# Can Gaia find planets around white dwarfs?

Hannah Sanderson1,2, Amy Bonsor2, Alexander Mustill3
1Department of Earth Sciences, South Parks Road, Oxford, OX1 3AN, UK
2Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge,
CB3 0HA, UK
3Lund Observatory, Dept. of Astronomy and Theorical Physics, Lund University,
Box 43, 22100 Lund, Sweden E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The Gaia spacecraft presents an unprecedented opportunity to reveal the
population of long period (a>1 au) exoplanets orbiting stars across the H-R
diagram, including white dwarfs. White dwarf planetary systems have played an
important role in the study of planetary compositions, from their unique
ability to provide bulk elemental abundances of planetary material in their
atmospheres. Yet, very little is known about the population of planets around
white dwarfs. This paper predicts the population of planets that Gaia will
detect around white dwarfs, evolved from known planets orbiting main-sequence
stars. We predict that Gaia will detect $8\pm 2$ planets around white dwarfs:
$8\pm\,3\%$ will lie inside 3 au and $40\pm 10\,\%$ will be less massive than
Jupiter. As surviving planets likely become dynamically detached from their
outer systems, those white dwarfs with Gaia detected planets may not have
planetary material in their atmospheres. Comparison between the predicted
planet population and that found by Gaia will reveal the importance of
dynamical instabilities and scattering of planets after the main-sequence, as
well as whether photoevaporation removes the envelopes of gas giants during
their giant branch evolution.

###### keywords:

white dwarfs; astrometry; planets and satellites: detection

††pubyear: 2022††pagerange: Can Gaia find planets around white dwarfs?–C

##  1 Introduction

The Gaia space satellite has provided an unprecedented level of information on
about 1.8 billion stars in the Milky Way (and beyond) leading to advances
across astrophysics from structure and evolution of the Milky Way to
asteroseismology to tests of General Relativity (Gaia Collaboration et al.,
2016, 2018, 2021; Gaia Collaboration et al., 2022a). In providing precision
astrometry for all nearby stars, Gaia will uncover many companions to those
stars, including exoplanets. Uniquely this detection method will access stars
at all stages of stellar evolution, including white dwarfs. Already, in data
release three (DR3), Gaia astrometric, spectroscopic and radial velocity
measurments have brought the number of known binary systems across the H-R
diagram to 800,000 (Gaia Collaboration et al., 2022b). Gaia has detected ten
exoplanet candidates via radial velocity measurements. Eleven main-sequence
exoplanets between 4-20 $M_{J}$ have been detected astrometrically: nine have
been validated against existing detections and two are new detections. Four
white dwarf substellar companion candidates have been detected, including the
detection of one super Jupiter candidate around a white dwarf. In future data
releases, the reduction in the errors and the longer baseline for measurements
will enable detection of many more systems.

Understanding the planetary systems around white dwarfs is key, because these
stars potentially reveal planetary composition. Polluted white dwarfs, white
dwarfs with metal lines in their spectra, trace the composition of tidally
disrupted planetesimals that have been accreted onto the star (Jura & Young,
2014). This can provides an opportunity to study the interiors of rocky
exoplanets (Harrison et al., 2018; Swan et al., 2021; Zuckerman et al., 2011)
and the architecture of planetary systems post-main sequence (Mustill et al.,
2018; Veras et al., 2011). Although polluted white dwarfs provide signatures
of tidally disrupted bodies, little is known about planets orbiting white
dwarfs. Despite, extensive transit and direct imaging surveys (Fulton et al.,
2014; Hogan et al., 2011; Faedi et al., 2011; Burleigh et al., 2002; Gould &
Kilic, 2008; Debes et al., 2005; Xu et al., 2015) a limited number of white
dwarf planet candidates have been discovered, including Sigurdsson et al.
(2003); Luhman et al. (2011); Gänsicke et al. (2019); Vanderburg et al.
(2020); Blackman et al. (2021) which were discovered prior to DR3. One
candidate has a semi-major axis around 3 au (Blackman et al., 2021), but the
other planet candidates are in extreme environments. Some are very close e.g.
orbital period $\sim 1$ day (Vanderburg et al., 2020) or close enough to be
photoevaporating (Gänsicke et al., 2019). Others are very far ($\sim\,2500$
au) (Luhman et al., 2011) from their host star or circumbinary around a white
dwarf and pulsar (Sigurdsson et al., 2003). Detailed observations of planets
around white dwarfs are lacking.

Most known planets orbit host stars that end their lives as white dwarfs, and
models suggest that outer planets should survive post-main sequence stellar
evolution (Veras, 2016; Mustill & Villaver, 2012). The final position of a
planet after post-main sequence stellar evolution is determined by the
competing effects of mass loss, which moves planets outwards, and tidal
forces, which move them inwards. Tidal forces are most significant for massive
planets on the AGB, when stars have more extended envelopes (up to several au
Mustill & Villaver, 2012). Close-in planets may be engulfed, whilst the high
luminosity of the AGB star can cause heating and mass loss from the exoplanet
atmosphere (photoevaporation Veras, 2016; Barker, 2020; Bear & Soker, 2011).
In multi-planet systems, these effects are complicated by the influence of
planets on each other and resulting instabilities can lead to ejection of
planets or star-planet collisions (Veras et al., 2013).

Although the broad effects are known, detailed modelling of post-main sequence
planetary evolution is difficult due to uncertainties in stellar evolution
models for the RGB and AGB (e.g. Bertolami, 2018; Matthews & Claussen, 2018),
different approaches to modelling tidal interactions (Ogilvie, 2014), problems
quantifying energy sources that contribute to planet common envelope evolution
(Ivanova et al., 2013), chaotic evolution of multi-planet systems e.g. Veras
et al. (2013) and the uncertainty in the main-sequence population of wide-
orbit planets around WD progenitors. Observationally probing the population of
planets around white dwarfs would provide us with direct evidence of planets
that have survived this post-main sequence evolution, which is vital for
improving models of this process.

With its high precision astrometry, Gaia presents an unprecedented opportunity
to uncover the population of planets orbiting white dwarfs. Crucially, Gaia
will bridge the gap between the small (< 1 au) semi-major axes of transit
surveys and the large (> 10 au) semi-major axes of direct imaging surveys.
Gaia is predicted to find tens of thousands of planets beyond 1 au (Casertano
et al., 2008; Perryman et al., 2014; Ranalli et al., 2018), including some
around white dwarfs. Silvotti et al. (2011) show that Gaia will uncover
planets with masses greater than $2M_{J}$ around the brightest white dwarfs.
Gaia has already demonstrated its potential to find white dwarf exoplanets
with the announcement of a new white dwarf planet candidate in DR3: a $\sim 9$
$M_{J}$ planet with a period of $33.65\pm 0.05$ days around WD0141-675 (Gaia
Collaboration et al., 2022b). This is also the first planet candidate
detection around a polluted white dwarf. The full catalogue, which will
include planets with periods up to $\approx 10$ years, will not be available
until DR5. Comparison between HIPPARCOS and Gaia proper motion anomalies can
also be used to get obtain a longer baseline (24.75 years between HIPPARCOS
and DR3 Kervella et al., 2022). Most white dwarfs were too faint to be
observed in detail by HIPPARCOS, but this technique has already hinted at
binarity in two white dwarf systems (LAWD 37, GD 140) Kervella et al. (2019);
Kervella et al. (2022).

This paper aims to provide a benchmark with which to assess those planets
detected around white dwarfs by Gaia. The aim is to make predictions for the
population of planets that Gaia will detect around white dwarfs by evolving
the population of planets seen around main-sequence stars. Comparison of these
predictions with the population of planets that Gaia detects will probe the
importance of additional processes, such as dynamical scattering, survival of
common envelope evolution or second generation planet formation. If planets
are to arrive in the habitable zone around white dwarfs at around 0.01 au, a
potential avenue for the origin of life, such processes are key (Agol, 2011;
Loeb & Maoz, 2013; Kaltenegger et al., 2020). This paper starts by predicting
the mass and semi-major axes of planets that Gaia can detect (§2.1, §3.1).
Predictions are then made for the fate of currently detected planets around
main-sequence stars (§2.2), which are used to obtain a prediction for the
distribution of planets in the Gaia detection region that have evolved from
the main-sequence via tides and stellar mass loss (§3.2). By combining the
Gaia detection probabilities and predicted white dwarf planet distribution
with the EDR3 white dwarf catalogue (Gentile Fusillo et al., 2021) as
described in §2.3 and §3.3, predictions are made for the number of planet
detections around white dwarfs with Gaia. The implications of these
predictions and how they can be used in the future in combination with Gaia
planet detections is outlined in §4, alongside a discussion of their validity.
Finally, we conclude in §5.

##  2 Methods

The aim is to predict the population of planets around white dwarfs, as seen
by Gaia, that evolve from the population of planets seen around main-sequence
stars. We consider the fundamental dynamical effects on the orbit of stellar
mass loss and tides. Further complications are discussed in §4 and comparison
between the predictions in this paper and Gaia detections will determine the
importance of these additional processes, including photoevaporation and
dynamical scattering.

In order to predict the number of planets Gaia will find around white dwarfs
that evolved from the main-sequence by tides and mass loss, we consider three
main steps:

  1. 1.

Calculation of Gaia detection probability as a function of planet mass and
semi-major axis.

  2. 2.

Prediction of a post-main sequence planet distribution as a function of mass
and semi-major axis.

  3. 3.

Convolution of the detection probability and post-main sequence planet
distribution as functions of mass and semi-major axis and summation over this
convolution to determine how many planets Gaia should find around white
dwarfs.

The method for each of these steps is described in this section.

###  2.1 Gaia detection probabilities as a function of mass and semi-major
axis

Figure 1: The astrometric signal for a planet with given properties (mass and
semi-major axis) compared to the astrometric noise for Gaia measurements of a
15th mag white dwarf at 20 pc (left) or 40 pc (right). The semi-major axis
range spans positions from the innermost initial position in our simulations
to the maximum semi-major axis detectable by Gaia due to its ten year mission
length (see §2.1). Planets with S/N>3 have an average detection probability of
0.93, whilst planets with S/N<1 have average detection probability of 0.14.
Figure 2: The astrometric signal of a Jupiter or Neptune mass planet on a
circular orbit at 1-4 au, compared to the astrometric noise in a Gaia
measurement for a star at a given distance. The maximum semi-major axis
corresponds to the maximum detectable by Gaia on a ten year mission observing
a $0.6M_{\odot}$, $G_{\textrm{mag}}=15$ white dwarf. Dot-dash lines are for a
Jupiter mass planet, dash lines are for a Neptune mass planet. A solid
horizontal line is for a $S/N=1$ below which the detection probability drops
to zero. Jupiter mass planets are only detectable out to 30 pc. Neptune mass
planets are not detectable, because their signal to noise drops below one at 3
pc and there are no white dwarfs within 5 pc (Hollands et al., 2018; Gentile
Fusillo et al., 2021).

In this paper, a signal to noise (S/N) criterion, relating astrometric signal
of a planet to the single along scan accuracy per field of view of a Gaia
measurement, is used to link planet mass and semi-major axis to detection
probability. This is the simplest approach to reliably understand which
planets have a strong enough signal to be detected by Gaia and is commonly
used as indicator when comparing different planet detection algorithms (e.g.
in Casertano et al., 2008; Ranalli et al., 2018; Perryman et al., 2014). Other
subtler signatures, such as RUWE, can also currently be used as an indicator
of binarity (Belokurov et al., 2020; Penoyre et al., 2022).

The astrometric signature, $\alpha$, is approximated by the ratio of planet
and stellar masses ($M_{\textrm{pl}}$ and $M_{\star}$) in solar masses, their
separation, $a$, and the distance to the star, $d$, as shown in Equation 1.

$\alpha=\left(\frac{M_{\textrm{pl}}}{M_{\star}}\right)\left(\frac{a}{1\,\textrm{au}}\right)\left(\frac{d}{1\,\textrm{pc}}\right)^{-1}as$
(1)

More massive planets or those further from the star have larger signatures,
because they displace the barycentre position further from the centre of the
star. The signature is inversely proportional to the distance to the system,
because Gaia measures angular positions of stars and the angle subtended by
the distance between the star and the system’s barycentre will decrease with
increasing distance to the system. We consider it sufficient to use an average
white dwarf mass $M_{\star}=0.6\,M_{\odot}$ (Hollands et al., 2018) in this
work, when calculating $\alpha$ , because the difference in white dwarf masses
is small and makes little difference to $\alpha$ in comparison to the large
change in astrometric noise with white dwarf magnitude. Additionally, there
are several sources of error on the mass values in the catalogue, as discussed
in §4.1.4.

Signal can be maximised by looking for planets at large semi-major axes, a.
However, the ability of Gaia to accurately recover a companion’s orbital
parameters is limited by mission length. For planets with orbital periods
longer than the expected mission length following extensions (ten years), the
accuracy of the planet orbital parameters which are recovered decreases
(Ranalli et al., 2018). Therefore in this work, a period of ten years has been
taken as the cut off for reliable planet detections. This corresponds to
$a=4.64$ au for a planet around a $1M_{\odot}$ star or $a=3.91$ au for a
planet around a $0.6M_{\odot}$ white dwarf.

The noise on Gaia’s astrometric measurement is the single along-scan accuracy
per field of view, $\sigma_{\textrm{fov}}$ (Perryman et al., 2014).
Importantly, $\sigma_{\textrm{fov}}$ is a function of a star’s magnitude. In
this work the S/N ratio will be defined as

$S/N=\frac{\alpha}{\sigma_{\textrm{fov}}}.$ (2)

Gaia errors for future data releases are publicly available as sky-averaged
parallax accuracy $\sigma_{\varpi}$ not as $\sigma_{\textrm{fov}}$.
$\sigma_{\varpi}$ can be converted to $\sigma_{\textrm{fov}}$ by considering
the sky-averaged number of field crossings per star ($150\pm 50$ for a ten
year mission (Ranalli et al., 2018)), the geometrical factor linking the sky-
averaged parallax accuracy with the error per field crossing (2.15) and a
science contingency margin (1.1 for EDR3 onwards), which covers any additional
errors in the modelling process (Perryman et al., 2014).

$\sigma_{\textrm{fov}}=\frac{\sqrt{150}\sigma_{\varpi}}{1.1\times 2.15}\approx
5.18\sigma_{\varpi}$ (3)

This paper uses the predicted errors for Data Release 5, because this data
release will have the longest timespan for observations so can detect planets
with periods up to ten years. These error estimates are based on Gaia EDR3 and
are the most up-to-date at the time of press. The errors are described as:
111Taken from Gaia Mission Science Performance for Data Release 5
https://www.cosmos.esa.int/web/gaia/science-performance.

$\sigma_{\varpi}[\mu\,as]=0.527(40+80z+30z^{2})^{\frac{1}{2}},$ (4)

where $z$ is given by

$z=MAX[10^{0.4(13-15)},10^{0.4(G-15)}]$ (5)

where G is the broad-band, white-light, Gaia magnitude. This S/N criterion
describes the accuracy of a measurement. Visibility periods are used as a
measure of reliability (Lindegren, 2018): the more visibility periods a source
has, the less the astrometric solution will be affected by a bad measurement.
This is used to filter sources before fitting the astrometric solution (e.g.
visibility_periods_used >11 Gaia Collaboration et al., 2022b). Once DR5 is
available, visibility periods for each source can be used to assess the
reliability of the astrometry before looking for companions.

S/N can be linked to detection probability. Ranalli et al. (2018) used Markov
chain Monte Carlo methods and three information criteria to determine
detection probabilities over a grid of S/N and orbital period. Planets with
S/N>3 have an average detection probability (across periods less than 10
years) of 0.93, whilst planets with S/N<0.7 cannot be detected. In this work,
$p_{ljk}(S/N_{ljk},P_{j})$ denotes the detection probability of a planet with
semi-major axis in bin $j$ with period $P_{j}$, mass in bin $k$ around white
dwarf $l$ and were calculated from the Ranalli et al. (2018) detection
probabilities.

###  2.2 Prediction of the distribution of planets which survive tidal
evolution and stellar mass loss to reach the white dwarf phase

The distribution of close-in planets around main-sequence stars is well known
from radial velocity and transit observations. Some of these observations,
alongside direct imaging surveys, now reach orbital periods of years, as
probed by Gaia, but the planets detected by Gaia around main-sequence stars
will always provide the most reliable probe of the planetary population of
interest on the main-sequence.

Radial velocity (Bryan et al., 2016), transit (Zhu & Dong, 2021; Santerne et
al., 2016) and direct imaging (Biller et al., 2013) measurements suggest the
frequency of giant planets decreases at distances greater than 2-3 au from the
star, such that a broken power law is the most appropriate model for the giant
planet occurrence rate around main-sequence stars (Fernandes et al., 2019;
Fulton et al., 2021). In this work, we use the EPOS, three parameter fit
(symmetric) broken power law of Fernandes et al. (2019)222The errors in
Fernandes et al. (2019) are asymmetric, but this work uses the average of the
upper and lower bounds as an estimate of the size of the error..

$\frac{\textrm{d}^{2}N}{\textrm{d\,log}P\>\textrm{d\,log}M}=c_{0}\left(\frac{P}{P_{\textrm{break}}}\right)^{p_{1}}\left(\frac{M}{10M_{\oplus}}\right)^{m_{1}}$
(6)

where $c_{0}=0.84\pm 0.17$, $m_{1}=-0.45\pm 0.05$, $p_{\textrm{break}}=1581\pm
643$ days and

$p_{1}=\begin{cases}0.65\pm 0.17&P<P_{\textrm{break}}\\\ -0.65\pm
0.17&P>P_{\textrm{break}}\end{cases}.$ (7)

This broken power law describes the number of planets per log semi-major axis
and log mass bin, $\frac{\textrm{d}^{2}N}{\textrm{d\,log}P\>\textrm{d\,log}M}$
and models the distribution of planets around main-sequence stars accounting
for observation biases. This planet occurrence rate predicts $4.9\pm 0.7\%$ of
stars have planets with masses $1-13\,M_{J}$ with semi-major axes < 20 au.
Non-detections are considered by weighting the number of detected planets in a
mass and semi-major axis bin by the inverse of the survey completeness. The
power law of Fernandes et al. (2019) was used because it reproduced Cumming et
al. (2008) earlier results, gave results comparable to direct imaging
observations at large semi-major axes and drew on multiple groups of
observations (the latest Kepler data release and radial velocity results from
Mayor et al. (2011)).

To predict the change in planetary orbits from the main-sequence to the white
dwarf phase, the orbital evolution of planets along the AGB was simulated,
following the prescription presented in Mustill & Villaver (2012). This model
breaks the effects of stellar evolution on planets into three key components:
expansion of the planetary system due to mass loss, inward movement of planets
due to tidal effects and engulfment of planets which are too close to the star
as the star expands. Tidal forces were modelled as viscous dissipation of the
equilibrium tide (Zahn, 1977). The model calculates the semi-major axis of the
orbits of isolated planets and the stellar envelope as a star evolves along
the AGB, following the models of Vassiliadis & Wood (1993). Any planet which
moves inside the stellar envelope at a given timestep is removed. The
limitations of this model are discussed in §4.1.1.

Extending the masses and initial semi-major axes of Mustill & Villaver (2012),
we simulated 13,051 planets around a $1M_{\odot}$ progenitor and 12,121
planets around a $1.5M_{\odot}$ progenitor333Fewer simulations were needed for
the higher mass progenitor because the larger expansion of the planetary
system moved most of the sample outside the Gaia detection range.. For
simplicity, all planets were assumed to be in single planet systems on
circular orbits (see §4.1.5). The planet masses sampled 31 possible values
evenly distributed in log space between Earth and 13 Jupiter masses. Initial
semi-major axes ranged from 1-10 au. All planets at semi-major axes smaller
than this would be engulfed by their host star and those beyond this would be
far beyond the Gaia detection region so were not considered. Two groups of
simulations were carried out. Simulation A was a coarse spacing of 0.03 au to
enable sampling of a large range of semi-major axis space. Simulation B and C
were a fine spacing of 0.001 au spanning $\pm 0.12$ au ($1\,M_{\odot}$) and
$\pm 0.09$ au ($1.5\,M_{\odot}$) from the initial position of the outermost
engulfed planet (determined from Simulation A) at each mass. This fine
sampling was to investigate behaviour near the initial semi-major axis of the
outermost engulfed planet, because in this region there is large variation in
the strength of the tidal forces (see Mustill & Villaver, 2012). The
parameters for these simulations are summarised in Table 1. The final masses
of the white dwarfs (needed for estimating the importance of tidal effects)
were $0.5702M_{\odot}$ and $0.6014M_{\odot}$ for the $1M_{\odot}$ and
$1.5M_{\odot}$ progenitors respectively.

Simulation | Progenitor mass/$M_{\odot}$ | Spacing/au | Planet mass/$M_{J}$ | Semi-major axis range/au
---|---|---|---|---
A | 1.0, 1.5 | 0.03 | $3.1\times 10^{-4}-13$ | 1-10
B | 1.0 | 0.001 | $3.1\times 10^{-4}-13$ | inner position of outermost engulfed planet $\pm 0.12$
C | 1.5 | 0.001 | $3.1\times 10^{-4}-13$ | inner position of outermost engulfed planet $\pm 0.09$
Table 1: Semi-major axis ranges for simulated planets for each planet mass. 31
planet masses were chosen equally spaced in log space between $M_{\oplus}$ and
$13\,M_{J}$. For each planet mass, the above semi-major axes arrays were used
for the initial conditions of the simulations. All planets were assumed to be
in single planet systems on circular orbits (see §4.1.5). For discussion of
choices of semi-major axis spacing see §2.2.

To combine the main-sequence planet occurrence rate with the results of our
simulations, mass and semi-major axis space were split into bins. There were
31 bins equally spaced in log$M$ from $1M_{\oplus}$ to $13\,M_{J}$ and the
semi-major axis bins have variable width corresponding to equispaced period
bins from (Ranalli et al., 2018) (average width $0.21$ au). This corresponded
to the range of initial positions in our simulation. We assumed the planets
were around a 0.6$M_{\odot}$ white dwarf to transform the main-sequence
distribution from $\frac{\textrm{d}^{2}N}{\textrm{d\,log}P\>\textrm{d\,log}M}$
to $\frac{\textrm{d}^{2}N}{\textrm{d\,log}a\>\textrm{d\,log}M}$ where $N$ is
the number of planets per star, $a$ is the semi-major axis, $P$ is the period
and $M$ is the planet mass. All logarithms are to base 10 so
$\textrm{d\,log}a=\frac{\textrm{d}a}{a}\textrm{log}_{10}(e)$ was used to
transform from $\frac{\textrm{d}^{2}N}{\textrm{d\,log}a\>\textrm{d\,log}M}$ to
$\frac{\textrm{d}^{2}N}{\textrm{d}a\>\textrm{d\,log}M}$ where d$a$ was the
corresponding bin width and $a$ was the midpoint of the bin. In the following
equations, the index i represents the initial semi-major axis bin of a planet,
j represents the final semi-major axis bin and k represents the mass bin.
Planetary mass loss was not modelled so a planet’s mass bin does not change
during the simulation. All summations are written explicitly.

The predicted occurrence rate of planets around white dwarfs in a given semi-
major axis, mass range,
$\frac{\textrm{d}^{2}N_{\textrm{WD}}}{\textrm{d}a\>\textrm{d\,log}M}$ is
calculated by

$\left(\frac{\textrm{d}^{2}N_{\textrm{WD}}}{\textrm{d}a\>\textrm{d\,log}M}\right)_{jk}=\Sigma_{i}R_{ijk}C_{ik}.$
(8)

$R_{ijk}$ is the number of planets of mass labelled by $k$ in the simulation
that started in the $i^{th}$ semi-major axis bin, but end the simulations in
the $j^{\textrm{th}}$ semi-major axis bin post-main sequence. $C_{ik}$ is the
weight of each simulated planet, which is the value of the main-sequence
planet distribution from Fernandes et al. (2019) for the centre of the bin,
divided by the number of simulated planets initially in the $i^{th}$ bin,
$B_{i}$.

$C_{ik}=\left(\frac{\textrm{d}^{2}N_{\textrm{MS}}}{\textrm{d}a\>\textrm{d\,log}M}\right)_{ik}\frac{1}{B_{i}}$
(9)

###  2.3 Predicted numbers of planet detections

####  2.3.1 Using the EDR3 catalogue

To predict the number of planets Gaia will detect around white dwarfs requires
the calculation of the parameter space available for planet detection around
known white dwarfs. The white dwarf sample, containing mass estimates, Gaia
broad-band magnitudes ($G_{\textrm{mag}}$) and parallaxes was obtained from
the Gentile Fusillo et al. (2021) catalogue from Gaia early Data Release 3,
which improved on their previous catalogue from Data Release 2 (Gentile
Fusillo et al., 2019). In Gentile Fusillo et al. (2021), each object in the
catalogue was assigned a probability of being a white dwarf $P_{\textrm{WD}}$
based on a series of absolute magnitude and colour critera. In Gentile Fusillo
et al. (2021), applying the criteria $P_{\textrm{WD}}>0.75$ recovered 359,000
high confidence white dwarf candidates with only 1$\%$ of contaminant objects.
Therefore in this work, we applied three filters to the catalogue to obtain a
catalogue of white dwarfs:

  * •

$P_{\textrm{WD}}>0.75$

  * •

$G_{\textrm{mag}}<20.7$ \- remove objects too faint to be detected by Gaia.

  * •

$M<0.663\,M_{\odot}$ \- remove white dwarfs with progenitor masses
$>1.5\,M_{\odot}$.

After applying these cuts, the sample is reduced to 282,718 white dwarfs out
to distances of 16,000 pc. Planet detections around the furthest away white
dwarfs in this sample will not be possible due to the high noise and small
astrometric signature. For a $G_{\rm mag}=20.7$ white dwarf, there will be a
50% relative uncertainty on position at approximately 800 pc. Although an a
priori distance cut was not applied, they will not have contributed to the
detection probability value as their S/N was so small.

White dwarfs with high progenitor masses were excluded from the catalogue,
because they lose a greater proportion of mass as they evolve off the main-
sequence (Cummings et al., 2018) and the surviving planet positions are beyond
the maximum semi-major axis detectable by Gaia for a ten year mission. This
will be discussed further in §3.2 (Figure 5(b)), where the chosen cut-off is
justified based on simulations that indicate that no planets directly evolved
from the main-sequence will be detected around stars within initial masses
higher than $M_{i}>1.5M_{\odot}$. The final mass cut off was determined using
the Cummings et al. (2018) MIST-based initial-final mass relation based on
stellar clusters. This gave $M_{f}=0.609\pm 0.054M_{\odot}$ for a
$1.5M_{\odot}$ progenitor, so an upper limit of $0.663\,M_{\odot}$ was
adopted. White dwarfs in the Gentile Fusillo et al. (2021) catalogue with
unknown masses were assumed to have a mass equal to $0.6\,M_{\odot}$ (the
average white dwarf mass Hollands et al., 2018).

####  2.3.2 Combining the predicted planet distribution with Gaia detection
probabilities

For each white dwarf in the filtered catalogue, labelled by the index $l$, the
probability of detecting a planet as a function of planet mass, $j$ and planet
semi-major axis, $k$, $p_{ljk}$ was calculated as described in §2.1. The semi-
major axis bins were determined from the period bins in Ranalli et al. (2018).
The mass bins were unchanged.
$\left(\frac{\textrm{d}^{2}N_{\textrm{WD}}}{\textrm{d}a\>\textrm{d\,log}M}\right)_{jk}$
calculated in §2.2 was recalculated to match these bins for the following
calculations.

The distribution of detected planets
$\frac{\textrm{d}^{2}N_{\textrm{det}}}{\textrm{d}a\>\textrm{d\,log}M}$
corresponds to

$\left(\frac{\textrm{d}^{2}N_{\textrm{det}}}{\textrm{d}a\>\textrm{d\,log}M}\right)_{jk}=\Sigma_{l}p_{ljk}\left(\frac{\textrm{d}^{2}N_{\textrm{WD}}}{\textrm{d}a\>\textrm{d\,log}M}\right)_{jk}.$
(10)

The number of detected planets is

$N_{\textrm{pl}}=\Sigma_{jk}\left(\frac{\textrm{d}^{2}N_{\textrm{det}}}{\textrm{d}a\>\textrm{d\,log}M}\right)_{jk}(\textrm{d}a)_{j}(\textrm{d\,log}M)_{k}.$
(11)

####  2.3.3 Error Analysis

The error on the number of planets was calculated by appropriately propagating
the errors on each of the quantities in equations 8-11. The error on
$\left(\frac{\textrm{d}^{2}N_{\textrm{MS}}}{\textrm{d}a\>\textrm{d\,log}M}\right)_{ik}$
came from the average of the upper and lower bounds on these values from
Fernandes et al. (2019). The error on $R_{ijk}$ (the number of planets of mass
labelled by $k$ in the simulation that moved from the $i^{\textrm{th}}$ to the
$j^{\textrm{th}}$ semi-major axis bin in the simulations) was assumed to be
Poissonian $\sigma(R_{ijk})=\sqrt{R_{ijk}}$. The dominant error in the
detection probabilities is from the error in the sky-averaged number of field
crossings per star ($150\pm 50$) (Ranalli et al., 2018). The resulting error
in the detection probabilities was combined fractionally in quadrature with
the error on
$\left(\frac{\textrm{d}^{2}N_{\textrm{WD}}}{\textrm{d}a\>\textrm{d\,log}M}\right)_{ik}$
to obtain the error on
$\left(\frac{\textrm{d}^{2}N_{\textrm{det}}}{\textrm{d}a\>\textrm{d\,log}M}\right)_{ik}$.
Another possible source of uncertainty is the exclusion white dwarfs with
$M>0.663\,M_{\odot}$. This is difficult to quantify so was not included in our
error calculations but is discussed in §4.1.4.

##  3 Results

###  3.1 What is the probability of Gaia detecting a planet as a function of
mass and semi-major axis?

Only planets with masses greater than or equal to Jupiter will be detectable
by Gaia around white dwarfs. The astrometric signal of a planet on a given
orbit with a given mass is calculated according to Equation 1 and the noise is
calculated by 3. Figure 2 indicates that Neptune mass planets have $S/N>1$ up
to 3 pc and beyond this only Jupiter mass planets have $S/N>1$. Since the
closest white dwarf, 40 Eri B, is at 5.04 pc (Hollands et al., 2018; Gentile
Fusillo et al., 2021), this suggests no Neptune mass planets will be
discovered around white dwarfs by Gaia. As shown in Figure 1, even Jupiter
mass planets have a low S/N and correspondingly a low detection probability at
small semi-major axes These calculations were done for a bright white dwarf
with $G_{\textrm{mag}}=15$. For fainter white dwarfs, the astronometric noise
would be larger, decreasing the S/N of a given planet mass and semi-major
axis.

Figure 3 shows an example map illustrating detection probability as a function
of mass and semi-major axis generated for a $G_{\textrm{mag}}=15$ white dwarf.
It clearly demonstrates that almost all planets less massive than Jupiter will
not be detected by Gaia. There is a significant region of parameter space
where planets with orbital periods longer than the mission length could be
detected, shown by the hashed region in Figure 3, however the recovery of
orbital parameters in this region is unreliable. For clarity, planet
detections in this region are not included in the predicted number of planets
Gaia will detect (§3.3).

Figure 3:  Detection probability, $p_{ljk}$ as a function of mass and semi-
major axis for an example $G_{\textrm{mag}}=15$ white dwarf at 30 pc. Here
detection probability refers to the likelihood of detecting a given planet if
it exists around the white dwarf and does not account for
$\frac{\textrm{d}^{2}N_{\textrm{WD}}}{\textrm{d}a\>\textrm{d\,log}M}$.The
hatched region indicates semi-major axes which have non-zero detection
probabilities, but periods longer than the mission length. The dark square
around 4.5 au and 3$\,M_{J}$ results from the stochastic method to determine
the detection probabilities in (Ranalli et al., 2018). Almost all planets
above $3\,M_{J}$ at a few au will be detected, whilst almost no planets less
massive than Jupiter will be detected.

###  3.2 What is the planet distribution around white dwarfs?

Predictions for the population of planets surviving to the white dwarf phase
are shown in Figure 5(a) for a $1\,M_{\odot}$ progenitor. These should be
compared to the initial population of planets on the main-sequence Fernandes
et al. (2019) shown in Figure 4 as
$\frac{\textrm{d}^{2}N_{\textrm{MS}}}{\textrm{d}a\>\textrm{d\,log}M}$. Over
the mass and semi-major axis range shown here the main-sequence distribution
gives a planet occurrence rate of $0.61\pm 0.01$ planets per star. This is to
be contrasted with the population surviving to the white dwarf phase, where
only $14\%$ of planets initially within 3.91 au (the detection limit due to
the length of Gaia mission) of $1\,M_{\odot}$ progenitor remain. The number of
planets per star in the Gaia detection region is lower than the main-sequence,
because planets are engulfed or move out of this region as their orbits expand
due to stellar mass loss post-main sequence. This is evident from the white
regions in Figures 5(a) where
$\frac{\textrm{d}^{2}N_{\textrm{WD}}}{\textrm{d}a\>\textrm{d\,log}M}$ is zero.
Even fewer planets ($0.1\%$) survive in the detection region around a
$1.5\,M_{\odot}$ progenitor, because larger mass progenitors have stronger
tidal forces and lose a greater proportion of their mass. The majority of the
region inside 3.91 au in Figure 5(b) is white. This means it is very unlikely
that planets will be detected around white dwarfs with $M>1.5\,M_{\odot}$
progenitors. This is the basis for the mass cut off applied to the EDR3
catalogue (§2.3), excluding white dwarfs with higher progenitor masses.

Tidal evolution of gas giants leads many planets to migrate inwards: some
migrate so far they are engulfed by the stellar envelope. This means that only
planets with initial positions sufficiently far out survive to the white dwarf
phase. This distance increases with planet mass, as higher mass planets feel a
stronger tidal pull from the star. This is demonstrated by the diagonal trend
of the boundary between zero and non-zero values of
$\frac{\textrm{d}^{2}N_{\textrm{WD}}}{\textrm{d}a\>\textrm{d\,log}M}$ in
Figures 5(a) and 5(b). Planets beyond a certain distance are too far away to
be affected by tides (see Appendix A). Planets below $10^{-2}\,M_{J}$ are not
massive enough to be affected by tidal forces and their orbits undergo a
purely adiabatic expansion, as shown by the vertical portion of the boundary
in Figure 5(a).

Figure 4: Number of planets per semi-major axis and log M bin
($\frac{\textrm{d}^{2}N_{\textrm{MS}}}{\textrm{d}a\>\textrm{d\,log}M}$) for
main-sequence stars from Fernandes et al. (2019). Lower mass planets are more
common.

(a) 1 $M_{\odot}$ progenitor. A small proportion of planets survive inside the
detection region to the white dwarf phase (see §3.2).

(b) $1.5\,M_{\odot}$ progenitor. Stronger tidal effects and greater mass loss
means fewer planets survive inside the detection region compared to the
$1M_{\odot}$ progenitor.

Figure 5: Predicted number of planets per semi-major axis and log M bin
($\frac{\textrm{d}^{2}N_{\textrm{WD}}}{\textrm{d}a\>\textrm{d\,log}M}$) around
a white dwarf with a 1$M_{\odot}$ and 1.5$M_{\odot}$ progenitor. This is based
on the evolution of an initial distribution around main-sequence stars
((Fernandes et al., 2019) see Fig 4) taking into account tidal evolution, mass
loss and engulfement by the AGB star (see §2.2 for further details). The semi-
major axis bins have variable width corresponding to equispaced period bins
from (Ranalli et al., 2018) and have average width 0.21 au. The mass bins are
$0.12$ wide in $\textrm{log}(M/M_{J})$. The solid black line denotes the semi-
major axis with a period equal to the length of the Gaia extended mission for
a planet orbiting a $0.6M_{\odot}$ white dwarf. The orbital parameters of
planets at semi-major axes greater than this cannot be recovered accurately.
The dashed and dot-dashed lines represent the detection cut off for a signal-
to-noise of three for a $G_{\textrm{mag}}=15$ white dwarf at 20 pc and 40 pc
respectively. Trends discussed in §3.2 have been labelled on Figure 5(a).

###  3.3 Predictions for the number of planets Gaia will detect around white
dwarfs

The best white dwarf candidates for finding planets are bright (smaller noise)
and close (larger signal). $50\pm 10\%$ of planet detections occur around
white dwarfs brighter than $G_{\textrm{mag}}=15$, even though they comprise
just 441 out of the 282,718 stars in the catalogue. There are far fewer white
dwarfs where detection of a close in Jupiter mass planet is likely compared to
a $13\,M_{J}$ planet at large semi-major axes (52 compared to 5519) as
demonstrated in Figure 6. The two example planets in this figure correspond to
some of the smallest ($1\,M_{J}$ at 2 au) and largest ($13\,M_{J}$ at 3.91 au)
signals for which there is $S/N>3$ (almost $100\%$ detection probability). For
$1\,M_{J}$ planets there are no $S/N>3$ candidates with magnitudes greater
than 15.4 and beyond $30\,pc$. These high detection probability candidates are
discussed further in Appendix C.

Figure 6: The properties of the white dwarfs ($G_{\textrm{mag}}$ and distance)
for whom detection of a $1\,M_{J}$ at 2 au (weak signal) and a $13\,M_{J}$ at
3.9 au is guaranteed if that planet exists around those white dwarfs
(detection probability = 1). Candidates with $S/N>10$ are denoted by the
purple +. There are an far fewer candidates, 3 (see Appendix C), for detecting
lower mass ($\sim M_{J}$) close in (<2 au) compared to 3770 for a $13\,M_{J}$
planet at 3.91 au. Also note the difference in semi-major axis and
$G_{\textrm{mag}}$ scales between the two plots.

We predict Gaia will detect $8\pm 2$ planets around white dwarfs that evolve
due to mass loss and tides post-main sequence. Of these planets $9\pm 2\,\%$
will lie inside 3 au (i.e. at most one), $90^{+10}_{-20}\,\%$ will lie between
3-3.91 au. $40\pm 10\,\%$ will be less massive than Jupiter. The predicted
distribution of planets Gaia can detect is shown in Figure 7. We predict Gaia
will not be able detect planets less massive than $0.13\,M_{J}$ or with semi-
major axes less than 1.6 au. The faded region corresponds to planets with non-
zero detection probabilities, but which have periods longer than the Gaia
extended mission length. The orbital parameters of these planets cannot be
recovered reliably (Ranalli et al., 2018) so they are not included in the
final predicted number of planet detections (see §4.1.3) and the maximum semi-
major axis where Gaia can detect planets is 3.91 au (for a $0.6\,M_{\odot}$
white dwarf). The detected planets are concentrated at the largest masses and
semi-major axes, because they have the strongest signals. Comparison with
Figure 5(a) highlights that Gaia will not detect any of the low mass planets
which should exist around white dwarfs, because they are below the Gaia
detection threshold.

Figure 7: Predicted number of detected planets per semi-major axis and log M
bin ($\frac{\textrm{d}^{2}N_{\textrm{det}}}{\textrm{d}a\>\textrm{d\,log}M}$).
Summing over the distribution gives the total number of planets we predict
Gaia will detect. The number of planets that we predict will be detected in
each region is written in black. The vertical dashed boundary is at $a=3$ au
and the horizontal dashed boundary is at $M=M_{J}$. The faded region
corresponds to planets with periods longer than the extended mission length,
which have non-zero detection probabilities, but for which orbital parameters
cannot be accurately recovered. The number of planet detections expected if
periods up to 40% longer than the mission length can be detected are shown in
the bounded faded region (see §4.1.3).

For periods longer than the extended mission length, detection probability
also decreases so there is a fall off in
$\frac{\textrm{d}^{2}N_{\textrm{det}}}{\textrm{d}a\>\textrm{d\,log}M}$. No
planets less massive than $0.03\,M_{J}$ or interior to 1.6 au will be
detected. The gaps and isolated non-zero bins at small semi-major axes result
from difference in tidal effects for planets at different semi-major axes
motion of planets during post-main sequence evolution (see gaps in Figure
5(a)).

##  4 Discussion

This work makes two key predictions: the number and distribution of planets
Gaia will find around white dwarfs that have evolved from the main-sequence by
tides and mass loss. We predict Gaia will find $8\pm 2$ planets between
1.6-3.91 au with masses between $0.03-13\,M_{J}$. Whilst the broad form of the
distribution (mass and semi-major axis of discovered planets) is robust, the
exact numbers of predicted planet detections depend crucially on the chosen
main-sequence planet occurrence rate, the adopted post-main sequence planetary
system evolution model and the ten year period cut-off for detected planets.

Nonetheless, we consider the numbers to be sufficiently reliable to be used as
a benchmark comparison with Gaia exoplanet detections in order to determine
how important photoevaporation, common envelope evolution and dynamical
scattering inwards are for planets around white dwarfs.

###  4.1 Validity of results

####  4.1.1 Post-main sequence planetary evolution model

This work focuses only on the effects of stellar mass loss and tidal evolution
on the fate of planets post-main sequence Mustill & Villaver (2012).
Crucially, two processes are neglected which could potentially lead to planets
on orbits interior to 1.6 au, the innermost orbit predicted by tidal evolution
in this work. Firstly, planets may survive inside and subsequently escape from
the stellar envelope (common envelope evolution Paczynski, 1976). While
traditional common-envelope models predict a minimum mass of
$10\mathrm{\,M_{J}}$ for the smallest planets that can survive common-envelope
evolution (e.g., Nordhaus et al., 2010), the orbit of the Jupiter-mass planet
WD 1856b can be explained by models that incorporate re-ionisation energy or
successive engulfment events (Lagos et al., 2021; Chamandy et al., 2021).
Common envelope evolution could drastically shrink the orbit, making the
planet invisible to astrometry (but more detectable by the transit method).
Secondly, multi-planet interactions or Kozai-Lidov interactions with stellar
companions can lead to planets scattered inwards post-main sequence stellar
evolution, as suggested by Ronco et al. (2020); O’Connor et al. (2021);
Stephan et al. (2021), amongst others. In the absence of tidal forces,
scattering can decrease the semi-major axis by up to a factor of 2 (Mustill et
al., 2014, Fig. 6). The detection of planets interior to 1.6 au (for which we
already have the candidates around WD 1856 at 0.02 au (Vanderburg et al.,
2020) and WD 0141 at 0.18 au (Gaia Collaboration et al., 2022b)) provides an
important test of the importance of common envelope evolution and/or dynamical
scattering.

The model also neglects photoevaporation, high stellar luminosity heating the
planet’s gaseous envelope such that it escapes, which could reduce a planet’s
size below the Gaia observable limit (Villaver & Livio, 2007). Villaver &
Livio (2007) predict a strong dependence of photoevaporation on the planet’s
mass and orbital properties. $1M_{J}$ planets initially within 2 au of their
$1\,M_{\odot}$ progenitor host star lose 50$\%$ of their envelopes, whilst
$5M_{J}$ planets initially at 3 au only lost 0.4-5$\%$ of their envelope.
Whilst the predicted Gaia planet detections will suffer from small number
statistics, if there is a lack of detections of $M_{P}\leq M_{J}$ planets it
could be an indication that photoevaporation is important in these systems.

This work models tidal forces as the dissipation of an equilibrium tide by
means of convective turbulence in the giant star’s envelope (Mustill &
Villaver, 2012; Zahn, 1977). The strength of this tide and scalings in the
model (especially the dependence on tidal strength with the forcing frequency,
i.e., the mean motion) are debated (see, e.g., Goldreich & Nicholson, 1977;
Zahn, 1977; Penev et al., 2009; Ogilvie & Lesur, 2012; Ogilvie, 2014). Mustill
& Villaver (2012) found that the maximum orbital radius for engulfment can
change by up to 1 au for reasonable changes in these parameters. In principle,
if we better understood the progenitor population of giant planets,
decetection of planets around white dwarfs could be used to better constrain
the tidal models. Unfortunately however, the number of planets Gaia will
detect around white dwarfs will be too small for this.

Another potential limitation to the models is the number of simulated planets.
Increasing the number of simulated planets would reduce the fractional error
on $R_{ijk}$, since $\sigma(R_{ijk})$ is Poissonian. This would be most
significant for bins closest to the star, where the number of planets
surviving in each bin is smallest. However, the predicted number of planet
detections inside 3 au is so low that increased resolution in this region
would be an unnecessary computational expense. The number of planets per bin
in the simulations sufficiently resolves the structure in the white dwarf
planet distribution.

####  4.1.2 Planet occurrence rate

The main-sequence planet occurrence rate contributes the most to the
uncertainties in the predicted number of detected planets (see §2.3.3).
However, once Gaia exoplanet detections are available for main-sequence stars,
these will allow predictions for the number of planets Gaia detect to be
updated. This will further enable the use of the Gaia white dwarf exoplanets
to investigate the processes affecting planets’ orbital evolution discussed in
§4.1.1.

Our chosen main-sequence planet occurrence rate from Fernandes et al. (2019)
crucially accounts for non-detections when calculating the planet occurrence
rate by weighting the sum over the number of detected planets by the inverse
of survey completeness at the mass and semi-major axis of each planet (see
Equation 1 in Fernandes et al., 2019). However, estimates of completeness are
still limited by the lack of planet observations beyond 1 au where most Gaia
planet detections will lie. Short period planets are easily observed by the
transit method and have been surveyed extensively by Kepler and TESS (Fulton
et al., 2017; Petigura et al., 2017). Planets beyond 1 au have been discovered
by radial velocity measurements e.g. Mayor et al. (2011); Wittenmyer et al.
(2016), microlensing e.g. Gaudi (2012) and direct imaging e.g. Biller et al.
(2013), but these are are limited in number, relative to close-in planets.

####  4.1.3 Probing planets with periods longer than the Gaia mission length

In order to unambiguously detect and characterise the orbit of a planet,
multiple orbital periods must be observed. This necessarily limits Gaia to
planets with periods shorter than the ten year mission length (interior to
3.91 au). The faded region in Figure 7 predicts that a substantial population
($180^{+70}_{-80}$) of planets evolved from the main-sequence orbit white
dwarfs into the four to eight au region. In this region the significant
astrometric signals detected by Gaia, would correspond to fractional orbital
periods only. However, initial results from DR3 comparing spectroscopically
and astrometrically determined periods (Pourbaix et al., 2022) suggests
periods may be recoverable that are up to 20-40% longer than the mission
length. $50\pm 20$ more planets could be found if detections between the
mission length and this upper limit are possible (see the faded bounded region
in Figure 7). There are also other techniques which probe the population of
planets on longer periods, based on astrometric detection of fractional
orbital periods (see e.g. Penoyre et al., 2020).

These planets have not been included in the predicted detected population in
this work, but these planets would provide a population to compare in detail
with individual post-main sequence planetary evolution models similar to
comparisons of the Kepler radius valley with photoevaporation models e.g.
(Owen & Wu, 2017). These planets would be sufficient in number that it becomes
possible to investigate how planet occurrence rate changes between more
massive, early main-sequence stars (which have now reached the white dwarf
phase) and lower mass main-sequence stars.

####  4.1.4 Planets only survive around higher mass progenitors on orbital
periods longer than the Gaia mission length

Figure 5(b) shows that planets end up beyond Gaia detection limit in planetary
systems with high mass progenitors. This is because the star loses a higher
proportion of its mass and has stronger tidal forces. In this work, we attempt
to account for the initial mass of the white dwarf progenitors, by using an
initial-final mass relationship to determine which white dwarfs had high mass
progenitors. No detectable planets are found around white dwarfs with initial
masses higher than $1.5\,M_{\odot}$, which using Cummings et al. (2018)
corresponds to a final mass of $M=0.609\pm 0.054M_{\odot}$

In practice, this meant that white dwarfs with masses higher than
$0.663\,M_{\odot}$ were not included in the analysis presented in §2.3.
However, this mass cut off introduces uncertainty in the predicted number of
detected planets, because there is error associated with the initial final
mass relation and the mass values in the catalogue.

The mixed masses from the Gentile Fusillo et al. (2021) catalogue were used in
this work. These were derived by fitting white dwarf atmospheric models and
evolutionary sequences to Gaia photometry and photometric estimates of log g
and $\boldmath{T_{\rm eff}}$ to obtain best fit values of mass and radius.
Reddening for white dwarfs beyond 100 pc was corrected for in the mass
calculations. These estimates are unreliable for very high ($>1.3\,M_{\odot}$)
or very low masses ($<0.2\,M_{\odot}$), but this is not a problem for the mass
cut off as the very lowest masses are not underestimated enough to need
excluding from the sample and the highest masses are far above the cut off.
Using only Gaia photometry introduces error in these masses, because as
discussed in Bergeron et al. (2019) due to the high temperatures of white
dwarfs it is important to include ‘bluer’ photometry such as SDSS ‘u’ to
constrain the peak of the blackbody curve and the value of $\boldmath{T_{\rm
eff}}$. White dwarfs in the catalogue that did not have masses were assumed to
have masses of 0.6 $M_{\odot}$ \- the average mass of a white dwarf (Hollands
et al., 2018). Excluding these white dwarfs instead does not change the
predicted number of planet detections within error.

In order to predict which white dwarfs would have no planets on orbital
periods shorter than the Gaia mission length due to their high progenitor
masses, it is necessary to deduce their evolutionary pathway and initial mass
based on current observed properties, notably their mass (log g). For this
process, we utilise initial to final mass relations which are notoriously
unreliable. Uncertainties arise because two stellar models of different
initial masses and metallicities or run through different stellar evolution
codes can have the same final mass and it is difficult to obtain the
metallicity of the progenitor mass from observations. In this work we used the
Cummings et al. (2018) MIST-based initial-final mass relation based on stellar
clusters, but the difference to the resulting predicted planet population if
another relation (e.g. El-Badry et al., 2018; Barrientos & Chanamé, 2021) were
used is small.

The resolution of the post-main sequence stellar evolution models is also
limited to $0.5\,M_{\odot}$ differences in progenitor mass. However, the slope
of the initial final mass relation in the $1-1.5\,M_{\odot}$ progenitor range
is so shallow and error on the relation so large that this outweighs the
uncertainty due to model resolution.

The multiple sources of uncertainty on the mass cut off do not have a
significant effect on the results, because the large number of white dwarfs
mean the resulting fractional change in catalogue size is small. Repeating the
analysis in this work using one and three $\sigma$ bounds on the white dwarf
masses and the cut off value from the initial final mass relation to increase
and decrease the number of white dwarfs in the catalogue do not change our
results within the errors. Aside from the mathematics of the cut off, there is
also uncertainty due to its initial validity. If we discover planets around
white dwarfs with higher mass progenitors it suggests these planets have been
scattered inwards or these white dwarfs have had a different evolutionary
pathway. For example, if a white dwarf underwent a merger, the planetary
systems may have a complex history, that proceeds in quite a different manner
to the simple picture of tidal evolution presented in this work.

####  4.1.5 Other limitations

In addition to the effects discussed above the Gaia detection probabilities,
the predicted magnitude of Gaia DR5 errors and the width of the bins in mass-
semi-major-axis space will have minor effects on the predicted number of
planet detections. The Gaia errors used in this work are predictions for DR5
based on EDR3 and we acknowledge this may be updated for future data releases.
As the errors decrease with every data release this may lower the mass
detection threshold. Revised detection probabilities may also change the exact
number of detected planets.

The assumption of circular orbits may also have a small effect on the
predicted surviving planet distribution and the detected signal. Planets with
initially eccentric orbits are more likely to be engulfed during post-main
sequence evolution as their pericentres may pass inside the stellar envelope
and eccentricity speeds up tidal semi-major axis decay. This could increase
the final semi-major axis of the innermost surviving planet by around 0.25 au
for Jupiter mass planets (Mustill & Villaver, 2012). This may shift the inner
boundary of the coloured region in Figure 7, however so few planets are
predicted to be detected interior to 3 au that this will not make a
significant difference to the total expected number of planets. Eccentricity
also affects the detectability. Whilst planets that experienced tidal decay
post-main sequence may have been circularised, multi-planet interactions may
increase eccentricities in the white dwarf phase.

As already demonstrated in DR3, this astrometric detection method will also
reveal larger companions to white dwarf, such as brown dwarfs or white dwarfs
(double degenerates) (Gaia Collaboration et al., 2022b). This provides an
opportunity to probe other white dwarf binary systems that have previously
been difficult to detect. These are distinguishable from the planets discussed
in this paper since the masses of these companions are larger than planets by
an order of magnitude or more.

###  4.2 Significance of the results

This paper highlights the importance of Gaia for the detection of exoplanets
on medium orbital periods across the HR-diagram (Perryman et al., 2014). White
dwarf exoplanets are key to investigating the effects of stellar evolution on
planetary systems. Whilst planetary systems around white dwarfs have been
hypothesised for years to explain white dwarf pollution (Jura & Young, 2014),
there are few known planets (Vanderburg et al., 2020; Blackman et al., 2021;
Luhman et al., 2011). Many unsuccessful surveys have searched for close-in
planets orbiting white dwarfs (e.g. Faedi et al., 2011; Agol, 2011; Burleigh
et al., 2002), but no current techniques are sensitive to planets on wide (
year) orbits. This work highlights how Gaia will shed light on these important
planetary systems.

Predictions are made for the planet population around white dwarfs, based on
the evolution of the currently known main-sequence planet population due to
tides and stellar mass loss, resulting in $8\pm 2$ planets detectable by Gaia.
Notably, no planets are predicted interior to 1.6 au, where they should have
been engulfed by their host star, providing a test for the importance of
common envelope evolution or dynamical scattering in the post-main sequence
evolution of planetary systems. If we see fewer Jupiter mass planets than
expected it suggests they may be removed by photoevaporation. Existing
observations and theoretical work (e.g. Veras, 2016; Vanderburg et al., 2020;
Muñoz & Petrovich, 2020; Veras & Gänsicke, 2015; Villaver & Livio, 2007;
Alonso et al., 2021; Lagos et al., 2021) already suggest these processes could
be important. If more planets are seen than expected it could provide evidence
for second-generation planet formation. Predicting a distribution of planets
unaffected by these processes can help us confirm which detected planets
result from these processes and can be used to test models of these
mechanisms.

The results in this paper are broadly inline with previous work. The number of
planet detections ($8\pm 2$) predicted by this paper is slightly lower than
the 13 predicted by Perryman et al. (2014). Perryman et al. (2014) used the
final mass of the white dwarf in a main-sequence planet occurrence rate, which
may have overestimated the number of planets around white dwarfs in the Gaia
observable range, because it did not account for the expansion of planetary
orbits and the engulfment of planets during post-main sequence stellar
evolution. The mass detection threshold, $0.03\pm 0.004\,M_{J}$, is also lower
than previous work ($2\,M_{J}$ Silvotti et al. (2011)). We attribute this to
improved estimates of Gaia errors and an enlarged catalogue of known white
dwarfs. The lowest mass planets can only be found around the closest white
dwarfs: including white dwarfs within 13 pc lowers the mass detection
threshold from $0.15\,M_{J}$ to $0.03\,M_{J}$.

###  4.3 Outer planets and the link with metals observed in the atmosphere of
white dwarfs

Observations of polluted white dwarfs provide the best means of studying outer
planetary systems to date. Most theories to explain the presence of metals in
the atmospheres of white dwarfs suggest that material is scattered inwards
from an outer planetary system that survived the star’s evolution (Farihi,
2016). The planets that Gaia will detect provide a crucial piece of the
puzzle: for the first time it will be possible to probe the outer planets
orbiting white dwarfs. Here, we highlight that a link between the Gaia planet
detections and metals in the atmospheres of white dwarfs is not straight-
forward and a one:one correlation is not anticipated.

In order to investigate a potential link between Gaia planet detections and
asteroids or comets scattered inwards to pollute white dwarfs, we consider a
simple scenario. Planets on the inner edges of planetesimal belts can scatter
planetesimals and their ability to scatter planetesimals can increase
following stellar mass loss. Many scattered bodies are ejected, but some can
be scattered inwards, potentially ending up on star-grazing orbits close to
the white dwarf (Bonsor et al., 2011; Frewen & Hansen, 2014; Debes et al.,
2012). Interior planets may be required to scatter planets along a chain
(Marino et al., 2018). Many of the planets detected by Gaia migrate less than
expected from an adiabatic expansion due to stellar mass loss due to the
effects of tides. If the planetary body directly exterior to them, be it a
planet or a planetesimal belt, migrates outwards adiabatically, a dynamical
‘gap’ in the planetary system is created, across which it becomes harder to
scatter bodies inwards. Here we highlight that many planets detected by Gaia
will have created such dynamical ‘gaps’ in their planetary systems, making the
scattering of material inwards harder, and thus, the pollution of their host
stars less likely.

To assess which planets in the predicted white dwarf planet distribution could
contribute to white dwarf pollution we grouped the simulated planets into four
categories (see Figure 8): planets engulfed by the star, planets that migrated
outwards adiabatically and could scatter material from an outer planetesimal
belt and planets whose orbits were influenced additionally by tides, either
migrating inwards or outwards (less far than adiabatically). Whilst the inward
migrating planets are less likely to be part of a planet chain scattering
material onto the white dwarf from the outer planetary system, they could
potentially interact with rocky material interior to their initial orbit, if
the material migrates outwards adiabatically. For the mathematics of the
categorisation see Appendix B.

Almost all surviving planets in the Gaia detection region have dynamical
‘gaps’ in their planetary systems. This is shown in Figure 8, where only
purple and green points lie above the Gaia detection curves and inside 3.91
au. All planets that migrate outwards adiabatically and could readily pollute
white dwarfs by scattering material inwards lie beyond 3.91 au and most of
them are not massive enough to be detected. Even if Gaia detects Jupiter mass
planets that have migrated inwards, they are unlikely to be responsible for
scattering planetary material from an outer system inwards. The detected
Jupiters will have migrated inwards, separating themselves dynamically from
the outer system. Additionally, Jupiter mass planets are more efficient at
ejecting material from a planetary system than scattering it inwards (Wyatt et
al., 2017). They also scatter material too rapidly to give the observed long-
term accretion of material onto some white dwarfs (Mustill et al., 2018). Low
mass planets are more likely to be a long term source of pollution in a
planetary system, because they scatter more material inwards and do so more
slowly than Jupiter (Mustill et al., 2018; O’Connor et al., 2021). These low
mass planets are not detectable by Gaia. Therefore, white dwarfs with Gaia
detected planets are less likely to be polluted than other white dwarfs with
lower mass planets, that Gaia does not detect. This is an important result: if
Gaia does not find a correlation between detected planets and polluted white
dwarfs, this does not rule out the possibility that planets pollute white
dwarfs.

The white dwarf planet candidate in DR3 has been found around WD 0141-675,
which is a polluted white dwarf (Debes et al., 2010). This planet is very
close to its star (P=$33.65\pm 0.05$ d) so cannot have survived post-main
sequence evolution as it is modelled in this work. It must have survived
common envelope evolution or been scattered in during the white dwarf phase,
as has been suggested for WD 1856b (Chamandy et al., 2021; Lagos et al., 2021;
Merlov et al., 2021; Muñoz & Petrovich, 2020; O’Connor et al., 2021). A giant
planet this close to a white dwarf would eject most material scattered inwards
from an outer disc. An alternative explanation for the pollution might be the
accretion of the planet’s gaseous envelope, as suggested by Gänsicke et al.
(2019) for WDJ0914+1914. Planets on such short orbital periods are likely to
be amongst the first detections, and it is not yet clear whether they are
special cases or the norm.

Figure 8: Final positions of simulated planets plotted as a function of their
final semi-major axis and mass colour coded according to whether or not they
could scatter material onto a white dwarf (see §4.3). Engulfed planets are
shown by the grey stars, inward migrating planets by the purple circles,
planets moving outward adiabatically where material could be scattered from an
outer planetesimal belt by the yellow pluses and outward moving planets with a
dynamical ‘gap’ in their system due to the effects of tides by the green
crosses. The minimum planet mass detected as a function of semi-major axis for
$S/N=1$ for a $G_{\textrm{mag}}=15$ white dwarf at 20 pc and 40 pc shown by
the black dashed and dotted lines. The solid vertical line indicates the
period cut off due to the length of the extended Gaia mission.

##  5 Conclusion

Gaia is revolutionary for almost every field in astrophysics, and exoplanets
is no exception. Astrometric detection of wide-orbit (few au) planets orbiting
stars across the HR diagram by Gaia will uniquely probe the outer regions of
planetary systems, crucially linked to the habitable zone around sun-like
stars, such as our own. Gaia planet detections around white dwarfs will probe
the fate of planetary systems, providing secure evidence as to the key
processes that influence planets post-main sequence.

This work predicts that Gaia will detect $8\pm 2$ white dwarf exoplanets that
evolved from the main-sequence only due to tides and the mass loss of the host
star during post-main sequence evolution. These planets will have masses
$0.03-13\,M_{J}$ and semi-major axes 1.6-3.91 au. Comparison of these
predictions with the Gaia white dwarf planet detections will tell us how
important dynamical scattering, common envelope evolution and photo-
evaporation are. Notably, any planets detected interior to 1.6 au, or around
stars with progenitors more massive than $1.5\,M_{\odot}$ must have been
subject to additional processes post-main sequence.

Metals observed in the atmospheres of white dwarfs are commonly linked to
outer planetary systems, in particular, comets or asteroids scattered inwards
by planets. This work highlights how many white dwarfs planets detected by
Gaia will have evolved due to tides, leaving dynamical gaps in their outer
planetary systems. These gaps render the scattering inwards of material to
pollute the white dwarf harder. White dwarf pollution may, therefore, be more
commonly associated with lower mass outer planets, hidden from Gaia
astrometry.

##  6 Acknowledgements

We would like to thank Piero Ranalli for sharing the exact values of his
detection probabilities from Ranalli et al. (2018). We would also like to
thank the reviewer for their insightful comments and Laura Rogers, Zephyr
Penoyre, Marc Brouwers, Andy Buchan, Elliot Lynch, Mark Wyatt and Rachel
Fernandes and Giorgia Busso for their helpful discussions. AJM acknowledges
support from the Swedish Research Council (grant 2017-04945). AB acknowledges
funding from a Royal Society Dorothy Hodgkin Research Fellowship, DH150130 and
a Royal Society University Research Fellowship, URF\R1\211421. HS acknowledges
funding on a NERC studentship NE/S007474/1 and an Exonian Graduate Scholarship
from Exeter College, University of Oxford. For the purpose of open access, the
authors have applied a Creative Commons Attribution (CC BY) licence to any
Author Accepted Manuscript version arising.

##  7 Data Availability

Additional data and the code used in this publication are available on Github
at https://github.com/Hannah-RS/Gaia-white-dwarf. The table of white dwarf
candidates in Appendix C will be available on Vizier after publication.

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##  Appendix A Effects of tides

As discussed in §3.2 and §4.1.1 tides have a strong effect on the final
position of planets above a certain mass. The strength of this effect can be
quantified by $f$, the ratio of the final position of a planet and the final
position expected from an adiabatic expansion:

$f=\frac{a_{\textrm{f}}(\textrm{true})}{a_{\textrm{f}}(\textrm{adiabatic})}.$
(12)

The final position of a planet can be written as

$a_{\textrm{f}}=\frac{M_{\star}(\textrm{initial})}{M_{\star}(\textrm{final})}fa_{\textrm{i}}=pfa_{\textrm{i}}.$
(13)

If $f<\frac{1}{p}$ then $a_{\textrm{f}}<a_{\textrm{i}}$ and tidal forces were
strong enough that the planet moved inwards. For a $1M-\odot$ star in our
simulations $p=0.5702$. Planets closer to their star experience stronger tidal
forces and more massive planets experience tidal forces over a wider range of
initial positions, as shown by Figure 9. These tidal forces cause planets with
a small range of initial positions (Figure 10) to spread over a wide range of
final positions. This contributes to the low number of predicted planet
detections, because the semi-major axis range of main-sequence planets which
will evolve to have periods less than ten years (the predicted Gaia extended
mission length) is small.

Figure 9: Final positions of simulated planets with periods less than ten
years colour coded by
$f=\frac{a_{\textrm{f}}(\textrm{true})}{a_{\textrm{f}}(\textrm{adiabatic})}$.
Yellow points indicate planets’ orbits which underwent an adiabatic expansion,
whilst green to blue points indicate planets affected by tidal forces. Values
of $f<0.57$ indicate the planet moved inwards. Figure 10: Initial positions of
planets with final positions inside the Gaia detection region colour-coded by
final position.

##  Appendix B Dynamical Gap Calculations

For definitions of $f$ and $p$ see Appendix A.

Planetary systems tend on average to be dynamically packed, or in other words
the planetary bodies settle down to a stable configuration in which they are
sufficiently separated that interactions occur on timescales longer than the
lifetime of the system (Barnes & Quinn, 2004; Shannon et al., 2016; Dawson,
2018). The inner edge of a planetesimal belt can be carved by a planet that is
orbiting just interior to it (Chiang et al., 2009), just like Neptune sculpts
the inner edge of the Kuiper belt (e.g. Morbidelli et al., 2008). If this is
the case, when the star undergoes mass loss, the planetesimals in the belt
migrate outwards adiabatically, but the planet interior to the belt may also
be influenced by tides from the AGB star. As a result, the planet migrates out
more slowly, or even in the opposite direction to the belt. In this scenario,
a dynamical gap is created, across which it is harder to scatter planetesimals
than in the main-sequence configuration, as discussed in §4.3. The width of
this gap can be calculated analytically, by considering the change in the
width of a planet’s chaotic zone to the orbit of an exterior planetesimal belt
undergoing an adiabatic expansion.

The width of this chaotic zone, $\delta a_{chaos}$ is defined as:

$\delta
a_{\textrm{chaos}}=1.3a_{\textrm{pl}}\left(\frac{M_{\textrm{pl}}}{M_{*}}\right)^{\frac{2}{7}},$
(14)

where $M_{\textrm{pl}}$ and $M_{*}$ are the planet and stellar mass
respectively (Wisdom, 1980). It extends from the semi-major axis of the planet
($a_{\textrm{pl}}$) to $a_{\textrm{pl}}\pm\delta a_{\textrm{chaos}}$.

As a star loses mass, the orbit of a planet and the outer edge of its chaotic
zone and any surrounding planetesimal belts expand. The inner edge of the
planetesimal belt, which is initially at the outer edge of the chaotic zone,
$a_{\textrm{in}}$, expands adiabatically

$a_{\textrm{in}}=pa_{\textrm{pl}}(1+1.3\left(\frac{M_{\textrm{pl}}}{M_{*}}\right)^{\frac{2}{7}}).$
(15)

The outer edge of the chaotic zone, $a_{\textrm{out}}$, is affected by the
altered planet-star mass ratio and the change in planet semi-major axis.

$a_{\textrm{out}}=fpa_{\textrm{pl}}(1+1.3\left(p\frac{M_{\textrm{pl}}}{M_{*}}\right)^{\frac{2}{7}}).$
(16)

If the outer edge of the chaotic zone becomes larger than the inner edge of
the belt, mass from the belt ends up in the chaotic zone and becomes unstable
and can be scattered inwards onto the star. However, if the planet is strongly
affected by tidal forces, the outer edge of the chaotic zone expands less than
the planetesimal belt and a dynamical ‘gap’ forms in the system. The critical
value of $f$ for this scenario is given in Equation 17.

$f_{\textrm{crit}}<\frac{1+1.3\left(\frac{M_{\textrm{pl}}}{M_{*}}\right)^{\frac{2}{7}}}{1+1.3\left(p\frac{M_{\textrm{pl}}}{M_{*}}\right)^{\frac{2}{7}}}.$
(17)

In this equation $M_{*}$ is the initial stellar mass. For a Jupiter mass
planet and $1M_{\odot}$ progenitor this corresponds to $f<0.975$ \- an almost
adiabatic migration. The points in Figure 8 were classified as follows:
purple, $f<0.5702$ (see Appendix A); green, $0.5702<f<f_{\textrm{crit}}$;
yellow, $f>f_{\textrm{crit}}$ and grey, engulfed.

Figure 11: Detection probability, $p_{ljk}$ (§2.1) as a function of mass and
semi-major axis for G29-38. Here detection probability refers to the
likelihood of detecting a given planet if it exists around G29-38 and does not
account for the likelihood of a given planet existing around the star.

##  Appendix C High detection probability white dwarfs

There are 52 white dwarfs for which the S/N of a $1\,M_{J}$ planet at 2 au is
greater than three (Figure 6). These objects are listed in Table 2. Of
particular interest is G29-38 - the first polluted white dwarf for which a
debris disc was observed (Tokunaga et al., 1990; Jura, 2003) and which has
variable photospheric calcium line strengths (von Hippel & Thompson, 2007).
Gaia measurements of G29-38 will build on existing observations with Keck, the
Hubble Space Telescope, Herschel and ALMA (e.g. Debes et al., 2005; Farihi et
al., 2014; Kuchner et al., 1998). These observations currently rule out
planets $>6\,M_{J}$ beyond 12 au and $>16\,M_{J}$ between 3-12 au (Debes et
al., 2005). Farihi et al. (2014) use ALMA and Herschel observations to rule
out the presence of dust in the 1-100 au region emitting with
$L_{IR}/L_{\star}>10^{-4}$ from an evolved Kuiper-belt analogue. Gaia could
rule out the existence of planets above approximately $1\,M_{J}$ between 1-4
au around this star (see Figure 11). A similar figure could be produced for
any white dwarf in the catalogue. This list also features WD 0141-675 around
which a $\approx 9\,M_{J}$ planet candidate has been found in DR3 (Gaia
Collaboration et al., 2022b) (see §4.3).

White dwarf name | Gaia DR3 source ID | Distance (pc) | Gmag | S/N for 1$M_{J}$ at 2 au | error on S/N | Ca/He | Ca/H
---|---|---|---|---|---|---|---
WDJ064509.30-164300.72 | 2947050466531873024 | 2.7 | 8.5 | 33.7 | 5.6 |  |
WDJ004909.90+052318.99 | 2552928187080872832 | 4.3 | 12.3 | 20.9 | 3.5 | -9.92 |
WDJ114542.92-645029.46 | 5332606522595645952 | 4.6 | 11.4 | 19.4 | 3.2 |  |
WDJ041521.80-073929.20 | 3195919254111315712 | 5.0 | 9.5 | 18.0 | 3.0 |  |
WDJ043112.57+585841.29 | 470826482637310848 | 5.5 | 12.3 | 16.3 | 2.7 |  |
WDJ174807.99+705235.92 | 1638979384378696704 | 6.2 | 13.8 | 10.8 | 1.8 |  |
WDJ084132.43-325632.92 | 5639391810273308416 | 8.5 | 11.8 | 10.6 | 1.8 |  | <-11.12
WDJ074020.79-172449.16 | 5717278911884258176 | 9.1 | 13.0 | 9.8 | 1.6 |  |
WDJ031031.02-683603.38 | 4646535078125821568 | 10.4 | 11.4 | 8.7 | 1.4 |  |
WDJ120526.67-233312.14 | 3489719481290397696 | 10.4 | 12.7 | 8.6 | 1.4 |  | -9.7
WDJ055509.53-041007.07 | 3022956969731332096 | 6.4 | 14.2 | 8.6 | 1.4 |  |
WDJ192034.92-074000.07 | 4201781696994073472 | 10.5 | 12.3 | 8.6 | 1.4 |  |
WDJ075308.14-674731.38 | 5273943488410008832 | 8.2 | 13.8 | 8.2 | 1.4 |  |
WDJ214241.01+205958.12 | 1792830060723673472 | 11.0 | 13.2 | 7.7 | 1.3 |  |
WDJ043747.41-084910.62 | 3186021141200137472 | 9.4 | 13.6 | 7.6 | 1.3 |  |
WDJ055625.46+052148.44 | 3320184202856435840 | 8.1 | 14.0 | 7.6 | 1.3 |  |
WDJ013759.39-045944.67 | 2480523216087975040 | 12.6 | 12.7 | 7.1 | 1.2 |  | <-11.53
WDJ014300.98-671830.35* | 4698424845771339520* | 9.7 | 13.7 | 7.1 | 1.2 |  |
WDJ133631.85+034045.94 | 3713594960831605760 | 8.3 | 14.4 | 6.2 | 1.0 |  |
WDJ195629.23-010232.67 | 4235280071072332672 | 11.6 | 13.6 | 6.2 | 1.0 |  | <-11.27
WDJ215140.11+591734.85 | 2202703050401536000 | 8.5 | 14.4 | 6.2 | 1.0 |  |
WDJ203421.89+250349.75 | 1831553382794173824 | 14.8 | 11.5 | 6.1 | 1.0 |  |
WDJ004126.03-222102.29 | 2349916559152267008 | 9.1 | 14.3 | 5.9 | 1.0 |  |
WDJ154730.02-375508.46 | 6009537829925128064 | 15.2 | 13.0 | 5.9 | 1.0 |  | <-10.28
WDJ133013.64-083429.47 | 3630035787972473600 | 16.1 | 12.4 | 5.6 | 0.9 |  |
WDJ201056.85-301306.63 | 6749419923164242816 | 16.2 | 12.3 | 5.6 | 0.9 |  | <-8.84
WDJ015202.96+470006.66 | 356922880493142016 | 16.5 | 12.5 | 5.5 | 0.9 |  |
WDJ211856.26+541241.24 | 2176116580055936512 | 17.3 | 12.4 | 5.2 | 0.9 |  |
WDJ001214.75+502520.74 | 395234439752169344 | 10.9 | 14.2 | 5.1 | 0.8 |  | <-11.15
WDJ113430.48-325002.40 | 3478127467639543296 | 9.6 | 14.5 | 5.1 | 0.8 |  |
WDJ232847.64+051454.24 | 2660358032257156736 | 17.5 | 13.1 | 5.0 | 0.8 |  | -6.58
WDJ041630.04-591757.19 | 4678664766393827328 | 18.3 | 12.5 | 4.9 | 0.8 |  |
WDJ234350.72+323246.73 | 2871730307948650368 | 18.6 | 13.0 | 4.8 | 0.8 |  | <-9.53
WDJ193713.75+274318.74 | 2025389380082340992 | 18.2 | 13.1 | 4.8 | 0.8 |  |
WDJ081227.07-352943.32 | 5544743925212648320 | 11.2 | 14.3 | 4.7 | 0.8 |  |
WDJ055119.48-001021.11 | 3218697767783768320 | 11.2 | 14.4 | 4.5 | 0.8 |  |
WDJ211316.85-814912.88 | 6348672845649310464 | 16.2 | 13.6 | 4.4 | 0.7 |  | -8.6
WDJ233850.74-074119.97 | 2439184705619919488 | 18.6 | 13.3 | 4.3 | 0.7 |  |
WDJ191858.63+384321.48 | 2052891361294411520 | 11.9 | 14.5 | 4.2 | 0.7 |  |
WDJ162825.00+364615.85 | 1331106782752978688 | 15.9 | 13.8 | 4.2 | 0.7 | -9.07 |
WDJ204234.75-200435.94 | 6857939315643803776 | 21.8 | 12.4 | 4.1 | 0.7 |  |
WDJ202025.46-302714.65 | 6797171060323993728 | 17.4 | 13.6 | 4.1 | 0.7 |  |
WDJ122642.02-661218.47 | 5860131207828395648 | 15.4 | 13.9 | 4.1 | 0.7 |  |
WDJ021228.98-080411.00 | 2486388560866377856 | 16.7 | 13.7 | 4.1 | 0.7 |  |
WDJ212657.66+733844.66 | 2274076297221555968 | 22.2 | 12.9 | 4.1 | 0.7 |  |
WDJ215225.38+022319.58 | 2693940725141960192 | 22.5 | 12.8 | 4.0 | 0.7 |  |
WDJ011800.08+161020.56 | 2591754107321120896 | 16.8 | 13.8 | 4.0 | 0.7 |  |
WDJ112412.97+212135.57 | 3978879594463300992 | 14.7 | 14.1 | 4.0 | 0.7 |  |
WDJ225353.39-064654.49 | 2611561706216413696 | 8.5 | 15.4 | 3.8 | 0.6 | -10.00 |
WDJ143307.64-812014.13 | 5772718006135360128 | 20.9 | 13.4 | 3.7 | 0.6 |  |
WDJ080653.75-661816.70 | 5274517467840296832 | 19.2 | 13.7 | 3.6 | 0.6 |  |
WDJ015151.14+642552.55 | 518201792978858880 | 17.3 | 13.9 | 3.6 | 0.6 |  | <-12.08
Table 2: White dwarfs in the EDR3 catalogue for which a 1 $M_{J}$ planet at 2
au would have a $S/N+\sigma(s/N)>3$. The data in the first four columns come
from Gentile Fusillo et al. (2021). The white dwarf names are WD J $+$ J2000
RA (hh mm ss.ss) $+$ Dec. (dd mm ss.s), equinox and epoch 2000, the distance
is the median of the geometric distance posterior (pc) (Bailer-Jones et al.,
2021) and $G_{\rm{mag}}$ is the corrected photometric G mean magnitude (Brown
et al., 2021). The S/N and error on the S/N were calculated by the authors.
The Ca/H and Ca/He data comes from the Montreal White Dwarf Database (Dufour
et al., 2017). The asterisk marks WD 0141-675, the white dwarf with a $\approx
9\,M_{J}$ planet candidate in DR3 (Gaia Collaboration et al., 2022b). This
data along with $P_{\textrm{WD}}$ and the white dwarf mass will be made
available on Vizier after publication.