11institutetext: Astronomical Observatory Institute, Faculty of Physics, Adam
Mickiewicz University, Słoneczna 36, PL60-283 Poznań, Poland

# Important stellar perturbers found during the StePPeD database update based
on Gaia EDR3 data

Piotr A. Dybczyński<EMAIL_ADDRESS>Filip Berski  Jakub Tokarek  Edyta
Podlewska-Gaca  Krzysztof Langner  Przemysław Bartczak

(Received XXXXXX; accepted XXXXX)

###### Abstract

Context. In 2020, the initial version of the Stellar Potential Perturbers
Database (StePPeD) was presented with the aim to deliver up-to-date
information on the stars and stellar systems that may perturb a long-period
comet motion. We used the minimal distance between a star and the Sun as a
selecting tool when compiling a list of interesting objects with close
encounters with the Solar System, and our selection for that study was based
on Gaia DR2 data.

Aims. When the Gaia EDR3 data release was published, it became necessary to
update this database. Additionally, we performed Monte Carlo simulations to
obtain uncertainties on the parameters of the closest approach to the Sun of
each object.

Methods. We recalculated the close approach parameters of all stars in the
previous StePPeD release, which resulted in removing approximately one-third
of the total. Then we searched for new candidates in the whole Gaia EDR3
catalogue. We also take into account the duplicity of the found stars and
additionally searched for double stars passing near the Sun which had been
overlooked in previous papers. We also found the necessary mass estimates for
new objects and updated this information for previously selected stars.

Results. After a careful checking of all the collected data, we composed a new
list of 155 potential stellar perturbers of the long-period comet motion. We
applied a new threshold of 2 pc for the minimum star-Sun distance. This list
consists of 146 single stars and nine multiple systems. For each object, we
also estimated the uncertainty of the parameters of their closest approach to
the Sun. Among these stars, we found a new potential strong past perturber,
HD~7977, and confirmed the plausibility of a similar action on the part of
Gliese~710 in the future.

###### Key Words.:

comets:general – stars:kinematics and dynamics – (Galaxy:) solar neighbourhood

##  1 Introduction

Stellar perturbations have been the only mechanism producing observable comet
orbits proposed by Oort (1950). The importance of the galactic perturbations
has been recognised since then as the second most important factor, but
growing knowledge on the nearby stars positions and movements causes the
necessity to simultaneously account for both these sources of perturbations in
the studies of long-period comets (hereafter LPCs) motion and origin.
Dybczyński & Breiter (2022) recently proposed a new and efficient method for
such calculations, which is partially based on the data on potential stellar
perturbers.

In Wysoczańska et al. (2020b) (WDP20) the authors introduced a publicly
available Stellar Potential Perturbers Database
(StePPeD)111https://pad2.astro.amu.edu.pl/StePPeD containing data on potential
stellar perturbers of LPCs motion. This database has had four revisions
between April and July 2020, which finally led to release 2.3. However, after
the Gaia Early Data Release 3 was made available (Gaia Collaboration et al.,
2021a), it appeared that the substantial update of the StePPeD database had
become necessary.

In this paper, we describe the update of the StePPeD database, which fully
accounts for the Gaia EDR3 results (Gaia Collaboration et al., 2021a).
Motivated by several important differences between astrometric data in the
last two Gaia data releases, as well as the large uncertainties for some
objects and the expected significant improvement in the near future, we
decided to temporary limit our list of potential perturbers by applying a
smaller threshold for the minimal star-Sun distance that is used to select
objects of interest. The previous threshold of 4 pc resulted in a list of 825
objects and now, using a limit of 2 pc, we shortened the list to 155 stars or
stellar system.

The update process first concentrated on a new data for the perturbers from
the old list and this step is described in Section 2. In Section 3, we
describe the procedure of finding new candidates among stars in Gaia EDR3
catalogue. It also presents how we deal with double stars and discusses
sources of additional data (e.g. stellar mass estimates). Moreover, we discuss
various sources of the uncertainties of our results. Section 4 presents the
obtained results for the closest star-Sun encounters. The most important case
of the star P0230 (HD 7977) is discussed in Section 5. The next section
answers the question on the influence of such a close passage of a star near
the Sun on the Solar System planets and small bodies. Section 7 briefly
describes other important stellar perturbers. The final section summarises our
work on the StePPeD database update.

##  2 Update part I: Refinement of the star list included in the StePPeD
release 2.3

The StePPeD release 2.3, made available on July 27, 2020, consisted of 825
potential perturbers, that is, 787 single stars and 38 multiple systems. Among
them, 648 objects might nominally pass the Sun closer than 4 pc. As it
concerns multiple systems, systemic position and velocities were calculated
from individual component data for 34 systems. For the remaining four (P1008,
P1024, P1037, P1069), the data were taken directly from dedicated papers (see
WDP20 for details). Astrometric data were gathered for 858 individual stars,
833 of them were taken from Gaia DR2 catalogue (Gaia Collaboration et al.,
2018), and for remaining 25 from other sources.

The first step in updating StePPeD database was to search for all these stars
in the Gaia EDR3 catalogue. It was rather troublesome since in many cases
cross-identifications were ambiguous – in many cases, we found two stars in
EDR3 possibly corresponding to one star in DR2. It resulted often from finding
a possible second component of a double system that was absent or overlooked
in previous investigations. Finally, we identified 43 new pairs as suspected
to be binaries, many of them with poor quality or incomplete data (missing
mass estimate or radial velocity).

The comparison of data between DR2 and EDR3 led to surprising results. We were
able to compare 836 stars (single perturbers and components of the stellar
systems) from our potential perturbers list (which are present in both of
these catalogues). Among these, we found 197 stars with a parallax difference
greater than 100% (and additionally nine stars with a negative parallax in
EDR3). For proper motions, the situation is only slightly better: 148 stars
with the proper motion difference in the right ascension greater than 100% and
154 stars for declination. Among these 836 stars, 350 have a renormalised unit
weight error (RUWE) parameter greater than 1.4, which indicates an unresolved
binarity or potentially less accurate data (Fabricius et al., 2021). For 43
stars on our list, there are no parallax neither proper motion in EDR3.

After a careful verification of our list, we ended up with 743 single stars
and 81 multiple systems. In the new StePPeD release 3.0, due to poor quality
or incomplete data, we decided to omit 16 single stars and 20 multiple
systems, usually because EDR3 reports much smaller parallax or significantly
changed proper motions with respect to the previous data or because they have
possible secondaries. Additionally, for the same reasons, for 11 systems, we
decided to use the data for only one component. Over 50 individual comments on
particular objects are available at the database web interface. All omitted
stars or components were kept in our database for an upcoming verification
when the new data become available.

Apart from the astrometry, we also collected radial velocities and mass
estimates. The velocities are mainly incorporated from Gaia DR2, but also from
Soubiran et al. (2018), along with tens of other individual papers. The mass
estimates are most often taken from Anders et al. (2019) and (Stassun et al.,
2019), as well as other relevant papers. It is worth mentioning that every
numeric input data included in the StePPeD database can be directly inspected
through the web interface where its source is also explicitly shown. There is
also a possibility to download all input data as the plain ASCII text file.
Based on the acquired data, we calculated the Galactic rectangular
heliocentric position and velocity components for 783 perturbers (single stars
or the centre of mass of multiples). For an additional five objects (P0403,
P1008, P1024, P1037, and P1069) we used systemic data taken from individual
papers. All the positions and velocities were moved to the common epoch of
2016.0 – which is that of the Gaia EDR3 catalogue.

The above described data for 788 potential LPCs motion perturbers were used as
the starting data for backward or forward numerical integration, taking into
account all mutual interactions and the overall Galactic potential. For
details of our dynamical model, used constants, reference frame, and Galactic
potential approximation, we refer to Dybczyński & Berski (2015).

Based on the nominal parameters of close encounters with the Sun in WDP20, the
authors selected 642 objects which, in the past or future, could approach
closer than 4 pc from the Sun. After some revisions, in the StePPeD release
2.3 there are 648 such perturbers. The significant change between Gaia DR2 and
EDR3 in astrometric data for large number of stars from our list resulted in
remarkable downsizing of this number. Using the same threshold of 4 pc, we
obtained only 407 objects. As many as 157 objects have their nominal closest
distance from the Sun greater than 100 pc and among them, 25 have passed or
will pass farther than 1 kpc. Some of them, especially the multiple systems,
require further verification when new data become available, but most of them
will be probably permanently removed from the StePPeD future releases.

A similar work on updating the StePPeD version 2.3 parameters of the closest
star-Sun approaches was published recently by Bobylev & Bajkova (2021) but it
is limited to stellar passages closer that 1 pc. They also ignore the
multiplicity of the passing stars and use different Galactic parameters and
potential model, so they obtained slightly different results. In November
2021, the radial velocities of a dozen stars have been updated and as a result
StePPeD version 3.1 was released (see footnote 1). For details see the
’Changelog’ available at the StePPeD Web page.

##  3 Update part II: Adding new stars based on Gaia EDR3

Taking into account a large percentage of significant differences in
parallaxes and proper motions between Gaia DR2 and EDR3, as well as the
resulting number of perturbers removed from our list, we expect that some new
candidates can be found in Gaia EDR3. To search for new stars, we decided to
repeat the whole procedure of finding stars approaching the Sun, similar to
what was described in WDP20, in addition to the stars that are already in our
list.

Using a linear approximation and the ADQL query similar to that proposed in
Bailer-Jones et al. (2018), we searched the whole Gaia EDR3 catalogue (Gaia
Collaboration et al., 2021a) for the stars that have visited or will visit the
solar neighbourhood. However, this time we did not restrict our calculations
to objects with a nominal linear minimal distance from the Sun
$D_{min}^{lin}<10$ pc, as proposed in some previous attempts (Dybczyński &
Berski, 2015; Bailer-Jones et al., 2018), but we take into account also stars
that approach the Sun (according to a linear model) up to a heliocentric
distance of 280 pc. The reason we would look for objects with
$D_{min}^{lin}>10$ pc is that the track of the Sun and stars in longer time
intervals are significantly different from a straight line and linear
approximation cannot provide reliable results. Unfortunately, the longer time
of motion of objects increases the error of our results due to the propagation
of the stellar data uncertainties. That issue and the low number of remaining
objects after numerical verification (only 14 stars with $D_{min}^{lin}>200$
pc and $D_{min}^{lin}\leq 280$ pc) prompted us to stop searching for stellar
perturbers with $D_{min}^{lin}$ values larger than 280 pc.

Using a linear approximation we find over 1.3 million of objects with
$D_{min}^{lin}\leq 280$ pc. Among them we find over three thousands stars with
$D_{min}^{lin}\leq 10$ pc and over nine thousands stars with
$10\leavevmode\nobreak\ pc<D_{min}^{lin}\leq 20$ pc. The number of potential
candidates grows significantly with the allowed $D_{min}^{lin}$. At
$D_{min}^{lin}=210$ pc this growing trend almost stops and a number of stars
per 10 pc bin stabilises around 60 thousands of objects, with a maximum value
of 62521 stars for $240\leavevmode\nobreak\ pc<D_{min}^{lin}\leq 250$ pc. The
details are shown in top panel of Fig. 1.

Figure 1: New stellar candidates statistics. Top histogram shows a number of
new candidates for close flyby stars from Gaia EDR3 catalogue based on a
linear motion approximation. The bottom panel shows a number of stars
remaining after numerical integration and applying a threshold of for the
nominal star distance $D_{min}^{nom}\leq 5$ pc. The horizontal axis describes
a minimal distance between the star and the Sun $D_{min}^{lin}$ calculated
from a linear approximation.

The next step is to numerically integrate the stellar motion and obtain much
more reliable proximity distance value. Due to a large number of objects
obtained from a linear approximation (over 1.3 million stars), we decided to
first use a simple model. It ignores any mutual action between a star and the
Sun, therefore, their motion depends only on the gravitational potential of
the Galaxy. To compensate the simplicity of this approach we decided to
slightly increase the threshold for star-Sun minimal distance of potentially
interesting objects from 4 pc used in the past to 5 pc. Our investigations
show that the relative error between this simple model and the model including
distance corrections and interactions between bodies is about 13.5%. For
example, for star EDR3 469547750615777280, the minimum distance changes from
4.61 pc to 3.07 pc when we use a more accurate model.

Unexpectedly, the numerical integration showed that many of objects, that pass
much further than 10 pc from the Sun according to the linear approximation,
can in fact approach the Sun at a very close distance. Please note that we
discuss here only the nominal results (see the bottom panel of Fig. 1).

For example, the star EDR3~4312257326836128896 (P2001), which in a linear
approximation has the minimal distance $D_{min}^{lin}=53.12$ pc appeared to
pass the Sun at $D_{min}^{nom}=0.38$ pc based on a numerical integration. The
star EDR3~6755981602406174720 (P2009) is even a more spectacular example: we
obtained a change from $D_{min}^{lin}=252.28$ pc to $D_{min}^{nom}=0.83$ pc.
We also recorded several examples of the opposite effect: for more that one
hundred of stars a linear approximation gives the minimal distance
D${}_{min}^{lin}<10$ pc, while the nominal distance from the numerical
integration is D${}_{min}^{nom}>100$ pc. The extreme example is the star
EDR3~450118246277153152 for which D${}_{min}^{lin}=7.9$ pc, but
D${}_{min}^{nom}=4541$ pc.

Following this step, we obtained 527 new star flybys near the Sun closer than
5 pc (all stars or stellar systems present on our previous list were
automatically omitted). After we performed the numerical integration, it
appeared that only about 10% of objects with $D_{min}^{lin}\leq 10$ pc have
$D_{min}^{nom}$ smaller than 5 pc; namely, it was 298 from 3121 tested stars.
In the next bin ($10\leavevmode\nobreak\ pc<D_{min}^{lin}\leq 20$ pc), we
found 64 objects from among over nine thousands of candidates. Finally, as
many as a half of all new perturbers have $D_{min}^{lin}>10$ pc and 90% of all
objects have $D_{min}^{lin}\leq 100$ pc. The largest value of $D_{min}^{lin}$,
which was reduced below the 5 pc threshold as a result of a numerical
integration is for the star EDR3~5616156518241119872. In this case, we have
$D_{min}^{lin}=258.68$ pc and $D_{min}^{nom}=4.01$ pc.

###  3.1 Searching for secondaries

At least 60 of our targets appeared to be members of binary systems listed in
the Gaia EDR3 catalogue of possible binary stars of El-Badry et al. (2021).
Furthermore, by inspecting carefully the vicinity of each star in our list we
found some additional possible binaries. For example, the star
EDR3~196124257030748544 (P2024) have probably a companion
EDR3~196124257030748032 at the angular distance of 13.7 arcsec. Thanks to the
fact that the secondary is relatively bright ($\sim$14 mag), we were able to
collect a complete set of data (masses, radial velocities) for both
components. However, the physical binarity of this system is highly uncertain.

In the case of a binary system, the knowledge of all parameters is crucial,
because it is necessary to calculate the spatial position and velocity of its
centre of mass. In such cases, the movement of a system as a whole might be
completely different from a single component motion in space. It is worth
mentioning that most of the binaries listed in (El-Badry et al., 2021) are
faint objects with a brightness fainter than 18 mag, so a complete set of
parameters is difficult to find. In particular, lacking are radial velocities
or mass estimates.

###  3.2 Unrecognised binaries passing close to the Sun

In general, there is also the possibility that we have completely overlooked a
binary passing close to the Sun when both components (treated separately) have
their minimal distance greater than our adopted threshold, but the mass centre
of such a system could approach the Sun closely. To our knowledge it is the
first attempt to find such systems.

We searched for candidates in (El-Badry et al., 2021). Our first requirement
is that radial velocities for both components are known. In applying this
criterion, we found 15 359 appropriate candidate binaries. For all systems on
this shortened list, we calculated their centres of masses and velocities
based on the dependence of the mass distribution varying from 0 to 1 with step
0.001, where: 1 means all the mass of the system is located in component A and
0 is that all the mass of the system is located in component B. At this point,
the real masses of components are not necessary, we only need a mass ratio to
calculate the centres of mass position and velocity. That allows us to
determine their trajectories in the Galaxy. We found only 156 systems of
interest. For them we searched for real mass estimates of components. A list
of the used mass sources is described in Section 3.3. We found mass estimate
for both components for only 108 systems, and for the remaining 48 systems, we
found no mass or mass for only one component. Using the real mass estimates we
found that from the binaries with complete data only 12 systems might pass
closer than 15 pc to the Sun, within their mass uncertainties (see Figure 2).
We used here a larger threshold of 15 pc because close approach parameters
strongly depend on the components’ masses, which often come with large
uncertainties. Using the nominal dynamical parameters for both components, we
calculated how the uncertainty in mass estimates impacts the minimal system
distance from the Sun. These systems parameters are presented in Table 2.
Consecutive columns content is as follows: ID in the StePPeD database; first
component Gaia EDR3 identification number; its mass with the 1$\sigma$
uncertainty; next two columns contained the same information for the second
component. In the last two columns we present our results as the system
minimum close-up distance and the time of that event. The positive time
indicates that the approach will happen in the future, the negative describes
the approaches in the past. The errors of last two parameters are here
calculated only from the uncertainties in mass estimations.

For all cases, we have studied the effect of the ratio of the masses of the
system components on the distance for transits through the Solar System. We
selected three potential candidate systems for which only one mass is
available. In these three cases we found the optimal mass distribution between
components, which allows them for a Solar System pass-by below 15 pc (pairs:
[Gaia~EDR3 2738129612130247680, 2738129616426300160], [Gaia
EDR3~3154610636614026624, 3154610636617646848], and [Gaia EDR3
3228938722162512256, 3228938722164251904]). This candidates will be valid
targets for further calculations of the approaches to the Sun if in future
both mass estimations will become available.

Figure 2: Minimum distance from the centre of mass of binary systems to the
Sun, including the possible error intervals arising from the uncertainties in
the mass estimates of the individual components (see Table 2)

In Figure 3, we show an example case of P5000 (BD-06~310, WDS~J01395-0612). In
this system, both components have mass estimates in the TESS-8 catalogue
(Stassun et al., 2018). For their nominal masses, we created 11 635 random
virtual stars (hereafter VSs) with different dynamical parameters randomly
drawn according to the covariance matrix from Gaia EDR3. Our calculations show
that in 6.8 million years, P5000 will fly past the Sun at a distance of 0.9
pc. Statistics of that and other closest binary systems flybys are presented
in Table 1. Due to the uncertainty in mass estimates, this cloud of VSs could
shift towards the Sun or in an opposite direction in the range shown in the
sixth column of Table 2.

Figure 3: Picture represents projection of 11 635 VSs of P5000 system (black
dots) in their closest encounter with the Sun (red dot). Nominal position of
P5000 is marked with green dot. Blue circle is showing outer Oort Cloud
boundary at 0.5 pc away from the Sun. All VSs always have the same masses, but
the dynamical parameters have been randomised. Table 1: Closest approaches of
new binary systems. The first column shows the ID number in the StePPeD
database, the second Gaia EDR3 identifiers for both components, the third
presents the distance D${}^{geom}_{min}$ from the Sun to the centroid of VSs
with its uncertainty (see text for detail description, Section 3.4), the next
three columns present a statistical description of the minimal distance,
D${}^{stat}_{min}$, time of the approach T${}^{stat}_{min}$ and the relative
velocity, V${}^{stat}_{rel}$, for the set of all VSs. In the last column the
mass estimate is included. Here and elsewhere, some extra significant digits
are given for uniformity.

StePPeD |  Gaia EDR3 | geometry | statistics | mass estimate
---|---|---|---|---
ID | ID | D${}^{geom}_{min}$ [pc] | D${}^{stat}_{min}$ [pc] | T${}^{stat}_{min}$ [Myr] | V${}^{stat}_{rel}$ [km s${}^{-1}]$ | [M⊙]
P5000 | 2479549701620909312 | 0.909$\pm$0.203 | $0.912^{+0.115}_{-0.097}$ | $6.841^{+0.311}_{-0.280}$ | 7.18 $\pm$ 0.29 | 0.65
2479549701621101184 | 0.64
P5001 | 4569088021684763904 | 0.920$\pm$1.791 | $1.199^{+0.754}_{-0.554}$ | $8.732^{+0.295}_{-0.279}$ | 60.32 $\pm$ 1.09 | 0.77 a
4569088021688882048 | 0.77
P5002 | 2182863462586498432 | 4.139$\pm$0.590 | $4.143^{+0.527}_{-0.515}$ | $-33.338\pm 0.039$ | 9.946 $\pm$ 0.013 | 1.33
2182863492643748096 | 1.08
P5003 | 4470126821233210112 | 4.437$\pm$0.885 | $4.511^{+0.202}_{-0.183}$ | $-14.637^{+0.491}_{-0.531}$ | 17.15 $\pm$ 0.61 | 1.42
4470126821224457344 | 1.42

222a The mass of this component was assumed equal to the second component.
This is a reliable assumption due to the comparable observed magnitudes of
both stars.

Table 2: Effect of mass uncertainty on the distance and time of approach to
the Sun. The first column shows StePPeD ID, the second and fourth Gaia EDR3
identifiers of the components. The third and fifth columns contain information
on masses and their uncertainties for the components. The last two columns
show the minimum distance and the time of approach of the centre of mass of
the system to the Sun. Uncertainties in the distance and time of approach to
the Sun are here only due to the uncertainties in the determination of the
mass components of the binary system.

StePPeD | First component | Second component | Mass centre approach to the Sun
---|---|---|---
ID |  Gaia EDR3 ID | mass est. [M⊙] |  Gaia EDR3 ID | mass est. [M⊙] | min dist. [pc] | min time [Myr]
P5000 | 2479549701620909312 | $0.65\pm{0.08}$ | 2479549701621101184 | $0.64\pm{0.08}$ | $0.90_{-0.62}^{+1.95}$ | $6.84_{-0.12}^{+0.09}$
P5001 | 4569088021684763904 | $0.77\pm{0.08}^{a}$ | 4569088021688882048 | $0.77\pm{0.08}$ | $0.88_{-0.21}^{+0.29}$ | $8.73_{-0.04}^{+0.04}$
P5002 | 2182863462586498432 | $1.33\pm{0.22}$ | 2182863492643748096 | $1.08\pm{0.13}$ | $4.16_{-3.62}^{+6.36}$ | $-33.37_{-1.26}^{+1.12}$
P5003 | 4470126821233210112 | $1.42\pm{0.25}$ | 4470126821224457344 | $1.42\pm{0.25}$ | $4.45_{-2.48}^{+2.40}$ | $-14.62_{-0.31}^{+0.31}$
P5004 | 1031783261988885376 | $1.15\pm{0.16}$ | 1031783639946006912 | $1.09\pm{0.14}$ | $6.95_{-2.81}^{+7.82}$ | $24.74_{-0.91}^{+1.03}$
P5005 | 6828177528743255552 | $0.73\pm{0.08}$ | 6828177528743255424 | $0.66\pm{0.08}$ | $10.724_{-0.001}^{+0.002}$ | $3.502_{-0.001}^{+0.004}$
P5006 | 287980516431427200 | $0.80\pm{0.10}$ | 287980512135232512 | $0.62\pm{0.08}$ | $5.623_{-0.01}^{+0.06}$ | $0.9845_{-0.0001}^{+0.0001}$
P5007 | 1361208348808624256 | $0.79_{-0.04}^{+0.02}$ | 1361208353105063296 | $0.66\pm{0.08}$ | $6.82_{-0.15}^{+0.23}$ | $-9.45_{-0.12}^{+0.16}$
P5008 | 6643765063414984832 | $0.65\pm{0.08}$ | 6643765067712454656 | $0.59\pm{0.02}$ | $11.12_{-0.05}^{+0.07}$ | $-4.66_{-0.01}^{+0.01}$
P5009 | 6243585681802135808 | $1.17\pm{0.16}$ | 6243585681802137728 | $0.91\pm{0.12}$ | $8.13_{-0.33}^{+0.42}$ | $-15.18_{-0.05}^{+0.05}$
P5010 | 2104485016711846656 | $1.23_{-0.34}^{+0.78}$ | 2104486489885306752 | $0.66\pm{0.02}$ | $10.78_{-0.07}^{+0.08}$ | $8.004_{-0.008}^{+0.006}$
P5011 | 1237090738916392704 | $0.96\pm{0.12}$ | 1237090738916392832 | $0.67\pm{0.08}$ | $6.372_{-0.003}^{+0.005}$ | $-0.396_{-0.007}^{+0.005}$

333a The mass of this component was assumed equal to the second component.
This is a reliable assumption due to the comparable observed magnitudes of
both stars.

###  3.3 Numerical integration: Second approximation

The next step towards more reliable nominal data on stellar encounters with
the Sun is to perform a numerical integration using a more adequate dynamical
model. To the overall Galactic potential, we add the mutual interactions
between all considered stars and between these stars and the Sun. We also used
more sophisticated distance estimations.

This approach make it necessary to have a mass estimate also for the single
stars. We updated the mass estimates for all of them, including those from the
previous list because a lot of new sources were made available since the
publication of WDP20. To this aim we start the search in the recent large data
sets: a catalogue TESS-8 (Stassun et al., 2018), and the stellar parameters
derived by the spectroscopic-astrometric-photometric Bayesian STARHORSE
software (Anders et al., 2019), provided in five catalogues (APOGEE DR16,
GALAH DR2, GES DR3, LAMOST DR5, and RAVE DR6) (Queiroz et al., 2020).
Recently, Anders et al. (2022) have used the data from Gaia EDR3 to improve
their previous estimations. Now, their distances are in much better agreement
with the results of (Bailer-Jones et al., 2021) that we have used.

The majority of stellar distance were adopted from Bailer-Jones et al. (2021),
preferring the photo-geometrical estimates. For a selected subset of nearby
single stars, we used distances from GCNS catalogue (Gaia Collaboration et
al., 2021c). In this catalogue, we found distance estimates for 138 stars in
main catalogue and for 28 objects in the rejected part of catalogue.

###  3.4 Dealing with the stellar uncertainties

It is very important to estimate the influence of the stellar data
uncertainties on the parameters of the close passage near the Sun. In a
separate manner, we checked the influence of the mass estimate uncertainty on
the binary system results as it is described in Section 3.2. Here, we describe
a procedure used for dealing with astrometric and kinematic uncertainties. To
this purpose, we used a star cloning technique similar to that used by
Dybczyński & Berski (2015). Here, the VS is drawn using the respective six-
dimensional covariance matrix described in the Gaia EDR3 documentation444see
the Gaia EDR3 documentation, paragraph 4.1.7.0.4 on page 196 from which we
obtain random values of $\alpha$, $\delta$, $\tilde{\omega}$, $\mu_{\alpha*}$,
$\mu_{\delta}$, $\mu_{r}$. There is only one exception: when the uncertainty
of the radial velocity is unknown or it is greater then 10%, we draw the VS
parameters from the five-dimensional covariance matrix, keeping the radial
velocity equal to the nominal value. To estimate the influence of one star
uncertainties on its parameters of the closest passage near the Sun we draw a
VS for this star and replace the nominal starting data with the VS parameters.
For the rest of the stars, the nominal starting data remain. Then we
numerically integrate the whole set of stars under the Galactic gravitational
potential, including their mutual interactions and the interaction with the
Sun. We repeated this for at least ten thousand of VSs. Such a procedure is
extremely time-consuming, but taking into account the mutual gravity of all
bodies seems to be necessary since we have observed several close approaches
between stars.

We describe the uncertainty in the close approach parameters (distance, epoch,
and relative velocity) using a standard deviation of the approximated Gaussian
distribution or, in cases when this approximation failed, we use three
percentiles: 16%, the median, and 84%. For checking whether the parameter
distribution might be treated as Gaussian, we used the test of Anderson &
Darling (1952). We also use a geometrical description of the VSs cloud
position with respect to the Sun (the details are described in Section 4).

Figure 4: Relative difference in percent of the minimum star-Sun distance
between our reference model Ia and four other Galactic potential models
described in the text. Horizontal axis describe the epoch of a close
encounter. Points in the same colour represent all 155 stars on our list, the
colours represent four models of Galactic potential as explained in the
picture.

###  3.5 Stellar close encounters in different Galaxy potential models

Another source of uncertainties is the adopted Galactic potential. Basing on
our previous experience we choose as a reference model the potential described
as Ia in Irrgang et al. (2013) with their numerical parameters. In a recent
paper by Bovy (2020), this model of a Galactic potential was presented as the
only one which is shown to be in agreement with the Solar System
Galactocentric acceleration deduced from the Gaia EDR3 data (Gaia
Collaboration et al., 2021b).

To examine the influence of different models of Galactic potential on the
minimum distance between stars and the Sun, we integrated all stars back and
forth for 50 million years using several different Galactic potential models.
The first model is our reference one, referred to as Ia. It is based on the
potential proposed by Allen & Santillan (1991), but with the revised parameter
values from Irrgang et al. (2013). The second model is based on the same
formulae, but with parameters taken from Bajkova & Bobylev (2017, here Model
Ib). Another pair is based on the Model III from Irrgang et al. (2013), which
is deduced from the halo density profile of Navarro et al. (1997). The former
uses parameters from Irrgang et al. (2013, which we call Model IIIa), and the
latter, from Bajkova & Bobylev (2017, here Model IIIb). Finally, we considered
the potential of McMillan (2017), which is the only non-axisymmetric model in
our comparison.

To estimate the effect of using different models of the Galactic potential, we
calculate the relative difference with respect to the reference model Ia. Our
results show clearly that the difference in minimum distance strongly depends
on the time that is necessary for a star to travel from the current position
to the closest approach point. It is a combination of effects due to the
velocity of the star and its current distance from the Sun. In Fig. 4 we show
the dependence of the relative difference on the proximity distance obtained
with four tested models with respect to this time interval. The largest
obtained disparity is for star P0504, the minimum distance obtain from
McMillan model is 0.1318 pc; whereas from model Ia, it is only 0.01154 pc, for
model Ib is 0.08831 pc, and for models IIIa and IIIb, respectively, it is
0.05533 pc and 0.06531 pc. We see that the maximum difference is about 0.12
pc, while the uncertainty of D${}^{nom}_{min}$ resulting from the stellar
astrometry errors is about 5.9 pc. We found only one more object that has a
relative error bigger than 100 percent, namely P5001 which is a binary star.
It is currently 517 pc from the Sun and the close encounter will take place in
8.7 Myr. Almost 90% of stars in our list have a relative error smaller than
10%. For example, P2002 has a relative difference, for the McMillan model, of
1.38 percent, and for models Ib, IIIa, and IIIb, respectively, it is 0.41,
0.94, and 0.02 percent.

Figure 5:  First four stars from Table 3: Comparison of their nominal minimal
distances from the Sun and the spread of VSs at their closest approach. Black
dots in the centre of each VSs cloud mark the nominal star position at the
closest Sun–star approach; see text on how this composite figure was obtained.
Label meanings: A: P0509, B: P0230 (HD~7977), C: P0506, D: P0508; see Table 3
for details on each star. The blue circle marks the usually adopted outer
limit of the Oort cloud and the cross in the middle marks the Sun’s position.
Figure 6: Distribution of P2001 VSs stopped at the closest approach to the Sun
and projected on the plane of their maximum scatter. The blue circle
represents the boundary of the Oort cloud at 0.5 pc, a red cross is the
position of the Sun and the blue dot (very close to the red cross) corresponds
to a nominal position of P2001. There are 2655 VSs shown from among 10000
calculated, while the remaining ones are beyond of the image border.

Figure 7: Spatial distribution of 10000 VS of P2004 at the moment of the
closest approach in the plane of their maximum scatter.

Blue dot represents the nominal position of the star at the closest proximity
and red cross represents the position of the Sun. The blue circle corresponds
to a boundary of the Oort Cloud at 0.5 pc. Omitted: 3974 VSs that are too far
from the Sun to be plotted.

##  4 Stellar close flybys near the Sun based on Gaia EDR3

For the StePPeD release 3.2 described in this paper, we prepared a
significantly updated list of potential perturbers. First, based on our
experience with cometary calculations (see e.g. Dybczyński & Breiter, 2022;
Dybczyński & Królikowska, 2022), we decided to temporary limit the list of
stars and stellar systems to those that nominally pass the Sun closer than 2
pc and ignoring their uncertainties. It appeared that more distant perturbers
practically do not disturb the LPC motion in a noticeable way. We
systematically (at each StePPeD update) repeated a scanning of the
calculations for the observed LPCs to check which stars perturb their motion
in a noticeable manner. The last result of such a test can be found in
Dybczyński & Breiter (2022, see their Table 2).

Moreover, due to continuous high levels of uncertainties in some stellar data,
in many cases, using a large list of potential perturbers seems inefficient
and inappropriate at the moment. We also decided to temporary omit the stars
that nominally pass close to the Sun but due to the current large distance
from the Sun (or the low velocity) they must spend a long time to travel
between their current position and the closest approach point, which strongly
increases the propagation of their uncertainties.

Using the minimum distance threshold of 2 pc and temporarily excluding some
promising stars on the basis of the subjective rank of the uncertainties
(described above), we shortened the whole list to 155 objects. It consists of
142 stars (including seven binaries) from the previous list updated with the
data from Gaia EDR3 and 11 new single stars and two binaries added on the
basis of the new Gaia EDR3 data.

In Table 3, we present ten of the closest past encounters based on current
stellar data. In Table 4, we present ten of the closest future flybys. In the
first two columns, we include our internal identifier and Gaia EDR3 number. We
describe the minimal Sun–star distance in two different ways. The value
presented in the third column, D${}_{min}^{geom}$, is calculated in a special
way: this is the distance from the Sun to the centroid of a cloud of 10 000
VSs drawn according to the data uncertainties and using the respective
covariance matrix. As its uncertainty, we present a radius of a sphere around
that centroid, which includes 68% of VSs. Uncertainties greater than the
D${}_{min}^{geom}$ directly indicate that the VSs cloud surrounds the Sun (see
for example Table 3, col. 3 for P0230 and its VSs cloud in Fig. 5). In the
fourth column, we present a formal statistical description of the Sun–star
distance set for all VSs (D${}_{min}^{stat}$) using the Gaussian mean and
1$\sigma$ uncertainty or the median and the upper and lower limits calculated
on the basis of percentiles: 16% and 84% in the cases where the distribution
is non-Gaussian. All these values are also expressed in parsecs. In exactly
the same way, we present the epoch of the closest approach T${}_{min}^{stat}$
(the fifth column, in Myr) and the relative velocity during the flyby
V${}_{rel}^{stat}$ (the sixth column, in km s-1. The last column in both
tables presents the mass estimate.

In the case of P0403 in Table 3, we use as the initial conditions for the
numerical integration the data from Dupuy et al. (2019) and due to the lack of
the covariance matrix the drawing VSs technique was not possible. We just
quoted here the uncertainties from this paper where available.

The comparison of the four representative examples of the VS clouds dispersion
is presented in Fig. 5. The first four stars from Table 3 are included in this
figure. The centroids of their corresponding clouds are shown, aligned along
the horizontal line keeping the correct nominal distance from the Sun and
maintaining the same scale of the VS clouds spread. It should be stressed that
the four stellar close approaches to the Sun presented in Fig. 5 took place in
different epochs spread over a 2.5 Myr interval (see the sixth column of Table
3). Without any doubt, the most important star in our new list is P0230 (HD
7977). It has moderately precise astrometric and kinematic data and
RUWE=2.015. We discuss its very close flyby near the Sun in the next section.

In comparison with the StePPeD release 2.3, several important changes in the
list of the closest stellar encounters to the Sun have appeared. What is
probably the most important is the complete removal of the perturber P1079
(ALS 9243), discussed in detail in WPD20. Its parallax changed from
10.5617$\pm$0.4027 to 0.16891$\pm$0.19025 mas, which moved this star to a
large distance, additionally with a high uncertainty. The closest star-Sun
approach of P0505 ( Gaia~DR2 955098506408767360) in the past is also cancelled
due to the parallax change from 34.5051$\pm$0.6146 to 1.53497$\pm$0.02945 mas.
These two stars were the most important perturbers in the StePPeD release 2.3
from the point of view of a dynamical evolution of LPCs, based on the Gaia DR2
results.

Table 3: Ten past closest stellar passages near the Sun. Column descriptions are the same as in Table 1.  StePPeD |  Gaia EDR3 | geometry | statistics | mass estimate
---|---|---|---|---
ID | ID | D${}^{geom}_{min}$ [pc] | D${}^{stat}_{min}$ [pc] | T${}^{stat}_{min}$ [Myr] | V${}^{stat}_{rel}$ [km s${}^{-1}]$ | [M⊙]
P0230 | 510911618569239040 | 0.014$\pm$0.040 | $0.032^{+0.027}_{-0.026}$ | $-2.471^{+0.026}_{-0.026}$ | $30.65\pm 0.29$ | 1.08
P0506 | 5571232118090082816 | 0.199$\pm$0.008 | 0.199 $\pm$ 0.006 | $-1.084\pm 0.004$ | 90.16 $\pm$ 0.32 | 0.77
P0508 | 2946037094762449664 | 0.264$\pm$0.376 | $0.385^{+0.205}_{-0.177}$ | $-0.978^{+0.080}_{-0.094}$ | $39.95^{+1.89}_{-1.92}$ | 0.25
P0509 | 52952720512121856 | 0.318$\pm$0.153 | $0.334^{+0.099}_{-0.095}$ | $-0.670^{+0.050}_{-0.057}$ | $31.25\pm 1.43$ | 1.46
P0403 | 3048443305671969152 | 0.335 | 0.333 $\pm$ 0.010 |  $-0.081\pm$ 0.001 | 82.5 | 0.16
P0533 | 3118526069444386944 | 0.504$\pm$0.155 | $0.510^{+0.120}_{-0.118}$ | $-3.179^{+0.085}_{-0.088}$ | $41.24\pm 1.01$ | 0.87
P0417 | 1281410781322153216 | 0.514$\pm$0.018 | 0.514 $\pm$ 0.013 | $-1.471\pm 0.003$ | 32.7338 $\pm$ 0.0006 | 0.85
P0514 | 6608946489396474752 | 0.572$\pm$0.034 | 0.572 $\pm$ 0.027 | $-2.751^{+0.036}_{-0.036}$ | 46.00 $\pm$ 0.58 | 0.75
P0524 | 1949388868571283200 | 0.621$\pm$0.028 | $0.621^{+0.020}_{-0.018}$ | $-0.672^{+0.014}_{-0.015}$ | $355.2\pm 7.0$ | 0.70
P0522 | 5261593808165974784 | 0.624$\pm$0.009 | $0.624^{+0.009}_{-0.008}$ | $-0.896^{+0.011}_{-0.011}$ | 72.71 $\pm$ 0.90 | 0.55
Table 4: Ten future closest stellar passages near the Sun. Column descriptions are the same as in Table 1.  StePPeD |  Gaia EDR3 | geometry | statistics | mass estimate
---|---|---|---|---
ID | ID | D${}^{geom}_{min}$ [pc] | D${}^{stat}_{min}$ [pc] | T${}^{stat}_{min}$ [Myr] | V${}^{stat}_{rel}$ [km s${}^{-1}]$ | [M⊙]
P0107 | 4270814637616488064 | 0.052$\pm$0.003 | 0.052 $\pm$ 0.002 | 1.290 $\pm$ 0.001 | 14.800 $\pm$ 0.006 | 0.65
P0551 | 1422721321394307456 | 0.173$\pm$0.313 | $0.279_{-0.136}^{+0.180}$ | $6.240_{-0.210}^{+0.228}$ | 39.27 $\pm$ 1.33 | 0.82
P2002 | 911145876981562496 | 0.405$\pm$0.266 | $0.448_{-0.151}^{+0.155}$ | $3.736_{-0.100}^{+0.103}$ | 31.27 $\pm$ 0.62 | 0.66
P0416 | 1952802469918554368 | 0.482$\pm$0.037 | $0.480_{-0.035}^{+0.040}$ | $0.070_{-0.005}^{+0.006}$ | $100.88_{-7.71}^{+7.90}$ | 0.20
P0414 | 729885367894193280 | 0.511$\pm$0.401 | $0.301_{-0.163}^{+0.418}$ | $0.505_{-0.183}^{+0.570}$ | $95.69_{-50.45}^{+53.14}$ | 0.08
P0504 | 4535062706661799168 | 0.548$\pm$5.923 | $4.375_{-2.295}^{+3.549}$ | $5.847_{-1.048}^{+1.494}$ | 29.69 $\pm$ 3.97 | 0.70
P0618 | 213090546082530816 | 0.672$\pm$0.262 | $0.711_{-0.147}^{+0.139}$ | $8.224_{-0.606}^{+0.730}$ | 23.75 $\pm$ 1.96 | 0.91
P0318 | 5469802896279029504 | 0.740$\pm$0.451 | $0.820_{-0.093}^{+0.179}$ | $13.763_{-1.141}^{+1.371}$ | $2.65_{-0.21}^{+0.21}$ | 0.65
P0520 | 1722612190157591680 | 0.757$\pm$0.157 | $0.750_{-0.130}^{+0.150}$ | $4.485_{-0.172}^{+0.189}$ | 37.84 $\pm$ 1.50 | 0.60
P0567 | 4536673181955253504 | 0.763$\pm$0.226 | $0.772_{-0.143}^{+0.156}$ | $1.335_{-0.069}^{+0.074}$ | 73.70 $\pm$ 2.35 | 0.60

For example, from among new stars, the closest nominal pass near the Sun we
obtained for P2001 (Gaia EDR3 4312257326836128896). It has a value of
$D_{min}^{stat}=0.26$ pc, while the minimal distance from a linear
approximation is $D_{min}^{lin}=53.12$ pc. Its present distance from the Sun
is 166.28 pc. Despite its rather good astrometric data ($RUWE=0.80$), we
cannot determine the flyby distance near the Sun reliably. The reason for that
is the combination of the present distance of P2001 and its small radial
velocity ($4.64$ $km\,s^{-1}$). The resulting journey time from the present
position to the closest approach to the Sun becomes $45.86$ Myr. The
uncertainties propagation on such a long interval of a numerical integration
makes the final result highly uncertain. Figure 6 demonstrates the lack of
reliable information on the close approach of this star. Green dots show the
VSs at their closest position to the Sun. Due to the unreliable results, this
star was removed from the final list.

On the other hand, we found several promising objects, but they are
characterised by a high level of inaccuracy on the part of their astrometry,
which is reflected by RUWE parameter. For example, P2004 (Gaia EDR3
3341225866116778880) displays $D_{min}^{stat}=0.94$ pc and
$T_{min}^{stat}=-9.06$ Myr. It has $RUWE=13.22$ and its current distance is
$373.33$ pc. As a result, the distribution of P2004 VSs shows that we cannot
determine this star flyby near Sun in a reliable manner (see Fig. 7).

##  5 Important case of P0230

In the StePPeD release 2.3, based on Gaia DR2, we obtained a minimal distance
between P0230 (HD 7977) and the Sun equal to 0.4 pc. The most important
difference between DR2 and EDR3 astrometric data for this star appeared in the
right ascension component of the the proper motion. In DR2, it was
0.559$\pm$0.040 mas yr-1; however, in EDR3 it, becomes 0.144$\pm$0.024 mas
yr-1. The reason for such a substantial change could be attributed to the
binarity of the star. However, we could not find any clues in this regard in
any catalogue or other source. All uncertainties in EDR3 are smaller than
those in DR2 and RUWE is only 2.015. Thus, it is clear that we should rely on
EDR3 data for the moment.

According to the new nominal data, P0230 passed the Sun 2.5 Myr ago at an
extremely small distance of 0.014 pc ($\sim$3000 au). However, the
uncertainties of its astrometric and kinematic data result in a considerable
spread of its VS cloud that surrounds the Sun (as depicted in Fig. 8).

There is a very important qualitative change resulting from the reduction of
star-Sun nominal minimal distance from 0.4 pc to 0.014 pc. With the larger
distance, passing near the outer border of the Oort cloud, this star can
become an significant perturber only for some LPCs that appear near the
stellar trajectory. Such a comet will be near its aphelion, so the orbit
change might be spectacular. One such case of the comet C/2002 A3 has been
described in detail in Wysoczańska et al. (2020a), based on Gaia DR2 data for
P0230.

When a star passes as close as a few thousand astronomical units (or closer)
to the Sun, the situation is different. In this case, the main perturbation is
gained by the Sun and, as a result, its Galactic trajectory and its velocity
are changed. This changes will be reflected in all heliocentric orbits of all
Solar System bodies but on different levels. As we show in the next section,
bodies in smaller orbits (e.g. planets) are practically not affected, but all
LPCs on their highly elongated orbits would be strongly perturbed. The example
cases of C/2014 UN271 and C/2017 K2 were recently discussed by Dybczyński &
Królikowska (2022).

Comparing this very close passage to that of the Scholz's star (P0403 in
StePPeD), discovered by Mamajek et al. (2015), we can list several important
differences. P0230 has passed ten times closer, with a much smaller relative
velocity and it is considerably more massive. All these properties make P0230
appear a much more efficient comet perturber than the Scholz’s star. Moreover,
the passage of the Scholz’s star appeared very recently (0.08 Myr ago) and if
this star produced observable comets, we have to wait for them for thousands
of years in the future. Conversely, P0230 passed 2.5 Myr ago (according to our
results) and a lot of observed LPCs might have experienced a strong orbit
change caused by this star (see e.g. Dybczyński & Królikowska, 2022).

Figure 8: Closest pass of P0230 and its 10 000 VSs near the Sun projected onto
the VS maximum scatter plane. Red dot represents the Sun position, while the
green one shows the nominal P0230 closest position. The blue circle depicts
the 0.5 pc outer border of the Oort cometary cloud.The statistical description
of the VS cloud can be found in Table 3.

##  6 Question of rejecting the possibility of such a close stellar passage

It is unlikely for stars to have passed too close to the Sun in the past,
because they would have disturbed the planetary orbits. The closest acceptable
star-Sun distance has been discussed in many papers. For example Morbidelli &
Levison (2004), following Ida et al. (2000), discussed the passage of a solar
mass star at a distance of 100 – 200 au as the scenario of the origin of a
strange orbit of Sedna. In another study, Adams et al. (2006) estimated that a
passage at a distance of 700-4000 au would not be disruptive for a planetary
system. But in a review paper on cometary dynamics by Dones et al. (2015),
these authors stated that a stellar passage at 400 au from the Sun would
excite the Neptune eccentricity above 0.1, while a passage at 200 au from the
Sun could have ejected Neptune from the Solar System.

Taking the parameters of the P0230 nominal encounter with the Sun and the
estimate of its mass, we decided to perform a series of simple tests to check
how close to the Sun P0230 could have passed. We should keep in mind that due
to the current P0230 data and their uncertainties the closest star-Sun
distance could be arbitrarily small (see Fig. 8).

Figure 9: Effect of close flybys on orbits of Neptune and Pluto. Each point in
the plots represents a single flyby of P0230 like star with different ecliptic
plane crossing point.

Figure 10: Effect of flybys on orbits of synthetic Kuiper Belt Object population. The initial orbits (top left) were simulated under gravitational effects of giant planets and Pluto without passing star (top right). Then we tested the same population adding flyby of P0230 at four different distances and plot differences in orbital elements caused by flyby compared to no flyby simulations (middle and bottom row, see the text for detailed descriptions). Table 5: Effect of a passing star on the Solar System planets. The simulation spans 100 000 years; a star passes its perihelion after 50 000 years of simulation. In the consecutive columns we show: initial value of orbital elements and final values in three cases: with no passing star, with star passing at nominal distance (2 940 au from Solar System barycenter) and with star passing ten times closer (294 au).  object |  | initial | after $10^{5}$ years
---|---|---|---
|  |  | no star | nominal | dist./10
| a | 5.162 | 5.1774475 | 5.1774485 | 5.1816781
Jupiter | e | 0.044 | 0.0386616 | 0.0386610 | 0.0380938
| i | 1.30 | 1.2510514 | 1.2510599 | 1.2524185
| a | 9.5258 | 9.5276163 | 9.5276170 | 9.5292422
Saturn | e | 0.056 | 0.0771802 | 0.0771808 | 0.0773365
| i | 2.488 | 2.4780500 | 2.4781168 | 2.4849129
| a | 19.1848 | 19.187124 | 19.187124 | 19.188364
Uranus | e | 0.047 | 0.0313471 | 0.0313486 | 0.0321796
| i | 0.77 | 1.3070023 | 1.3073480 | 1.3390967
| a | 30.066 | 30.058894 | 30.058895 | 30.080055
Neptune | e | 0.0086 | 0.0117391 | 0.0117399 | 0.0128642
| i | 1.77 | 1.8837079 | 1.8843004 | 1.9093493
| a | 39.49 | 39.50569 | 39.50571 | 39.88493
Pluto | e | 0.2490 | 0.2468883 | 0.2469112 | 0.2437073
| i | 17.14 | 17.260103 | 17.260642 | 17.272419

To estimate the possible impact of the flyby as close as the nominal distance
of P0230 on the Solar System, we used the Rebound N-body integrator software
(Rein & Liu, 2012). In our simulation we created a seven-body system with the
Sun, four giant planets, Pluto, and a passing star. For initial positions and
velocities we used a present-day Solar System JPL Horizon ephemeris and we
added a passing star with the mass equal to the estimated P0230 mass (1.08
M⊙). We choose a velocity value during the flyby that is equal to the value
calculated for the nominal of P0230 and set the initial position of the star,
so that it crossed the ecliptic plane halfway through the simulation duration.
The first tests showed that the flyby at nominal distance has negligible
effect on the giant planets and the impact on the Pluto orbit is also small
(see Table 5). The perturbation in orbital elements for Solar System objects
created by star gravity is much smaller than the change in orbital elements
caused by gravitational interactions between the giant planets. This results
shows that such a close flyby is quite possible and would not affect the
dynamics of planet in a significant way. With this result, we decided to check
the impact of even closer stellar flybys to determine the minimum realistic
distance at which this star could pass without disrupting our planetary
system. We set the position of the star at the ecliptic plane at specific
$(x,y,0)$ coordinates and moved it backward assuming a Keplerian hyperbolic
orbit. We used the resulting position as the initial data in the simulation.
We simulated the flybys with ecliptic crossing coordinates $x$ and $y$ ranging
from $-500$ to $500$ au, creating the grid with step of $10$ au, excluding
very close passages when a distance from the Solar System barycenter during
ecliptic plane crossing is smaller than $50$ au. The obtained final
eccentricity and semi-major axis of Neptune and Pluto are shown in Fig. 9.
These results show that even very close flybys are possible without any
resulting disruption with regard to the orbits of the outer Solar System
objects. We can assume that the safe minimum distance between the P0230 and
the Solar System barycenter (measured when the star crosses the ecliptic
plane) is about $300$ au – ten times closer than the nominal value we obtained
for P0230. At such a distance, the flyby causes the difference in the semi-
major axis below 0.1 au when compared with the simulation without a passing
star. The corresponding difference in eccentricity is smaller than 0.01. It is
also worth noting that even a flyby at a smaller distance can be quite safe
when occurring within the bounds of a favourable geometry.

During the simulation the Pluto was used as a simple indicator of the effect
of P0230 on Kuiper Belt Objects (KBOs), but we decided to check more carefully
the impact of such a stellar passage on the KBO population. To this aim, a
randomised set of massless objects with a semi-major axis between 30 and 100
au on near circular orbits (eccentricities $e<0.1$) with small inclinations to
the ecliptic plane ($i<10^{\circ})$ generated using the uniform distribution
of the orbital parameters was added to the simulation. This creates a
synthetic disc representing the Kuiper Belt. As a point of reference the
initial population (top left of Fig. 10) was first simulated under
gravitational effects of giant planets and Pluto and the resulting population
is shown in top right of Fig. 10. We note that some of the objects with a
small semi-major axis increased their eccentricity even when the passing star
was absent. This is the effect of orbital resonances or a close encounter with
Pluto.

After this initial test, we simulated the same synthetic population, but we
added the P0230 star and plotted the difference in orbital elements between
the simulations with and without flyby. We used nominal encounter scenario
(middle left of Fig. 10) and then we modified it multiplying the nominal
minimal distance to the Sun by 0.5 (middle right of Fig. 10), 0.2 (bottom left
of Fig. 10), or 0.1 (bottom right of Fig. 10). The results shows that for
majority of population the change in semi-major axis is smaller than 1 au and
the change in eccentricity is smaller than 0.01 when the flyby distance is
greater than 600 au. Even the closest approach we tested at about 300 au is
not enough to affect the population in a significant way ($75.6$% of the
objects have a difference in the semi-major axis less than 2 au and less than
0.02 in eccentricity).

These simulations show that objects similar to P0230 could possibly fly near
the Sun at a distance of about $3000$ au without creating a noticeable
disturbance. Very close flybys (at a distance smaller than $500$ au) are also
possible, however, the precise determination of their impact on Solar System
will require more careful investigation considering the long-term Solar System
stability and statistical properties of minor body population. Such an
investigation is beyond the scope of this article.

##  7 Other important potential perturbers in our list

In addition to P0230 among the past close approaches to the Sun we should
mention P0506, P0508, P0509, and P0417 as the stars that were recognised as
the frequent perturbers of the observed LPCs past motion (Dybczyński &
Breiter, 2022). While for P0506 and P0417, we have quite precise stellar data,
the uncertainties of P0509 are unsatisfactory large and of P0508 make its
influence on comets practically undetermined (Dybczyński & Królikowska, 2022).
We refer to Fig. 5 for more details as well. We should note here that both
P0508 and P0509 are absent in Gaia EDR3, so we copied their astrometry from
DR2.

Among the stars that will make the closest pass near the Sun in the future the
record-holder is still P0504. This star is absent in Gaia EDR3 and we use DR2
astrometry. Its nominal smallest distance is 0.012 pc ($\sim$2400 au) but this
result is highly unreliable. The cloud of its VSs is so great that that to
enclose 68% of them, we have to use a radius of almost 6 pc. The cloud is
slightly asymmetric, which causes its centroid to be 0.55 pc from the Sun and
the close passage parameters strongly dependent on the applied model of a
Galactic potential. Detailed statistics of this approach can be found in the
first row of Table 4.

In the second row of this table one can found parameters of the close passage
of the well known star Gliese 710 (P0107 in StePPeD). This star was recognised
many years ago as approaching closely to the Sun in the future, see for
example Mullari & Orlov (1996); Dybczyński & Kankiewicz (1999); García-Sánchez
et al. (2001). The parameters of its predicted close passage were updated
based on the Gaia DR1 catalogue in Berski & Dybczyński (2016). In the StePPeD
version 2.3, based on Gaia DR2, they obtained a passage at 0.054 pc in 1.29
Myr. In the present paper the updated values are almost identical, only the
minimum distance decreased to 0.052 pc. This star has the smallest
uncertainties among the studied stars so precise calculations of its future
influence on cometary orbits can be performed (see e.g. Dybczyński &
Królikowska, 2022).

In our current list, there is also another very promising candidate for a
significant comet motion perturber, namely, P0551. According to the available
data, it will pass near the Sun nominally at a distance below 0.2 pc in 6 Myr,
however, the uncertainties of its astrometry are far too large to treat this
result as definitive. It would be also valuable to obtain more precise radial
velocity for another candidate, P0414; however, its small mass makes it rather
marginally important from a point of view of perturbing comet motion. Another
interesting case is P0318. It will pas the Sun in over 13 Myr but a very good
astrometry and precise radial velocity allowed us to obtain rather reliable
parameters of its future approach to the Sun.

##  8 Summary, conclusions, and prospects

In this paper, we describe several aspects of the update of the StePPeD
database of the potential comet motion perturbers accounting for the Gaia EDR3
data. The first step was to update parameters of the close approaches to the
Sun for all stars included in the previous StePPeD releases. We use the
minimal star-Sun distance as a filtering tool when searching for potential
stellar perturbers. It appeared that significant changes in astrometric data
for a large percentage of these stars resulted in the exclusion of over one
third of the total from further consideration.

In this situation, it was necessary to search the whole Gaia EDR3 catalogue
for new candidates. Basing on our previous experience, we decided to repeat
the whole procedure described in WDP20 but with one important change: using
the linear approximation as the first filter, we allow for a much greater
star-Sun distance than the 10 pc threshold used in some previous papers
(Dybczyński & Berski, 2015; Bailer-Jones et al., 2018). This allows us to find
a lot of potential new candidates. After the second filtering, by means of the
numerical integration of each star with the Sun under the Galactic potential,
we obtained over 500 stars passing closer than 5 pc from the Sun and more than
half of them have a minimal distance obtained from a linear approximation
larger than 10 pc.

Next, we closely examined each star from this shorter list searching for
probable binary or multiple system. To this aim, we mainly used data from El-
Badry et al. (2021) but we also manually searched for secondaries directly in
the Gaia EDR3 catalogue. More than 10% of the stars appeared to be members of
multiple systems, which makes it necessary to calculate their centre of mass
motion and check again for their proximity to the Sun.

Using the very rich material presented in El-Badry et al. (2021), we also
decided for the first time to search for binaries overlooked in all earlier
papers on this subject because none of their components, treated as a single
star, did not appear in the list of stars approaching the Sun. During this
first attempt, we isolated over 15 000 systems for which the radial velocity
of both components is known. Then we filtered this list by checking whether it
is possible to obtain a close passage near the Sun for some mass ratio between
the components. This left only 156 candidates on our list. For these, we
searched for available mass estimates and we analysed in detail the
possibility of the close approach by varying the component masses within their
confidence intervals. This complicated and time-consuming procedure resulted
in only two new potential perturbers added to the final list.

We also studied how our results might depend on the chosen model of the
Galactic potential. Our test with five other models show that for the majority
of studied objects the differences are below 10%.

When preparing the final list of potential stellar perturbers of the LPCs
motion we decided to apply the new and more restricted threshold of 2 pc as
the maximal allowed distance from the Sun at the moment of the closest
approach. This decision resulted from several cometary orbital calculations
showing that more distant passages are rather irrelevant, (see e.g. Dybczyński
& Breiter, 2022; Dybczyński & Królikowska, 2022). Using this new threshold, we
compiled stars from the previous list, new single stars, and new multiple
systems and we obtained a list of 155 potential perturbers for the new release
3.2 of the StePPeD database.

In this new release, apart from the large amount of information previously
presented, we contribute two important new results: for all objects, we
calculated the uncertainty of the parameters of the close approach to the Sun
and we include a graphical presentation of this uncertainty by means of the
picture of 10 000 VSs of the star in question drawn according from the
appropriate covariance matrix and stopped at their closest approach to the
Sun. All the presented results are based on the Gaia EDR3 data (where
available) and all other information (e.g. mass estimates) are also updated to
accommodate the latest published values.

In the new list of potential perturbers of the LPCs motion, we found several
interesting objects. The most important seems to be the star HD 7977 (P0230 in
the StePPeD database) which nominally passed the Sun 2.5 Myr ago at an
extremely small distance of 0.014 pc ($\sim$3000 au). We performed several
simple tests checking the effect of such a close star passage on the outer
planetary system bodies. It seems that the outer Solar System planets are save
even during much closer passages but LPCs were all strongly perturbed. A
similar situation will occur in the future during the passage of Gliese 710
(P0107).

Concerning the completeness of the StePPeD database, we know that there are
missing stars in Gaia EDR3, both faint red and highly luminous. For a number
of years, we monitored the Simbad database and Vizier catalogues for stars
with known parallax and radial velocity and checked all new objects found.
Generally we are not particularly interested in the smallest objects (e.g.
brown dwarfs) because their small masses make them very inefficient perturbers
of cometary motion. Instead we carefully check all massive stars absent in the
Gaia catalogue, for example, we carefully studied the ’missing’ list of stars
presented with the Gaia Catalogue of Nearby Stars (Gaia Collaboration et al.,
2021c).

We are fully aware that a new release of data from the Gaia mission is
expected in a few months. We hope to see a great improvement in the astrometry
precision and a large number of the new radial velocity measurement. The
expected large amount of new data will require a considerable time to obtain
an updated list of potential perturbers. With this in mind, we decided to
prepare and publish the StePPeD update based on the currently available data.
All the methods developed for this purpose and tested practically in WDP20 and
the current paper might be used in the next update iteration.

###### Acknowledgements.

We would like to express our thanks to the anonymous referee for so many so
detailed and so helpful suggestions and comments. These allowed us to greatly
improve this paper. The calculations which led to this work were partially
supported by the project ”GAVIP-GC: processing resources for Gaia data
analysis” funded by European Space Agency (4000120180/17/NL/CBi).

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# A short proof of the chromatic Smith Fixed Point Theorem.

Nicholas J. Kuhn<EMAIL_ADDRESS>Department of Mathematics
University of Virginia
Charlottesville, VA 22903

(Date: December 9, 2021.)

###### Abstract.

We give a short and much simplified proof of the main theorem of the recent
study, by T. Barthel, M. Hausmann, N. Naumann, T. Nikolaus, J. Noel, and N.
Stapleton, of the Balmer spectrum for $A$–equivariant stable homotopy when $A$
is a finite abelian $p$–group. This theorem says that if $A$ is a finite
abelian $p$-group of rank $r$, and $X$ is a finite $A$–space that is acyclic
in the $(n+r)$th Morava $K$–theory, then its space of fixed points, $X^{A}$,
will be acyclic in the $n$th Morava $K$–theory. It is a chromatic homotopy
version of P. A. Smith’s classic theorem about the mod $p$ homology of the
fixed points of a finite $A$–space.

###### 2010 Mathematics Subject Classification:

Primary 55M35; Secondary 55N20, 55P42, 55P91.

##  1\. Introduction

If $G$ is a finite group, say that $G$–space $X$ is a finite $G$–space if it
is a retract of a finite $G$–CW complex in the $G$–equivariant homotopy
category. We let $X^{G}$ denote its subspace of fixed points.

The point of this note is to give a very short and simplified proof of the
main theorem of [6A19], which can be stated as follows.

######  Theorem 1.1.

Let $A$ be a finite abelian $p$–group of rank $r$, and let $X$ be a finite
$A$–space. If $\widetilde{K}(n+r)_{*}(X)=0$ then
$\widetilde{K}(n)_{*}(X^{A})=0$.

Here $K(n)_{*}$ denotes the $n$th Morava $K$–theory at the prime $p$.

A number of remarks are in order.

######  Remark 1.2.

(a) A finite space is acyclic in mod $p$ homology if and only if it is acyclic
in $K(n)_{*}$ for all large $n$. Thus Theorem 1.1 (specialized to $A=C_{p}$
and then iterated) both implies and generalizes a classical theorem of P. A.
Smith [S41]: if $G$ is a finite $p$–group and $X$ is a finite $G$–space, then
$\widetilde{H}_{*}(X;{\mathbb{Z}}/p)=0$ implies that
$\widetilde{H}_{*}(X^{G};{\mathbb{Z}}/p)=0$.

(b) [6A19, Theorem 2.1] (or, when $A=C_{p}$, [BS17, Proposition 7.1]) is
stated in terms of the geometric fixed point functor applied to a compact
object in $A$–spectra, but is easily seen to be equivalent to the theorem as
stated above. (For more detail, see [KL20, §3.1].)

(c) By iteration, the theorem for cyclic $p$–groups implies the theorem for
abelian $p$–groups, however the proof seems no easier when specialized to the
cyclic case. More generally, iterated use of the cyclic case of the theorem
implies the following theorem: whenever $H$ is a subgroup of a finite
$p$–group $G$ and $X$ is a finite $G$–space,

$\widetilde{K}(n+r)_{*}(X^{H})=0\Rightarrow\widetilde{K}(n)_{*}(X^{G})=0$

where $r$ is minimal such that there exists a sequence of subgroups
$H=K_{0}\lhd K_{1}\lhd\dots\lhd K_{r}=G$ with each $K_{i-1}$ normal in $K_{i}$
and $K_{i}/K_{i-1}$ cyclic. (Again, this is discussed in more detail in
[KL20].)

(d) [KL20, Theorem 2.2] goes as follows: given a finite $p$–group $G$, $H<G$,
$m$, and $n$ such that, for all finite $G$–spaces $X$,

$\widetilde{K}(m)_{*}(X^{H})=0\Rightarrow\widetilde{K}(n)_{*}(X^{G})=0,$

then, if $X$ is any finite $G$–space, one has

$\dim_{K(m)_{*}}K(m)_{*}(X^{H})\geq\dim_{K(n)_{*}}K(n)_{*}(X^{G}).$

Combined with Theorem 1.1 and specialized to $G=C_{p^{k}}$, one deduces that

$\dim_{K(n+1)_{*}}K(n+1)_{*}(X)\geq\dim_{K(n)_{*}}K(n)_{*}(X^{C_{p^{k}}}),$

which both implies and generalizes a classic theorem of E.E.Floyd [F52]: if
$X$ is a finite $C_{p}$–space, then

$\dim_{{\mathbb{Z}}/p}H_{*}(X;{\mathbb{Z}}/p)\geq\dim_{{\mathbb{Z}}/p}H_{*}(X^{C_{p}};{\mathbb{Z}}/p).$

See [KL20, KL21] for some applications.

One can also use the [KL20] result in contrapositive form, and look for
examples that show that the constraints in remark (c) are best possible. This
is the strategy implemented in [KL20], but whether these constraints are best
is still an open conjecture for many pairs $H<G$. See [KL20] for what is known
about this.

Our proof of Theorem 1.1 makes use of work of Stapleton in [S13], which
generalized the character theory of [HKR00]. Ideas from [S13] are certainly
used in [6A19], but our proof uses this work more explicitly, while avoiding
all discussion of equivariant spectra, geometric fixed point functors,
generalized Tate constructions, and classifying spaces of families, among
other things.

After recalling what we need of Stapleton’s work in §2, we give our proof of
Theorem 1.1 in §3.

The author would like to thank William Balderrama for conversations about this
work.

##  2\. Stapleton’s partial character map

###  2.1. Fixed Point Functors

The character maps of [HKR00] and [S13] both involve certain fixed point
constructions, which we now describe.

If $\Gamma$ is a profinite group and $G$ is a finite group, we let
$\operatorname{Hom}(\Gamma,G)$ denote the set of continuous homomorphisms. $G$
acts on $\operatorname{Hom}(\Gamma,G)$ via conjugation, and we define a
functor

$\operatorname{Fix}_{\Gamma,G}:G\text{--spaces}\rightarrow G\text{--spaces}$

by letting $\operatorname{Fix}_{G}(X)$ be a sub-$G$–space of
$\operatorname{Hom}(\Gamma,G)\times X$ as follows:

$\operatorname{Fix}_{\Gamma,G}(X)=\\{(\alpha,x)\ |\ \alpha(\gamma)x=x\text{
for all }\gamma\in\Gamma\\}\subset\operatorname{Hom}(\Gamma,G)\times X.$

As described in [K89, §6], these functors satisfy various nice properties; in
particular, they are homotopy functors that preserve $G$–equivariant pushouts.
Consequently, if $h^{*}$ is a generalized cohomology theory then

$X\mapsto h^{*}(EG\times_{G}\operatorname{Fix}_{\Gamma,G}(X))$

will be an $G$–equivariant cohomology theory on finite $G$–spaces.

It is useful to write this a bit differently. Let
$\operatorname{Rep}(\Gamma,G)=\operatorname{Hom}(\Gamma,G)/G$. Then there is a
natural homeomorphism of $G$–spaces

$\operatorname{Fix}_{\Gamma,G}(X)=\coprod_{\alpha\in\operatorname{Rep}(\Gamma,G)}G\times_{C_{G}(\alpha(\Gamma))}X^{\alpha(\Gamma)},$

and thus a natural homeomorphism

$EG\times_{G}\operatorname{Fix}_{\Gamma,G}(X)=\coprod_{\alpha\in\operatorname{Rep}(\Gamma,G)}EC_{G}(\alpha(\Gamma))\times_{C_{G}(\alpha(\Gamma))}X^{\alpha(\Gamma)},$

and finally a natural decomposition

$h^{*}(EG\times_{G}\operatorname{Fix}_{\Gamma,G}(X))=\prod_{\alpha\in\operatorname{Rep}(\Gamma,G)}h^{*}(EC_{G}(\alpha(\Gamma))\times_{C_{G}(\alpha(\Gamma))}X^{\alpha(\Gamma)}).$

Now suppose that $\alpha(\Gamma)=G$. Then the corresponding factor in the
product is $h^{*}(BC_{G}(G)\times X^{G})$. We specialize this observation to
the case when $\Gamma={\mathbb{Z}}_{p}^{r}$, the product of $r$ copies of the
$p$-adic integers ${\mathbb{Z}}_{p}$. Since there exists a continuous
surjective homomorphism $\alpha:{\mathbb{Z}}_{p}^{r}\twoheadrightarrow A$ if
and only if $A$ is an abelian $p$–group of rank at most $r$, we deduce the
following.

######  Lemma 2.1.

If $A$ is an abelian $p$–group of rank at most $r$, and $h^{*}$ is a
generalized cohomology theory, then $h^{*}(BA\times X^{A})$ is naturally a
direct summand in
$h^{*}(EA\times_{A}\operatorname{Fix}_{{\mathbb{Z}}_{p}^{r},A}(X))$ for all
$A$–spaces $X$.

###  2.2. Transchromatic characters

We describe what we need from [S13]. It is convenient to fix $n$ and $r$, and
then let $E=E_{n+r}$, the Morava $E$–theory of height $(n+r)$, and
$LE=L_{K(n)}E_{n+r}$, the Bousfield localization of $E_{n+r}$ with respect to
the $r$th Morava $K$–theory.

It is illuminating to recall ([S13, §2.1]) that

$E^{0}=\mathbb{W}({\mathbb{F}}_{p^{n+r}})[[u_{1},\dots,u_{n+r}]]$

and ([H97, Theorem 1.5.4]) that

$LE^{0}=\mathbb{W}({\mathbb{F}}_{p^{n+r}})[[u_{1},\dots,u_{n+r}]][u_{n}^{-1}]^{\wedge}_{I_{n}},$

where $I_{n}=(p,u_{0},\dots,u_{n-1})$.

######  Lemma 2.2.

$LE$ is complex oriented of height $n$. If $A$ is a finite abelian $p$–group,
then $LE^{*}(BA)$ is a finitely generated free $LE^{*}$–module, and thus there
is a Kunneth ismorphism

$LE^{*}(BA)\otimes_{LE^{*}}LE^{*}(Y)\simeq LE^{*}(BA\times Y)$

for all spaces $Y$.

The first of these properties is clear, and the second is well known ([HKR00,
Proposition 5.2]) and implies the last statement.

Stapleton constructs a faithfully flat $LE^{0}$–algebra $C$ and proves the
following theorem.

######  Theorem 2.3.

[S13, Main Theorem] For any finite group $G$ and finite $G$–space $X$, there
is a natural algebra isomorphism

$\chi:E^{*}(EG\times_{G}X)\otimes_{E^{0}}C\xrightarrow{\sim}LE^{*}(EG\times_{G}\operatorname{Fix}_{{\mathbb{Z}}_{p}^{r},G}(X))\otimes_{LE^{0}}C.$

The case $n=0$ (when $LE$ is a rational theory) was shown in [HKR00].

##  3\. Proof of Theorem 1.1

Let $A$ be a finite abelian $p$–group of rank $r$, and let $X$ be a finite
$A$–space. Our goal is to show that if $\widetilde{K}(n+r)_{*}(X)=0$ then
$\widetilde{K}(n)_{*}(X^{A})=0$.

As in §2.2, we let $E=E_{n+r}$, $LE=L_{K(n)}E_{n+r}$, and $C$ be Stapleton’s
flat $LE^{0}$–algebra as in Theorem 2.3.

Our assumption that $\widetilde{K}(n+r)_{*}(X)=0$ is equivalent to saying that
the constant map $X\rightarrow*$ induces an isomorphism

$K(n+r)_{*}(X)\xrightarrow{\sim}K(n+r)_{*}.$

This then implies that the induced map $EA\times_{A}X\rightarrow BA$ induces
an isomorphism

$K(n+r)_{*}(EA\times_{A}X)\xrightarrow{\sim}K(n+r)_{*}(BA).$

This is an immediate consequence of the fact that, given any finite group $G$,
$G$–space $X$, and generalized homology theory $h_{*}$, there is a natural
spectral sequence converging to $h_{*}(EG\times_{G}X)$ with
$E^{2}_{*,*}=H_{*}(BG;h_{*}(X))$.

As $K(n+r)_{*}$–isomorphisms are $E^{*}$–isomorphisms, we conclude that the
map $EA\times_{A}X\rightarrow BA$ induces an isomorphism

$E^{*}(BA)\xrightarrow{\sim}E^{*}(EA\times_{A}X).$

Now consider the commutative diagram

$\textstyle{E^{*}(BA)\otimes_{E^{0}}C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi}$$\textstyle{E^{*}(EA\times_{A}X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\otimes_{E^{0}}C}$$\scriptstyle{\chi}$$\textstyle{LE^{*}(EA\times_{A}\operatorname{Fix}_{{\mathbb{Z}}_{p}^{r},A}(*))\otimes_{LE^{0}}C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{LE^{*}(EA\times_{A}\operatorname{Fix}_{{\mathbb{Z}}_{p}^{r},A}(X))\otimes_{LE^{0}}C.}$

We have just seen that the top map is an isomorphism. So are the two vertical
maps by Theorem 2.3. Thus the bottom map is also an isomorphism.

Thanks to Lemma 2.1, we can conclude that there is an isomorphism

$LE^{*}(BA)\otimes_{LE^{0}}C\xrightarrow{\sim}LE^{*}(BA\times
X^{A})\otimes_{LE^{0}}C$

and, thanks to Lemma 2.2, this can be rewritten as an isomorphism

$LE^{*}(BA)\otimes_{LE^{0}}C\xrightarrow{\sim}LE^{*}(BA)\otimes_{LE^{0}}LE^{*}(X^{A})\otimes_{LE^{0}}C.$

Applying $LE^{*}\otimes_{LE^{*}(BA)}\text{\text@underline{\hskip 7.22743pt}}$
to this, we deduce that there is an isomorphism

$LE^{*}\otimes_{LE^{0}}C\xrightarrow{\sim}LE^{*}(X^{A})\otimes_{LE^{0}}C,$

and since $C$ is a faithful $LE^{0}$–module, we further deduce that there is
an isomorphism

$LE^{*}\xrightarrow{\sim}LE^{*}(X^{A}).$

Finally, since $LE^{*}$ has height $n$, an $LE^{*}$–isomorphism will be a
$K(n)_{*}$–isomorphism, so we conclude that $K(n)_{*}(X^{A})\rightarrow
K(n)_{*}$ is an isomorphism, and thus $\widetilde{K}(n)_{*}(X^{A})=0$.

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