$ [d] |  | $5.4196^{+0.6766}_{-0.0030}$ |  |
$K_{\rm c}$ [ms-1] |  | $3.81^{+0.43}_{-0.50}$ |  |
$e_{\rm c}$ |  | <0.064 |  |
$\omega_{\rm c}$ [rad] |  | $2.8^{+2.2}_{-1.7}$ |  |
$a_{\rm c}$ [$\rm AU$] |  | $0.06514^{+0.00469}_{-0.00041}$ |  |
$m_{p}$ sin $i$ c [M⊕] |  | $12.2^{+1.4}_{-1.6}$ |  |
Quasi Periodic GP parameters
$\eta_{1}$ |  |  | $2.4^{+1.3}_{-1.8}$ | $2.88^{+1.09}_{-1.03}$
$\eta_{1,2}$ |  |  |  | $-8.3^{+1.8}_{-2.3}$
$\eta_{2}$ |  |  | $20.1^{+4.9}_{-5.0}$ | $19.9^{+5.0}_{-4.8}$
$\eta_{3}$ |  |  | $2.42^{+0.25}_{-0.24}$ | $2.51^{+0.09}_{-0.10}$
$\Gamma~{}(\eta_{4}=\sqrt{2/\Gamma}$) [fixed] |  |  | $1~{}(\sqrt{2})$ | $1~{}(\sqrt{2})$
Systemic RVs and white noise parameters
$\gamma_{\textrm{RV,HN}}$ [ms-1] | $-0.93^{+0.55}_{-0.56}$ | $-0.55^{+0.64}_{-0.70}$ | $-0.50^{+1.79}_{-1.57}$ | $0.2^{+1.3}_{-1.2}$
$\rm{Offset_{RV,SO}}$ [ms-1] | $0.63^{+0.75}_{-0.74}$ | $0.37^{+0.97}_{-0.76}$ | $0.54^{+2.06}_{-1.94}$ | $-0.49^{+2.5}_{-2.4}$
$\sigma_{\textrm{RV,SO}}$ [ms-1] | $2.60^{+0.48}_{-0.45}$ | $0.41^{+0.92}_{-0.37}$ | $1.25^{+0.52}_{-0.64}$ | $1.50^{+0.40}_{-0.33}$
$\sigma_{\textrm{RV,HN}}$ [ms-1] | $1.75^{+0.45}_{-0.47}$ | $1.15^{+0.51}_{-0.77}$ | $1.49^{+0.40}_{-0.32}$ | $1.29^{+0.49}_{-0.44}$
$\sigma_{\textrm{BIS,SO}}$ [ms-1] |  |  |  | $2.80^{+0.68}_{-0.56}$
$\sigma_{\textrm{BIS,HN}}$ [ms-1] |  |  |  | $4.69^{+1.05}_{-0.96}$
Fitting statistics
BF | >7676 | 2397 | 111.4 | 1038
log $L$ | -168.4 | -147.9 | -162.8 | -394.6
Table 4: Posterior RV Model parameters and derived planet masses with 68.3%
confidence uncertainties.The log $L$ statistics in the final row are given for
the best fitting (maximum posterior) models. Models with 1-Keplerian
($N_{\textrm{p}}=1$) and 2-Keplerians ($N_{\textrm{p}}=2$) are tabulated in
columns 1 and 2. Column 3 is the $N_{\textrm{p}}=1$ \+ GP solution using only
the RVs. Column 4 tabulates parameters for the $N_{\textrm{p}}=1$ \+ GP
solution which uses simultaneous RVs and BIS. Eccentricities are all
consistent with 0 and therefore upper limits are quoted at 2$\sigma$.

(a)

(b)

Figure 8: Single Keplerian RV solution ($N_{\textrm{p}}=1$) for combined
SOPHIE (SO) and HARPS-N (HN) data sets taken in semesters 15A, 15B, 16A, 16B
and 19A. (a) RVs vs observation time and (b) phased RVs showing fit with 68%
and 95% model uncertainties. The light blue and pink data uncertainties
indicate the combined formal uncertainties and additive white noise.

(a)

(b)

Figure 9: As in Figure 8 for the 2-Keplerian ($N_{\textrm{p}}=2$) solution.
(a) The RVs with solution curve and (b) the phased RVs plotted for the $5.419$
d candidate signal (top) and the $2.457$ d signal (bottom).

(a)

(b)

(c)

(d)

(e)

Figure 10: Single Keplerian solution ($N_{\textrm{p}}=1$) with Gaussian
Process (GP). (a) Solution using only the RVs with a GP. The RV component is
represented by a short dashed line and the GP by a long dashed line. As in
Figure 8, the RV + GP curve shaded regions show 68% and 95% model
uncertainties. (b) The phased RVs corresponding to panel (a). (c) RVs for the
RV + BIS solution with a GP and (d) corresponding phased RVs. (e) The BIS
timeseries for the RV + BIS solution showing the GP as a long dashed line.

##  5 RV analysis

We began by assessing the coherence of periodic RV signals over time as more
data points were added. Figure 6 shows the stacked Bayesian general Lomb-
Scargle periodograms of the RVs and the residual RVs after accounting for the
first Keplerian (Mortier et al., 2015; Mortier & Collier Cameron, 2017).
Increasing the number of observations over the 6 observing campaigns, spanning
2015 and 2016 with the additional night in 2019 steadily increases the
significance of each signal. There is some aliasing of peaks, which split into
sub-peaks with addition of the final data points taken at HARPS-N in 2019. The
long-term signal coherence suggests that the periodicities are Keplerian in
origin and can be attributed to planets. Nevertheless, since the periodicities
found in the BIS data are similar to those of the second RV signal, we
investigate a number of scenarios below.

###  5.1 Maximum likelihood searches

We carried out likelihood period analyses on the combined SOPHIE and HARPS-N
RVs, assuming circular Keplerian orbits with $e=0$. The Keplerian model
includes offset terms for each instrument. In addition to a period search on
the RVs only (RV model), we also performed a period search that incorporated
the BIS values into the likelihood model via a linear correlation coefficient
(RV+linBIScorr model). Keplerians are added recursively into the model when
searching for additional signals. The likelihood model is described in detail
in Anglada-Escudé et al. (2013) and Anglada-Escudé et al. (2016) and includes
white noise terms to account for noise in addition to the formal uncertainties
and systematic RV offset terms for each data set.

Fig. 7 (top left) shows that periodicity at $P=3.498$ d is found for the
1-Keplerian models ($N_{\textrm{p}}=1$) for both RV and RV+linBIScorr models.
Table 3 lists the likelihood values, log$L$, and change in likelihood,
$\Delta$log$L$, of recursively added periodicities. The corresponding
estimates of the Bayesian Information Criteria,
$\Delta\textrm{BIC}=\textrm{BIC}_{k}-\textrm{BIC}_{k-1}$ were used to estimate
Bayes Factor values,
$\textrm{BF}=e^{(\textrm{BIC}_{k-1}-\textrm{BIC}_{k})}/2$. The ratio of
Bayesian evidence of competing models, the Bayes Factor (BF), provides a
measure of support in favour of one model over another and is obtained
directly from the samples obtained. We use the scale defined by Trotta (2008)
and discussed in Standing et al. (2022), where $3\leq\textrm{BF}<12$ implies
weak evidence, $12\leq\textrm{BF}<150$ implies moderate evidence and
$\textrm{BF}\geq 150$ indicates strong evidence. We note that these BF values
are approximations and should not be directly compared with the values
reported from posterior sampling in the following sections. For
$N_{\textrm{p}}=1$, $\Delta\textrm{BIC}$ for the RV+linBIScorr model is
smaller than for the RV model. Nevertheless, both $\Delta\textrm{BIC}$ and BF
indicate very highly significant periodicity at $P=3.498$ d for both models.

For the 2-Keplerian solutions, with $N_{\textrm{p}}=2$, significant peaks at
$2.459$ d are found (Table 3). For the RV model, $\Delta$ Log$L=17.3$
($\Delta\textrm{BIC}=-21.9$). While this periodicity also contains the highest
power in the RV+linBIScorr model, the significance is decreased to
$\Delta$Log$L=14.6$. The corresponding $\Delta\textrm{BIC}=-14.3$ and
$\textrm{BF}=1274$ still indicates very strong evidence for this model
(Raftery, 1995), though the RV model is preferred over the RV+linBIScorr
model. Although the linBIScorr correlation coefficient for HARPS-N is $-1.1\pm
0.5$, the coefficient for the SOPHIE data is consistent with zero at $-0.2\pm
0.7$. This suggests that the linear correlation coefficient is not adequate
for modelling long baseline timeseries.

The changing correlation between the RVs and simultaneous BIS values at
different epochs, as evidenced by Pearson’s $r$ statistic in Table 2, might be
indicative of an active star with continually evolving activity. Since DMPP-4
shows evidence of low level photometric modulation and evidence for BIS
variability at a period that matches the signal at $\sim 2.5$ d, we
investigated whether Gaussian Process (GP) modelling can be used to
characterise the effects of activity on the RVs.

###  5.2 Signal recovery and model comparison using posterior sampling

We used kima (Faria et al., 2018) to search for Keplerian signals and
investigate various model scenarios. We include the maximum likelihood values
in our model solutions in Table 4, but note that they should not be use to
distinguish models since they do not take into consideration the number of
model parameters or data points. Kima uses Diffusive Nested Sampling (Brewer
et al., 2011) to sample from the joint posterior distribution and enables the
marginal likelihood (Bayesian evidence) of a model to be directly estimated
(in contrast to the BF estimates derived from the BIC values in §5.1). This
has the benefit of correctly enabling inter-comparison of models (Brewer &
Donovan, 2015; Feroz et al., 2011). With kima, it is possible to either fix
the number of Keplerians, $N_{\textrm{p}}$, or include $N_{\textrm{p}}$ as a
free parameter. The likelihood model includes the systemic velocity, $\gamma$,
of a reference data set (in our case, the HARPS-N RV data,
$\gamma_{\textrm{RV,HN}}$) and relative offsets for other data sets. Since our
RV data are derived via matched spectrum template derived from the
observations, these parameters are expected to be close to
$0\,\textrm{ms}^{-1}$. As with our likelihood periodogram searches, additive
white noise, $\sigma$, are included as free parameters for each data set.
Further details can be found in Standing et al. (2022).

####  5.2.1 1-Keplerian ($N_{\textrm{p}}=1$) solution

Using kima to obtain planetary parameters as described in Standing et al.
(2022), we find $P_{\textrm{b}}=3.49733^{+0.00093}_{-0.23513}$ when we
restrict the search to 1 Keplerian ($N_{\textrm{p}}=1$), in agreement with the
likelihood period search. The skewed uncertainties (i.e. a relatively large
negative uncertainty) arise from posterior sampling of the alias peaks, which
can be seen at $P=3.19$ and $P=3.26$ d in Fig. 7 (a). The peak aliasing is a
consequence of the sampling, since observations span $\leq 5$ nights at any
given observing epoch with SOPHIE or HARPS-N. The $N_{\textrm{p}}=1$ solution
parameters are shown in Table 4, column 1; the RVs and solution curve are
shown in Fig. 8. We find $\textrm{BF}>7676$ in favour of a single-planet model
compared with a zero-planet model. (N.B. since no samples were obtained during
the run for the $N_{\textrm{p}}=0$ model, we estimate a lower limit on the BF
by setting the number of samples with $N_{\textrm{p}}=0$ equal to 1 (Standing
et al., 2022; Triaud et al., 2022; Baycroft et al., 2023; Standing et al.,
2023).

####  5.2.2 2-Keplerian ($N_{\textrm{p}}=2$) solution

We allowed kima to sample posterior space with $N_{\textrm{p}}$ as a free
parameter. In this way it is possible to recover the preferred number of
Keplerians. We place a maximum $N_{\textrm{p}}=3$, and find the solution with
$N_{\textrm{p}}=2$ has the highest posterior evidence, with
$\textrm{BF}=2397$, compared with the $N_{\textrm{p}}=1$ model, and is shown
in Fig. 9. There is not sufficient evidence for further Keplerians. However,
the two preferred periods differ from the those found by the likelihood model,
which adds Keplerian signals recursively. Instead of the
$P_{\textrm{b}}=3.498$ d and $P_{\textrm{c}}=2.459$ d identified in §5.1, the
$N_{\textrm{p}}=2$ model recovers
$P_{\textrm{b}}=2.45703^{+0.00293}_{-0.00082}$ d and
$P_{\textrm{c}}=5.4193^{+0.0060}_{-0.0032}$ d (Table 4, column 2). With the
additive white noise terms, $\sigma_{\textrm{RV,SO}}$ and
$\sigma_{\textrm{RV,HN}}$, the effective uncertainties in the
$N_{\textrm{p}}=2$ model are $1.68\,\textrm{ms}^{-1}$ and
$1.36\,\textrm{ms}^{-1}$, suggesting that the formal uncertainties,
particularly for HARPS-N (see §4), may be under-estimated.

We investigated the difference between the periods identified by the
$N_{\textrm{p}}=2$ recursive likelihood method in §5.1 and the posterior
sampling method of kima. We performed MCMC posterior sampling on the two
periods we identified in §5.1 and, then performed the same analysis on the
periods identified by kima. Moderate evidence in favour of the two
periodicities recovered by kima was found over the recursively added second
periodicity, with $\Delta\textrm{BIC+}=-5.04$ and $\textrm{BF}=12.2$.
Similarly, we tried restricting the first signal to the $N_{\textrm{p}}=1$
period identified by kima and used kima to detect any further signals. The
posterior sampling yields a second period at $2.23$ d with a moderate
$\textrm{BF}=29.2$, somewhat lower than for the preferred $N_{\textrm{p}}=2$
solution.

The discrepancy between the periods recovered by the two approaches arises
because of the data sampling, which is not optimal for recovering multiple
periodicities with short observing epochs spanning only a few days. In this
scenario, the multimodal posterior sampling used by kima may perform better
than a method that adopts recursive addition of signals.

###  5.3 Quasi-periodic Gaussian Process

kima has also been adapted to enable the use of a Gaussian Process (GP). The
hyperparameters of the GP are inferred together with the orbital parameters
(Faria et al., 2023). The GP quasi-periodic kernel is defined in the form
given by Rasmussen & Williams 2006(see also Haywood et al. 2014) as

$\gamma_{\rm QP}(t_{i},t_{j})=\eta_{1}^{2}{\rm
exp}\left\\{-\frac{(t_{i}-t_{j})^{2}}{2\eta_{2}^{2}}-\frac{2}{\eta_{4}^{2}}sin^{2}\left[\frac{\pi(t_{i}-t_{j})}{\eta_{3}}\right]\right\\}$
(1)

which describes the correlation between RVs at times $t_{i}$ and $t_{j}$. Four
hyperparameters define the covariance matrix. The amplitude, $\eta_{1}$,
describes the deviation of the GP models from the mean function. The
parameter, $\eta_{2}$ is the long term evolution timescale, while $\eta_{3}$,
the characteristic period of the GP, can be related directly to the stellar
rotation period. As noted by Rajpaul et al. (2015) and Barragán et al. (2022),
it is desirable that $\eta_{3}<\eta_{2}$, to ensure the validity of using the
quasi-periodic GP with a periodic component. The factor, $\eta_{4}$, is
proportional to the inverse of the square root of the harmonic complexity,
$\Gamma$ (e.g. see Rajpaul et al. 2015, Barragán et al. 2022 and Nicholson &
Aigrain 2022). From equation 1, $\Gamma=2/\eta_{4}^{2}$, and describes the
roughness or complexity of variations within each characteristic period,
$\eta_{3}$.

####  5.3.1 Keplerian + quasi-periodic Gaussian Process

Because our data comprise only a few nights per observing run, a careful
choice of priors for the GP hyperparameters, is required. We tested solutions
informed by the photometry and activity signatures investigated in §3 and §4.
Following our finding of photometric periodicity (Section 3), we first
trialled a GP prior for $\eta_{3}$ of $\mathcal{G}[11.6\pm 1.5]$ d (i.e. a
Gaussian distribution). Given the likely activity signatures at $P_{\rm
rot}/2$ and $P_{\rm rot}/4$ discussed above in §4.3, we expect characteristic
activity signatures on timescales shorter than than $P_{\rm rot}$. This
suggests some degree of harmonic complexity is required to account for any
structure related to the characteristic period, $\eta_{3}$. Nicholson &
Aigrain (2022) note, for instance, that analysis of stellar lightcurves
typically yield harmonic complexity of $0.5<\Gamma<2$. To investigate the
posterior distribution of the harmonic complexity, we used a fairly wide prior
of 0.1 < $\eta_{4}$ < 10, corresponding to harmonic complexity of
$0.02<\Gamma<200$.

With this configuration, we do not find evidence for any planetary signals.
Although the posterior distribution returns significant periodicities at $\sim
2.5$ d, $3.5$ d and $5.4$ d, the posterior samples for $N_{\textrm{p}}=1,2$
and $3$ return low respective Bayes Factors of BF = $1.2$, $1.3$ and $1.4$. We
found that the prior distributions are recovered in the posteriors for
$\eta_{2}$ and $\eta_{3}$. This not surprising since the few-day span of each
observing epoch is much less than the observationally informed priors. The
posterior distribution for $\eta_{4}=0.15^{+0.17}_{-0.04}$ corresponds to
harmonic complexity of $\Gamma=89^{+76}_{-69}$. This high harmonic complexity
thus accounts for most of the RV variability. However, the sampling rate of
the observations leads us to reject solutions that fit RV variability on very
short timescales. There are too many fitting parameters and not enough data
points to constrain a credible solution. Essentially, the model is too
flexible.

####  5.3.2 A simplified quasi-periodic Gaussian Process

The intensive sampling strategy we adopted to enable searches for shorter
period planets means that several observations were made over a few hours on
each night. By contrast, the significant periodicities in the data are
typically a few days, with the effect that the 71 data points essentially
behave as 21 high S/N data points. Since the $N_{\textrm{p}}=1$ \+ GP model in
§5.3.1 contains 13 fitting parameters, the flexibility of the GP model and
thus the degeneracy with planetary signals is high (Nicholson & Aigrain,
2022). Thus it is not surprising that the hyperparameter priors either
dominate the posteriors in §5.3.1, or lead to over-fitting when relatively
uninformative and wide.

In simulations, Nicholson & Aigrain (2022) noted that there is no single
starspot parameter in their models that relates directly to $\Gamma$. They
adopted initial guesses of $\Gamma=1$ when fitting simulated lightcurves, as
this value represents the transition between low and high harmonic complexity
(Barragán et al., 2022). Informed by the simultaneously derived BIS
periodicity of $2.54$ d, we obtained posterior samples with a prior on
$\eta_{3}$ of $\mathcal{G}[2.5\pm 0.5]$ d. In light of the posterior
distribution for $\eta_{4}$ that we found in §5.3.1, we followed Nicholson &
Aigrain (2022), but fixed $\Gamma=1$ (i.e. we fixed $\eta_{4}=\sqrt{2}$). With
these priors, posterior sampling leads to a preference for $N_{\textrm{p}}=1$,
with $P=3.4982^{+0.0015}_{-0.0027}$ d and evidence of $\textrm{BF}=111.4$. As
in §5.3.1, the posterior distribution is dominated by the priors for
$\eta_{2}$, though the data appears to have had slight influence on the
posterior GP periodic component $\eta_{3}=2.42^{+0.25}_{-0.24}$. The RV
solution curves and folded RVS are plotted in Fig. 10 (a&b). Hence, there is
moderate evidence for a single Keplerian with the same period as the purely
Keplerian $N_{\textrm{p}}=1$ model. The posteriors show that inclusion of the
GP resulted in reduced significance of the periodicities at $2.457$ d and
$5.419$ d found for the purely Keplerian $N_{\textrm{p}}=2$ model.

###  5.4 Alternative models using activity indicators

With the evidence for moderate activity correlations between the RV and BIS
data, we used kima to obtain simultaneous posterior samples using both data
sets and a simultaneous single GP model. Here an additional GP hyperparameter,
$\eta_{1,2}$ is required to model the BIS amplitude. The number of fitting
parameters for the model is 17. This is similar to the approach adopted by
Barragán et al. (2022) although we do not consider the time derivative of the
GP amplitude terms here. Further, adding the BIS timeseries doubles the size
of the data set to 142, or if we consider the effective data set size, to 42
as in §5.3.2. The flexibility of the model is thus reduced with inclusion of
the activity timeseries.

####  5.4.1 Model with simultaneous BIS activity indicator

We used the same model priors as in §5.3.1. We again found that the $\eta_{2}$
and $\eta_{3}$ posterior distributions matched the priors while a high
harmonic complexity is preferred, with $\eta_{4}=0.15^{+0.05}_{-0.04}$
($\Gamma=89^{+76}_{-39}$). The posterior samples demonstrate an over-density
at $P\sim 3.5$ d, though for $N_{\textrm{p}}=1,2$ and $3$, respective Bayes
Factors of BF = $1.0$, $0.8$ and $0.8$ indicate very weak evidence for any
Keplerian signals. Even with the inclusion of the BIS timeseries, the model is
too flexible.

####  5.4.2 Model with simultaneous BIS activity and fixed harmonic complexity

Finally, with fixed $\Gamma=1$ ($\eta_{4}=\sqrt{2}$), as in §5.3.2, we used a
prior on $\eta_{3}$ of $\mathcal{G}[2.5\pm 0.5]$ d. The solution is shown in
Table 4, column 4 and in Fig. 10 (c, d and e). A $\textrm{BF}=1038$ indicates
strong evidence in favour of a $N_{\textrm{p}}=1$ solution with with
$P=3.4982$ d. The inclusion of BIS thus appears to have provided further
evidence for the $2.5$ d activity periodicity, yielding a higher BF compared
with the same model and priors on the RV-only timeseries in §5.3.2.

Fig. 10 (e) demonstrates that there is considerable scatter in the BIS at some
epochs, particularly in the poorer quality data in the SO-15A run. The white
noise terms for the BIS data (Table 4) reflect this. Modelling the RV and BIS
data simultaneously has also likely resulted in the higher RV white noise
terms compared with the $N_{\textrm{p}}=2$ solution; accounting for
$\sigma_{\textrm{RV,SO}}$ and $\sigma_{\textrm{RV,HN}}$ yields respective
effective uncertainties of $2.22\,\textrm{ms}^{-1}$ and
$1.48\,\textrm{ms}^{-1}$.

##  6 Summary and Discussion

Although there is clear evidence for a single candidate Keplerian
($N_{\textrm{p}}=1$) with $P_{\textrm{b}}~{}=3.49791^{+0.00046}_{-0.00143}$ d
and $m_{\textrm{b}}$ sin $i$ = $11.6_{-1.5}^{+1.6}$ M⊕, the purely Keplerian
model indicates strong evidence favouring a 2-Keplerian ($N_{\textrm{p}}=2$)
solution, with a relative Bayes Factor, $\textrm{BF}=2397$. The preferred
$N_{\textrm{p}}=2$ solution does not include the $N_{\textrm{p}}=1$ period,
but rather recovers periods of
$P_{\textrm{b}}~{}=~{}2.4570^{+0.0026}_{-0.0462}$ d and
$P_{\textrm{c}}~{}=~{}5.4196^{+0.6766}_{-0.0030}$ d with respective derived
minimum masses of $m_{\textrm{b}}\,\textrm{sin}\,i~{}=8.0_{-1.5}^{+1.1}$ M⊕
and $m_{\textrm{c}}\,\textrm{sin}\,i~{}=12.2_{-1.6}^{+1.4}$ M⊕. The
$N_{\textrm{p}}=2$ solution derived from likelihood periodogram searches, with
$P_{\textrm{b}}~{}=~{}3.498$ d and $P_{\textrm{c}}~{}=~{}2.459$ d, suggests
that there may be conditions where the common approach of recursively adding
signals does not lead to the best solution; here, it is likely that the length
of the candidate periods and relative length of the observing runs has
resulted in ambiguity. Our finding may also in part be a limitation of the
period search algorithm, which does not explore maximum likelihood space for a
global maximum, but rather, iteratively adjusts the Keplerians already
identified when searching for additional signals. In this respect, an
algorithm capable of finding a global marginalised maximum likelihood in a
potentially multimodal posterior space is to be preferred, particularly when
considering multiple Keplerian signals. It is important to note that the
posterior samples from kima also contained a significant number of samples
around the $3.498$ d period (around 40% compared with the $2.4570$ d period
and 50% compared with the $5.4196$d period).

Despite the low log($R^{\prime}_{\rm HK})=-5.24$ that we used to select DMPP-4
for follow-up RV observations, there is some evidence for activity-induced RV
variability. This is consistent with the intrinsic stellar activity
corresponding to a higher value of log($R^{\prime}_{\rm HK}$) than observed.
The hypothesis underlying DMPP is that circumstellar absorption depresses the
log($R^{\prime}_{\rm HK}$), so this is as expected. Because the
$N_{\textrm{p}}=2$ solution recovers periods that are close to one half and
one quarter integer factors of the inferred rotation period
($P_{\textrm{rot}}/2$ and $P_{\textrm{rot}}/4$), as discussed in detail in
§4.3, we were prompted to investigate RV models that contained an activity
component.

The GP models we investigated for DMPP-4 were necessarily restrictive because
of the nature of the data sampling, the relatively short observing runs and
the effective size of the data sets. When a quasi-periodic GP is included in
the model, we find moderate evidence ($\textrm{BF}=111.4$) for a single planet
in the RVs with $P_{\textrm{b}}~{}=3.4982^{+0.0015}_{-0.0027}$ d and
$m_{\textrm{b}}\,\textrm{sin}\,i~{}=12.6^{+1.6}_{-1.8}\textrm{M}_{\oplus}$.
Including the simultaneous BIS measurements augments the Bayesian evidence,
yielding $P_{\textrm{b}}~{}=3.4982^{+0.0022}_{-0.2327}$ d and
$m_{\textrm{b}}\,\textrm{sin}\,i~{}=12.2^{+1.8}_{-1.9}\textrm{M}_{\oplus}$
with a $\textrm{BF}=1038$.

Thus, although the purely Keplerian model prefers $N_{\textrm{p}}=2$, the
model restricted to $N_{\textrm{p}}=1$ and the models with a GP and no
$N_{\textrm{p}}$ restriction all provide moderate to strong evidence for a
single planet. The periods and minimum masses are consistent within the
uncertainties for these models. For the Keplerian models with a GP, with and
without simultaneous BIS data, either greater model flexibility, the adopted
tighter priors, or both, result in lower overall Bayesian evidence. Without
further data it is thus difficult to distinguish between the preferred
$N_{\textrm{p}}=2$ model and models that return single Keplerians in the
presence of variable activity. Nevertheless, there is (i) evidence for
activity periodicities that coincide with the identified RV periodicities in
the purely Keplerian $N_{\textrm{p}}=2$ model. Further, there is (ii) a not-
insignificant appearance of the the $3.498$ d periodicity in the model
posterior samples for the $N_{\textrm{p}}=2$ model. And finally, (iii) a
preferred periodicity at $3.4982$ d is found in the Keplerian + GP models.
These three observations might be taken as reasonable evidence to prefer the
solutions with a single $3.498$ d Keplerian.

We based our choices of GP hyperparameter priors on starspot distribution
hypotheses and photometrically inferred periodicities. Recently however,
Nicholson & Aigrain (2022) have suggested that harmonic complexity priors
derived from photometry are not necessarily appropriate. Simplifying the model
by removing the harmonic complexity, $\Gamma$, and relying only on
simultaneously derived and modelled BIS periodicities makes intuitive sense,
especially in light of the relatively few available effective data points.
Nicholson & Aigrain (2022) also investigated the effect of reducing the number
of data points and hence increasing the flexibility of the model. The
resulting increased degeneracy highlights the need for sufficient data points.

The following sections further investigate the nature of the planet
candidates. Despite a preference for the models with a single planet, we also
considered the the $N_{\textrm{p}}=2$ solution.

Figure 11: The Planet mass vs orbital period diagram for planets with known
mass. Each planet is colour coded according to its estimated equilibrium
temperature, $T_{\rm eq}$ (assuming a Bond Albedo of $A_{\rm B}=0.36$). The
preferred $N_{\textrm{p}}=1$ solution in column 4 of Table 4 for DMPP-4 b is
shown by the large square symbol. The $N_{\textrm{p}}=2$ candidate planets,
DMPP-4 b and c, are shown by the large upright triangle and circle. DMPP-1 b,
c, d & e, DMPP-2 b and DMPP-3A b are also highlighted.

###  6.1 Orbital stability of planets in the $N_{\textrm{p}}=2$ model

We checked the stability of the orbital solution for the $N_{\textrm{p}}=2$
model using the IAS15 integrator within the N-body orbital integrator, rebound
(Rein & Liu, 2012; Rein & Spiegel, 2015). Simulations were started with $e=0$
and the semi-major axes listed in Table 4 for the $N_{\textrm{p}}=2$ solution.
For both planet candidates, only very small fluctuations in $a$ and $e$ are
seen. For DMPP-4 b, we find semi-major axis r.m.s. variability over 25,000
years of $5.5\times 10^{-7}$ AU (i.e. 0.0014% of the semi-major axis). A very
low degree of eccentricity of $e_{\rm b}=0.00020\pm 0.00009$ is established.
For the putative DMPP-4 c, we find corresponding r.m.s. semi-major axis
variability of $4.7\times 10^{-7}$ AU (i.e. 0.0007% of the semi-major axis)
and eccentricity $e_{\rm c}=0.00007\pm 0.00003$. For planets that formed in
circular orbits, or evolved into non-eccentric orbits via steady tidal
circularisation, we expect a stable configuration. Guided by the upper limit
to the eccentricities we found in Table 4, we conducted simulations that
started with $e=0.065$. The respective semi-major axis r.m.s. variabilities
for DMPP-4 b and DMPP-4 c are 0.0023% and 0.0019% and the corresponding
eccentricities were $e_{\textrm{b}}=0.071\pm 0.030$ with a full range of
$0.017<e_{\textrm{b}}<0.107$ and $e_{\textrm{c}}=0.055\pm 0.019)$ with a range
of $0.024<e_{\textrm{c}}<0.079$. Apart from the $680$ year periodic
eccentricity variability, the orbits did not show any further evidence of
evolution on the 25,000 year timescale of the simulations. The stability of
the orbits is not surprising given that the closest approach of the planets is
$0.021$ AU, while the respective Hill radii of DMPP-4 b and c are
$r_{H}=0.0007$ AU and $0.0013$ AU.

###  6.2 Fundamental properties of the exoplanet candidates

Fig. 11 shows the period-mass diagram for planets with known mass determined
with $\leq$ $20\%$ uncertainty. Only archival
planets777https://exoplanetarchive.ipac.caltech.edu with true dynamical mass
estimates and tabulated $T_{\textrm{eff}}$ and $R_{*}$ are plotted (e.g.
planets with lower mass estimates, $m_{\textrm{p}}\,\textrm{sin}\,i$, are
excluded). The $N_{\textrm{p}}=1$, DMPP-4 b, solution using RV and BIS data
(Table 4, column 4) and the $N_{\textrm{p}}=2$ solution (Table 4, column 2)
are shown. The planet candidates are closer to the Neptune desert than the
planets orbiting DMPP-1, -2 and -3 (Staab et al., 2020; Haswell et al., 2020;
Barnes et al., 2020). All DMPP planets possess lower mass estimates to better
than 20% with the exception of DMPP-1 e, which has an uncertainty of 22%. The
points in Fig. 11 are colour coded according to the planet equilibrium
temperatures, $T_{\rm eq}$. The value obtained depends on the assumed albedo.
In the Solar System Bond albedos, $A_{\rm B}$ vary between $\sim 0.088$ for
Mercury (Mallama, 2017), through a mean value of $0.36$ for gas giant planets
(Li et al. 2018; Hanel et al. 1983; Pearl et al. 1990; Pearl & Conrath 1991)
up to $0.76$ for Venus (Haus et al., 2016). Using these values in the equation
$T_{\rm eq}=T_{\rm eff}\sqrt{R_{*}/2a}(1-A_{\rm B})^{0.25}$ therefore gives a
range of plausible equilibrium temperatures. For the $N_{\textrm{p}}=1$,
respective $T_{\rm eq}\sim 1608$, $1472$ and $1152$ K are implied for DMPP-4
b. For $N_{\textrm{p}}=2$, the corresponding temperatures for DMPP-4 b are
$T_{\rm eq}\sim 1809$, $1655$ and $1295$ K and for DMPP-4 c are $T_{\rm
eq}\sim 1388$, $1270$ and $994$ K. The equilibrium temperatures for the
Mercury-like, low albedo cases are thus around $400-500$ K higher.

Using the probabilistic Forecaster (Chen & Kipping, 2017) and the minimum mass
for our preferred $N_{\textrm{p}}=1$ solution (Table 4), we estimate the
minimum radius of DMPP-4 b as $R_{\rm
b}=3.49^{+1.47}_{-1.04}\,\rm{R}_{\oplus}$. For the $N_{\textrm{p}}=1$
solution, minimum radii of $R_{\rm b}=2.72^{+1.17}_{-0.81}\,\rm{R}_{\oplus}$
and $R_{\rm c}=3.49^{+1.46}_{-1.02}\,\rm{R}_{\oplus}$ are predicted. DMPP-4 b
and c would thus be likely to possess few per cent $H_{2}$ dominated
atmospheres (Zeng et al., 2019). The planets may have lost much of their
atmospheres, through radiation-driven mass loss, evolving downwards through
the Neptune Desert (Fig. 11). Given this probable evolutionary history, and
the ongoing high irradiation from the F7V host star, DMPP-4 b and c may
deviate from expectations based on the general planet population. If DMPP-4 b
and DMPP-4 c are rocky ‘Chthonian’ planets stripped of their atmospheres, they
are likely hot enough that their surfaces comprise liquid magma oceans.
Temperatures of 1100 – 1500 K are typical for molten magma on the Earth
(Sigurdsson et al., 2015), as noted by Staab et al. (2020). This more extreme
scenario could mean that DMPP-4 b and c lie below the exoplanet radius valley,
with radii of less than $1.7\,-\,2$ R⊕ (Fulton et al., 2017; Zeng et al.,
2021), in turn implying densities of between $1.6\times\rho_{\textrm{Earth}}$
and $2.5\times\rho_{\textrm{Earth}}$ if $R\leq 1.7$ R⊕. This could potentially
affect mass loss since the surface gravities would be $2.8-4.2$ greater, but
needs detailed modelling to quantify. If substantial mass-loss does not occur,
and the planets are indeed atmosphereless, to satisfy the DMPP hypothesis, we
might expect that there are additional interior and as yet undetected mass-
losing planets in the DMPP-4 system.

Direct size determinations from transits would reveal bulk properties and
potential atmospheric compositions. If the rotation period of DMPP-4 is 11.6 d
and the orbital angular momentum of the planets is aligned with the stellar
rotational angular momentum, there is a high transit probability (Haswell et
al., 2020). We used the Lightcurve Analysis Tool for Transiting Exoplanets
(latte) to search for transits in the 2 minute cadence tess photometry (Eisner
et al., 2020). latte corrects the data for residual systematics using an
iterative non-linear filter (Aigrain & Irwin, 2004). A Box-Least-Squares
period search is performed on the flattened lightcurves. We do not find
evidence for transits in any of the tess sectors (see §3) observed to date. Of
course these planets may not be aligned exactly edge-on, so may not transit.
In this case information from phase curves, e.g. using JWST could be
revealing.

##  7 Conclusion

Because it is so bright, DMPP-4 is an important target for follow-up
characterisation. Although the current data do not enable us to unambiguously
distinguish between a system with one or two planets, there is significant
evidence to favour models with a single planet. Our currently preferred
solution indicates a $12.2~{}\textrm{M}_{\oplus}$ planet in a $3.498$ d orbit.
The potential activity contributions, amplitude modulations and periods of the
DMPP-4 RV signals we observe indicate that an intensive RV monitoring campaign
is needed. This would better enable us to distinguish between multi-planet
solutions and activity signals on the timescales of the stellar and planet
candidate periods. It may also reveal additional planets below our current
mass detection threshold.

## Acknowledgements

J.R. Barnes, M.R. Standing and C.A. Haswell were funded by STFC under
consolidated grant ST/T000295/1 and ST/X001164/1. The radial velocity data
were obtained at through The OPTICON Trans-National Access Programme (FP7II,
2013-2016). This work also used observations from the Las Cumbres Observatory
global telescope network. This paper includes data collected by the TESS
mission. Funding for the TESS mission is provided by the NASA’s Science
Mission Directorate. Data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia) have been used. The Gaia data were processed
by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This research made use of
the NASA Exoplanet Archive, which is operated by the California Institute of
Technology, under contract with the National Aeronautics and Space
Administration under the Exoplanet Exploration Program (see
https://exoplanetarchive.ipac.caltech.edu for further details of data
sources).

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author. The reduced radial velocities and activity indices are
tabulated in Appendix A.

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##  Appendix A Data

Table 5: SOPHIE RVs and activity indices with their respective uncertainties. BJD | RV | $\sigma_{\textrm{RV}}$ | $\kappa$ | $\sigma_{\kappa}$ | FWHM | $\sigma_{\textrm{FWHM}}$ | BIS | $\sigma_{\textrm{BIS}}$ | Sindex | $\sigma_{\textrm{Sindex}}$
---|---|---|---|---|---|---|---|---|---|---
-2450000 | [ms-1] | [ms-1] |  |  | [ms-1] | [ms-1] | [ms-1] | [ms-1] |  |
7141.601584 | -4.39524 | 2.28894 | -0.17840 | 0.04610 | -4.72111 | 2.25224 | -3.28979 | 2.25224 | 0.00344 | 0.00057
7141.653250 | 5.10336 | 2.13098 | -0.22447 | 0.04449 | 10.93894 | 1.91173 | 10.71020 | 1.91173 | 0.00123 | 0.00047
7143.531089 | 8.48712 | 2.55042 | 1.01775 | 0.07311 | 92.78629 | 2.45182 | 11.35026 | 2.45182 | 0.01114 | 0.00081
7143.573198 | 1.51026 | 2.47348 | -0.01728 | 0.03295 | 16.32131 | 2.50031 | -9.45646 | 2.50031 | 0.00367 | 0.00049
7143.602950 | -0.92205 | 2.11206 | -0.27088 | 0.02228 | 18.29004 | 1.91026 | -5.22389 | 1.91026 | 0.00225 | 0.00069
7144.557581 | -8.80357 | 2.58044 | 0.59530 | 0.08676 | 53.22427 | 3.01222 | 0.21021 | 3.01222 | 0.01168 | 0.00094
7144.646502 | -0.32047 | 2.22101 | -0.58017 | 0.04569 | -21.83060 | 2.29269 | 5.87688 | 2.29269 | 0.00260 | 0.00056
7336.228379 | 0.51648 | 1.29009 | 0.24119 | 0.02395 | -11.59571 | 1.61949 | -4.94574 | 1.61949 | -0.00402 | 0.00040
7337.233243 | -2.94198 | 1.18561 | -0.12274 | 0.02887 | -42.31921 | 1.35967 | -3.51409 | 1.35967 | -0.00274 | 0.00033
7337.268115 | -1.63685 | 1.15477 | -0.07434 | 0.02799 | -25.78468 | 1.35227 | 6.47557 | 1.35227 | -0.00374 | 0.00034
7337.319238 | 2.88576 | 1.33344 | 0.93347 | 0.07464 | 48.30336 | 1.33257 | 7.05742 | 1.33257 | 0.00457 | 0.00049
7338.222375 | 8.00376 | 1.07203 | 0.31575 | 0.02922 | -12.68507 | 1.35549 | -3.62720 | 1.35549 | -0.00288 | 0.00034
7338.250928 | 5.09088 | 1.00829 | 0.56340 | 0.03536 | 2.26548 | 1.35064 | -13.84313 | 1.35064 | -0.00259 | 0.00036
7339.228004 | -1.18240 | 1.13805 | -0.04812 | 0.02567 | -11.14233 | 1.23464 | 10.23551 | 1.23464 | -0.00143 | 0.00031
7339.257860 | -1.20310 | 1.07228 | -0.11327 | 0.02462 | -15.81676 | 1.24214 | 11.78683 | 1.24214 | -0.00263 | 0.00030
7340.227075 | -2.35796 | 1.58780 | 0.46295 | 0.03608 | -22.86128 | 1.65094 | 0.54354 | 1.65094 | -0.00113 | 0.00042
7340.263479 | -1.70372 | 1.59813 | 0.48781 | 0.03798 | -17.09651 | 1.76456 | 5.34441 | 1.76456 | -0.00339 | 0.00046
7340.319923 | -2.06757 | 1.57236 | 2.25613 | 0.10606 | 45.08994 | 1.87082 | -4.61932 | 1.87082 | 0.00246 | 0.00064
7485.596614 | 7.17563 | 1.33677 | 1.39810 | 0.07858 | 53.23429 | 1.31924 | -10.08458 | 1.31924 | 0.00737 | 0.00044
7485.637771 | 4.89580 | 1.21995 | 0.45280 | 0.03302 | 31.25865 | 1.15977 | -13.16582 | 1.15977 | 0.00284 | 0.00033
7485.653947 | 3.79001 | 2.45915 | -0.28637 | 0.04497 | -16.98751 | 2.33639 | -15.78980 | 2.33639 | -0.00280 | 0.00057
7486.602113 | -5.52468 | 1.14257 | 1.00287 | 0.05604 | 55.68851 | 1.25160 | 4.81432 | 1.25160 | 0.00206 | 0.00036
7486.620842 | -5.85710 | 2.34468 | 0.89867 | 0.09027 | 40.23651 | 2.11967 | -0.12312 | 2.11967 | 0.00115 | 0.00060
7486.651285 | -3.88557 | 1.26869 | 0.29901 | 0.03039 | 15.83827 | 1.23164 | 11.53410 | 1.23164 | -0.00099 | 0.00031
7488.568901 | 0.77768 | 2.15324 | 1.65057 | 0.17660 | 59.88006 | 2.28861 | -4.28979 | 2.28861 | 0.00672 | 0.00077
7488.589914 | 1.46278 | 2.71892 | 0.69793 | 0.10059 | 34.68188 | 2.33840 | -0.62312 | 2.33840 | -0.00030 | 0.00066
7488.628896 | 2.29283 | 2.69899 | 0.92556 | 0.10769 | 54.12248 | 3.48791 | -11.95650 | 3.48791 | -0.00659 | 0.00099
7488.660901 | 4.47383 | 3.22298 | 0.06100 | 0.05604 | -2.11315 | 3.23877 | 6.87688 | 3.23877 | -0.00663 | 0.00081
7489.572033 | -0.72295 | 1.36631 | 2.11115 | 0.08152 | 73.07962 | 1.29323 | -6.62897 | 1.29323 | 0.01013 | 0.00046
7489.605550 | -1.16558 | 2.26300 | 1.52788 | 0.13017 | 49.23823 | 2.24038 | 7.87688 | 2.24038 | 0.00647 | 0.00072
7489.647550 | 1.75240 | 1.23957 | 0.16331 | 0.03525 | -3.23986 | 1.55964 | -8.13880 | 1.55964 | -0.00186 | 0.00040
7723.218982 | 1.87889 | 0.96311 | -0.41169 | 0.02352 | -42.80423 | 1.04460 | -8.06201 | 1.04460 | -0.00316 | 0.00024
7723.253581 | 2.09855 | 1.40480 | -0.16232 | 0.03324 | -29.67417 | 1.34340 | -7.40832 | 1.34340 | -0.00076 | 0.00033
7723.295186 | 3.68135 | 1.25362 | 0.06011 | 0.03636 | -24.18203 | 1.32260 | -10.86468 | 1.32260 | -0.00045 | 0.00034
7723.350564 | 0.53657 | 2.27489 | 0.03555 | 0.04458 | -12.12334 | 2.25218 | -0.45646 | 2.25218 | 0.00014 | 0.00058
7724.224800 | 4.01791 | 1.25577 | -0.49720 | 0.03038 | -34.87339 | 1.34514 | 4.48724 | 1.34514 | -0.00296 | 0.00032
7724.261675 | -0.38338 | 1.04110 | -0.02925 | 0.02706 | -20.21665 | 1.03263 | 2.43856 | 1.03263 | -0.00084 | 0.00026
7724.313087 | 1.24111 | 1.22726 | -0.06321 | 0.02821 | -28.89020 | 1.31397 | -2.56293 | 1.31397 | -0.00118 | 0.00033
7725.233619 | -6.52162 | 1.35877 | -0.20534 | 0.02654 | -16.29436 | 1.33851 | 10.43826 | 1.33851 | -0.00185 | 0.00033
7725.270120 | -8.54515 | 0.99990 | -0.18854 | 0.02127 | -15.14537 | 1.14938 | 8.07956 | 1.14938 | -0.00152 | 0.00029
7725.323129 | -9.30088 | 1.27309 | 0.22961 | 0.02543 | 2.19168 | 1.29251 | 9.42790 | 1.29251 | 0.00006 | 0.00034
7727.215701 | -0.80584 | 1.17479 | -0.25188 | 0.02333 | -32.46199 | 1.15532 | 7.07822 | 1.15532 | -0.00319 | 0.00028
7727.247700 | 1.01686 | 1.25185 | -0.07080 | 0.02174 | -16.33009 | 1.14465 | -1.37031 | 1.14465 | -0.00205 | 0.00029
7727.291796 | 1.05599 | 1.00530 | 0.03403 | 0.02116 | -24.66017 | 1.30235 | 2.35188 | 1.30235 | -0.00232 | 0.00033
7727.339162 | -2.57386 | 1.21831 | 0.78096 | 0.04290 | 11.39416 | 1.26968 | -0.66842 | 1.26968 | 0.00350 | 0.00038
Table 6: HARPS-N RVs and activity indices with their respective uncertainties. BJD | RV | $\sigma_{\textrm{RV}}$ | $\kappa$ | $\sigma_{\kappa}$ | FWHM | $\sigma_{\textrm{FWHM}}$ | BIS | $\sigma_{\textrm{BIS}}$ | Sindex | $\sigma_{\textrm{Sindex}}$
---|---|---|---|---|---|---|---|---|---|---
-2450000 | [ms-1] | [ms-1] |  |  | [ms-1] | [ms-1] | [ms-1] | [ms-1] |  |
7584.425790 | -0.08931 | 0.86354 | -0.67475 | 0.04260 | -9.42621 | 0.85020 | -4.10532 | 0.85020 | -0.00065 | 0.00025
7584.582598 | -4.81271 | 0.75718 | -0.01508 | 0.01218 | 0.21320 | 0.74623 | -6.39534 | 0.74623 | -0.00036 | 0.00021
7584.613142 | -4.79456 | 0.80999 | -0.75987 | 0.02489 | -0.89548 | 0.81160 | -8.69211 | 0.81160 | -0.00054 | 0.00023
7584.700103 | -1.18472 | 0.61352 | -2.71648 | 0.06898 | 1.59290 | 0.62110 | -12.34777 | 0.62110 | -0.00156 | 0.00018
7584.729063 | -4.98745 | 0.85418 | -1.62125 | 0.01980 | 5.90301 | 0.69821 | 0.21189 | 0.69821 | -0.00121 | 0.00021
7585.385055 | -7.99774 | 0.66480 | 2.79469 | 0.02110 | 14.57599 | 0.65326 | 9.93206 | 0.65326 | 0.00010 | 0.00019
7585.415764 | -5.17417 | 0.80530 | 2.55094 | 0.02791 | 7.35933 | 0.76919 | 13.07475 | 0.76919 | -0.00008 | 0.00021
7585.680900 | -3.39515 | 0.80822 | 7.07734 | 0.19464 | -5.43576 | 0.85222 | 8.84914 | 0.85222 | 0.00258 | 0.00024
7585.710733 | -2.81989 | 0.98120 | 6.11499 | 0.12538 | -1.54140 | 0.88054 | 0.54579 | 0.88054 | 0.00115 | 0.00025
7586.385670 | 2.35233 | 0.61816 | 2.99824 | 0.03486 | -6.74626 | 0.63072 | 0.73490 | 0.63072 | 0.00082 | 0.00018
7586.416934 | 0.45226 | 0.81986 | 2.90381 | 0.07114 | -7.84784 | 0.86834 | 0.34820 | 0.86834 | 0.00077 | 0.00024
7586.695960 | 8.46464 | 0.74669 | 6.98437 | 0.20156 | -5.80447 | 0.72816 | -3.43951 | 0.72816 | 0.00193 | 0.00020
7586.725758 | 7.86117 | 1.02063 | 7.11873 | 0.27284 | 7.95727 | 0.97756 | -2.46851 | 0.97756 | 0.00112 | 0.00028
7587.387682 | -0.17318 | 1.02249 | 3.76661 | 0.10800 | 18.76964 | 0.95000 | -1.25455 | 0.95000 | 0.00108 | 0.00028
7587.421027 | -2.49737 | 1.24193 | 4.96372 | 0.17343 | 14.83074 | 1.11321 | 0.43501 | 1.11321 | 0.00080 | 0.00032
7587.617955 | 0.02368 | 0.78475 | 4.22303 | 0.10651 | 13.25287 | 0.73595 | 2.22708 | 0.73595 | 0.00007 | 0.00020
7587.647257 | 0.21431 | 0.96391 | 4.62988 | 0.12810 | 12.59398 | 0.86395 | 5.54800 | 0.86395 | 0.00096 | 0.00023
8698.388858 | 1.01422 | 0.51697 | 2.76669 | 0.04796 | 20.75482 | 0.45840 | -1.71191 | 0.45840 | 0.00022 | 0.00013
8698.404002 | -1.17661 | 0.84289 | 0.85213 | 0.10445 | 22.52849 | 0.74091 | -4.47112 | 0.74091 | -0.00048 | 0.00020
8698.464088 | 0.57934 | 0.38104 | 1.41506 | 0.05223 | -15.96659 | 0.44190 | -1.44664 | 0.44190 | -0.00049 | 0.00011
8698.486873 | -0.06746 | 0.44804 | 2.37282 | 0.04728 | -11.83323 | 0.52289 | -2.76537 | 0.52289 | -0.00043 | 0.00014
8698.500097 | 1.79272 | 0.77203 | -1.51143 | 0.22837 | -30.51439 | 1.31841 | 0.01580 | 1.31841 | -0.00162 | 0.00032
8698.623381 | 0.61839 | 0.35713 | -1.20359 | 0.07589 | -18.28953 | 0.45398 | 5.98544 | 0.45398 | -0.00115 | 0.00011
8698.646135 | 0.78231 | 0.35929 | -0.98089 | 0.07068 | -13.75577 | 0.42110 | 2.96804 | 0.42110 | -0.00084 | 0.00010
8698.697318 | 1.64556 | 0.38615 | 2.33327 | 0.03592 | 7.44778 | 0.39674 | 2.23658 | 0.39674 | -0.00008 | 0.00011
8698.714388 | 1.30359 | 0.52988 | 3.74413 | 0.05086 | 15.54402 | 0.51448 | 1.23481 | 0.51448 | 0.00046 | 0.00015