Gaia Search for stellar Companions of TESS Objects of Interest IV

We present the latest results of our ongoing multiplicity study of (Community) TESS Objects of Interest, using astrometric and photometric data from the ESA-Gaia mission to detect stellar companions of these stars and characterize their properties.\linebreak A total of 134 binary, 6 hierarchical triple, and two quadruple star systems are identified among 1106 targets whose multiplicity is investigated in the course of our survey, located at distances closer than about 500pc around the Sun. The detected companions and targets are at the same distance and have a common proper motion, as expected for components of gravitationally bound stellar systems, as demonstrated by their accurate Gaia DR3 astrometry. The companions have masses from about 0.11 to 2$M_\odot$ and are most abundant in the mass range between 0.2 and 0.5$M_\odot$. The companions have projected separations from the targets between about 50 and 9700au. Their frequency is the highest and constant from about 300 up to 750au, decreasing at larger projected separations. In addition to main sequence stars, four white dwarf companions are detected in this study, whose true nature is revealed by their photometric properties.


INTRODUCTION
In 2020 we initiated a new survey at the Astrophysical Institute and University Observatory Jena with the aim to explore the multiplicity of (Community) TESS Objects of Interest ((C)TOIs), i.e. stars photometrically monitored by the Transiting Exoplanet Survey Satellite (TESS, Ricker et al., 2015), which show promising dips in their light curves, possibly caused by exoplanets orbiting these stars.
In our survey, stellar companions of (C)TOIs are detected and their properties are determined with astrometry and photometry, originally from the 2nd data release (Gaia DR2 from hereon, Gaia Collaboration et al., 2018) and later from the early version of the 3rd data release (Gaia EDR3 from hereon, Gaia Collaboration et al., 2021) of the ESA-Gaia mission. The first results of the survey were presented by Mugrauer and Michel (2020), Mugrauer and Michel (2021), and Mugrauer, Zander, and Michel (2022), who have already examined the multiplicity of more than 4100 (C)TOIs, all of which are listed in the (C)TOI release of the Exoplanet Follow-up Observing Program for TESS (ExoFOP-TESS) 1 . Meanwhile, several of these (C)TOIs discovered as members of multiple star systems in the course of our survey have already been confirmed as exoplanet host stars by follow-up observations, e.g. TOI 179, TOI 277, TOI 954, TOI 1228, TOI 1246, TOI 1410, TOI 1452, TOI 1516, TOI 1690, TOI 1710, TOI 1749, TOI 1797, TOI 1801, TOI 2152, TOI 2193, and TOI 2459

FIGURE 1
The histograms of the individual properties of the targets in this study. The histograms of distance ( ), total proper motion ( ), and G-band magnitude are based on the Gaia DR3 data of all 1106 targets. Masses and effective temperatures ( ef f ) have been taken from the Starhorse catalog where available, which is the case for 1078 targets.
Thanks to the successful execution of the TESS mission and the photometric analysis of its data, the number of (C)TOIs, and thus the number of targets for our survey, is continuously growing. In this paper, we search for companions of more than thousand (C)TOIs reported by the ExoFOP-TESS, whose multiplicity has not yet been studied in our survey, using the latest data release from the ESA-Gaia mission.
In the following section we describe in detail the properties of the selected targets and the search for companions around these stars. In Section 3 we present all (C)TOIs with detected companions and characterize the properties of these stellar systems. Finally, we summarize the current status of our survey and give an outlook on the project in the last section of this paper.

SEARCH FOR STELLAR COMPANIONS OF (C)TOIS BY EXPLORING THE GAIA DR3
The companion search presented here uses astrometric and photometric data from the 3rd data release of the ESA-Gaia mission (Gaia DR3 from hereon, Gaia Collaboration et al., 2022), obtained with the instruments of the ESA-Gaia satellite during the first 34 months of its mission. This data release contains astrometric solutions, i.e. position ( , ), parallax , and proper motion ( cos( ), ) of about 1.5 billion sources down to a limiting magnitude of 21 mag in the G-band, which means white light observations exploiting the entire spectral sensitivity range of the used CCD detectors.
Parallaxes are measured with an uncertainty in the range of about 0.02 milliarcsec (mas) for bright ( < 15 mag, with a lower limit of ∼ 1.7 mag) up to 0.5 mas for faint ( = 20 mag) detected sources. Proper motions are determined with an accuracy of about 0.02 mas/yr for bright objects, deteriorating to 0.6 mas/yr for = 20 mag. In addition, the G-band magnitude of all sources is recorded with a photometric uncertainty ranging from about 0.3 millimagnitude (mmag) for the brightest to 6 mmag for faint sources.
In the study presented here, stellar companions of the investigated (C)TOIs are firstly identified as sources that are at the same distance as the targets and secondly have a common proper motion with these stars. In order to unambiguously detect co-moving companions and confirm their equidistance to the (C)TOIs, we consider in our study only those sources listed in the Gaia DR3 for which there is a significant measurement of parallax ( ∕ ( ) > 3) and proper motion ( ∕ ( ) > 3). Sources with a negative parallax are ignored.
Since our survey was originally based on Gaia DR2 data, with a typical parallax uncertainty of 0.7 mas for faint sources down to = 20 mag, it is restricted to (C)TOIs within 500 pc of the Sun (i.e. > 2 mas), to ensure that ∕ ( ) > 3 also applies to the faintest detectable companions. This distance constraint is slightly relaxed to + 3 ( ) > 2 mas, i.e. the parallax uncertainty of the (C)TOIs is also taken into account. Although we are now using Gaia DR3 data, which exhibit a smaller parallax uncertainty, we retain the chosen distance constraint for the survey for continuity.
In this paper, we explore the multiplicity of 1106 targets 3 that have not yet been investigated in the course of our survey with Gaia DR3 data, that meet the distance constraint described above and are therefore selected as targets for the study presented here. Figure 1 illustrates the properties of the selected targets with histograms. The distance ( ) and total proper motion ( ) of all targets are determined using their precise Gaia DR3 parallax ( [pc] = 1000∕ [mas]) and proper motion in right ascension and declination. The G-band magnitude of all targets is listed in the Gaia DR3, while their mass and effective temperature ( ef f ) are taken from the Starhorse catalog (SHC from here on, Anders et al., 2019) where available, which is the case for 1078 stars, i.e. the vast majority (∼ 97 %) of all 1106 targets. The targets have distances from the Sun between about 6 to 840 pc.
The distance distribution of the target sample examined here is rather flat compared to the distribution of the distance of the targets investigated earlier in this survey, which shows a peak at about 100 pc. This difference in distance distribution results from the unequal distance distributions of the TOIs and CTOIs in the target sample studied here. While the CTOIs have a peak in their distance distribution at about 100 pc, similar to the targets whose multiplicity was examined earlier in this survey, the TOIs are located at greater distances from the Sun and are most frequently found at about 350 pc. Due to their smaller distance, the CTOIs also appear brighter (by about 1 mag on median) and have proper motions that are on median about 1.5 times higher than those of the TOIs.
The targets studied here have proper motions in the range between about 1 and 1660 mas/yr 4 , G-band magnitudes from 3.4 to 17 mag 5 masses between about 0.14 and 3.8 ⊙ , and effective temperatures ranging from about 2700 up to 10100 K. Based on the cumulative distribution functions of the individual properties, the targets are most commonly located at distances between about 50 and 500 pc have typical proper motions from about 5 to 30 mas/yr, and G-band magnitudes from about = 11 to 14 mag. The targets are mainly solarlike stars with masses in the range between about 0.7 and 1.3 ⊙ . This population is also evident in the ef f -distribution of the targets at intermediate temperatures around 4500 to 6500 K. An additional but weaker clustering of targets is found in this distribution at lower effective temperatures between about 3400 and 4400 K, namely the early M to mid K dwarf population.
As defined and described in Mugrauer and Michel (2020), our survey is restricted to companions with projected separations up to 10000 au, which on the one hand guarantees an effective companion search, but on the other hand also detects the vast majority of all wide companions of the selected targets. This results in an angular search radius for companions around the targets of [arcsec] = 10 [mas], with the Gaia DR3 parallax of the (C)TOIs.
All sources listed in Gaia DR3 that are located within the search radius used around the targets and have a significant parallax and proper motion are considered companion-candidates. A total of about 33000 such objects are detected around 891 targets, whose multiplicity is investigated in this study. The companionship of all these candidates is tested using their precise Gaia DR3 astrometry and that of the associated (C)TOIs, following exactly the procedure described in Mugrauer and Michel (2020). The vast majority of these sources can be excluded as companions because they do not have have a common proper motion with the (C)TOIs and/or are not at the same distance as these stars. In contrast, 149 candidates can be unambiguously confirmed as companions of the (C)TOIs with their accurate Gaia DR3 astrometry. The properties of these companions and the associated (C)TOIs are described in detail in the next section of this paper.

(C)TOIS AND THEIR DETECTED STELLAR COMPANIONS
The mass, effective temperature and absolute G-band magnitude of most of the (C)TOIs with detected companions presented here are listed in the SHC, and we show these stars in a ef f -G -diagram in Figure 2. In this diagram we plot the main sequence of Pecaut and Mamajek (2013) 6 for comparison.
Most targets with detected companions are main sequence stars. Only a few (C)TOIs are (significantly) located above the main sequence, and these stars also have surface gravities of log( [cm∕s −2 ]) < 3.8, as listed in the SHC, and are therefore classified as (sub)giants.
The ef f -G -diagram of (C)TOIs with detected companions presented here. (C)TOIs listed in the SHC with surface gravities log( [cm∕s −2 ]) < 3.8 are shown as red circles, those with larger surface gravities with black circles, respectively. The main sequence is plotted as a grey dashed line.
The parallax, proper motion, apparent G-band magnitude, and extinction estimate of the (C)TOIs and their companions detected in this study are summarized in Table 3, which lists a total of 134 binary, 6 hierarchical triple, and two quadruple star systems.
We determine the angular separation ( ) and the position angle ( ) of all detected companions to the associated (C)TOIs, using the precise Gaia DR3 astrometry of each object. The derived relative astrometry of the companions is listed in Table 4, together with its uncertainty, which remains below about 1.3 mas in angular separation or 0.08 • in position angle. Table 4 also summarizes the parallax difference Δ between the (C)TOIs and their companions, together with its significance -Δ , which is also calculated taking into account the astrometric excess noise of each object. The same table lists for each companion its differential proper motion rel relative to the associated (C)TOI with its significance, and its -7 . The parallaxes of the individual components of the stellar systems presented here are not significantly different from each other ( -Δ < 3) when the astrometric excess noise is taken into account. This clearly proves the equidistance of the 7 The common proper motion ( ) index, as defined in Mugrauer and Michel (2020), characterizes the degree of common proper motion of a detected companion with the associated (C)TOI. detected companions with the (C)TOIs, as expected for components of physically associated stellar systems. All but one of the detected companions have a -≥ 5 and more than 95 % of them even have a -≥ 10, i.e. the detected companions and the associated (C)TOIs clearly form common proper motion pairs, as expected for gravitationally bound stellar systems. Note that for systems with small -and parallax, the possibility of random pairing is increased.
As we use astrometric data from the latest Gaia data release in this study, it is worth noting that the equidistance and common proper motion of all but two of the companions previously detected in this survey using Gaia DR2 astrometry (see Mugrauer & Michel, 2020), are confirmed by Gaia DR3 astrometry. While the -of almost all companions remains constant, the parallax difference between the detected companions and the (C)TOIs is reduced to about half with the more accurate Gaia DR3 astrometry, reinforcing the conclusion that these co-moving companions and the (C)TOIs are components of physically associated stellar systems. For TOI 737 B and TOI 851 B, their companionship could not be tested with Gaia DR3 astrometry because only their equatorial coordinates, but not their proper motion or parallax, are listed in the latest Gaia data release. However, according to their Gaia DR2 astrometry, both companions have a high degree of common proper motion ( -of 29 and 57, respectively), and their differential proper motion does not significantly exceed the expected escape velocity. We therefore conclude that also these two stars are gravitationally bound companions of the associated TOIs.
The absolute G-band magnitudes of all detected companions are from the SHC or, if not available, from the SHC2 (Anders et al., 2022), indicated with the SHC2 flag in Table 5.
The SHC is based on several photometric catalogues as well as on Gaia DR2 astrometry and photometry, the SHC2 on the corresponding data of the Gaia EDR3. Where possible, we use the data from the SHC instead of the SHC2, since the properties of a larger number of detected companions but also (C)TOIs are listed in this catalogue than in the SHC2.
If the absolute magnitude of the companions is not listed in these catalogs, it is derived using the apparent G-band photometry of the companions as well as the parallax of the (C)TOIs, and the Apsis-Priam G-band extinction estimate listed in the Gaia DR2 if available, otherwise the G-band extinction from the SHC. The companion extinction estimate is used if available, otherwise that of the (C)TOIs.
The projected separation of all companions is determined from their angular separation from the associated (C)TOIs and the parallax of these stars.
The mass and effective temperature of the companions presented here, including their uncertainties, are from the SHC or the SHC2 (indicated by the SHC2 flag in Table 5) where TABLE 1 Photometry of three white dwarf companions detected in this study. For each companion we list the color difference Δ( P − P ) and the G-band magnitude difference Δ to the associated (C)TOI, its apparent ( P − P ) color, as well as its derived intrinsic color ( P − P ) 0 and effective temperature ef f . FIGURE 3 This ef f -G -diagram shows all detected companions whose effective temperatures are listed in either the SHC or SHC2, or for which Apsis-Priam temperature estimates are available. Companions, which are the primary components of their stellar systems, are shown as green star symbols. The main sequence is plotted as a dashed grey line for comparison.
available, which is the case for about 73 % of all companions. We plot these companions in Figure 3 in a ef f -G -diagram, along with the companions for which an Apsis-Priam estimate of their effective temperature is available 8 , as indicated by the PRI flag in Table 5. The photometry of all these companions is in good agreement with that expected for main sequence stars.
For the remaining 40 companions their mass and effective temperature are derived from their absolute G-band magnitude by interpolation (inter flag in Table 5) using the G -mass-and Gef f -relation from Pecaut and Mamajek (2013), assuming that these companions are main sequence 8 As recommended by Andrae et al. (2018), we only use Apsis-Priam temperature estimates in this survey if their flags are equal to 1A000E where A and E can have any value.
stars. To test this hypothesis, we compare the obtained effective temperature of the companions either with their Apsis-Priam temperature estimate, if available, or with the effective temperature of the companions inferred from their ( P − P ) color and reddening estimate ( P − P ) 9 , using the ( P − P ) 0ef f -relation from Pecaut and Mamajek (2013).
For all but three of these companions their effective temperature, determined from their absolute magnitude under the assumption that they are main sequence stars, agrees well with their Apsis-Priam temperature estimate or with the temperature derived from their color. The typical deviation of the different temperature estimates is about 430 K, which is in good agreement with the precision of the derived effective temperatures. We therefore conclude that these companions are all main sequence stars.
In addition, we also compare the Gaia DR3 ( P − P ) color of the (C)TOIs and their companions (if any), indicated by the BPRP flag in Table 5. For main sequence stars companions, fainter/brighter than the (C)TOIs, are expected to appear redder/bluer than the stars, and this is true for most of the detected companions, except for TOI 3714 B, TOI 3984 B, and CTOI 13073396 B. These three companions are also observed with the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS) and their y-, i-, g-band color-composite images are shown in Figure 4. In these images the faint companions are clearly visible as bluish sources next to the associated much brighter (C)TOIs.
The photometric properties of these companions are summarized in Table 1. The companions are several magnitudes fainter than the (C)TOIs but appear bluer than these stars. The temperatures of the companions inferred from their colors are significantly higher (by about 2700 to 6900 K) than the temperatures, derived from their absolute G-band magnitudes, assuming that they are main sequence stars. 9 The reddening of an object is estimated from its extinction G using the relation G ∕ ( P − P ) = 1.89 from Wang and Chen (2019).  T eff [K] FIGURE 5 ef f -G -diagram of the stellar systems with white dwarf components detected in this study. The main sequence is shown as a grey dashed line and the evolutionary mass tracks of DA white dwarfs with masses of 0.5 and 0.6 ⊙ as black dotted lines. The primaries of the systems are plotted as white circles, the white dwarf secondaries as black circles, respectively. Figure 5 shows these companions together with the other components of their stellar systems in a ef f -G -diagram. For comparison, we plot the main sequence from Pecaut and Mamajek (2013) in this diagram, and the mass tracks for DA white dwarfs from the Bergeron et al. evolutionary models of white dwarfs (for further details see Bédard, Bergeron, Brassard, & Fontaine, 2020;Bergeron et al., 2011;Blouin, Dufour, & Allard, 2018;Holberg & Bergeron, 2006;Kowalski &Saumon, 2006 andTremblay, Bergeron, &Gianninas, 2011) 10 . While the brighter primary components of these systems are all main sequence stars, the faint secondaries are well below the main sequence, and their Gaia photometry best matches that expected for white dwarfs. We therefore conclude that these companions are white dwarfs, as indicated by the WD flag in Table 5.
Another white dwarf companion is revealed in this project due to its intrinsic faintness. TOI 4301 B has an absolute Gband magnitude of G = 13.21 +0.08 −0.1 mag and a magnitude difference of Δ = 8.458 ± 0.005 mag to the associated potential exoplanet host star TOI 4301 A. Although there is no color information for TOI 4301 B in the Gaia DR3, the white dwarf nature of this companion can be deduced from its photometric properties. While TOI 4301 B is visible in the optical spectral range and is listed in the Gaia DR3, it is not detected in the 2MASS Ks-band image which is shown in Figure 6. In this figure, the expected position of the companion at the 2MASS observing epoch (February 2000) is indicated by a black circle, calculated using the Gaia DR3 astrometry of the companion and the epoch difference between the 2MASS observation and the Gaia DR3 reference epoch (2016.0). The 2MASS image has a ∕ = 10 detection limit of about 14.9 mag. Near the position of the companion, the star 2MASS 11050267-4602149 ( = 14.6 ± 0.1 mag, Skrutskie et al., 2006) is well detected in the 2MASS image, consistent with the given detection limit. If TOI 4301 B were a main sequence star, we would expect its apparent Ks-band magnitude to be = 14.5 ± 0.1 mag, derived from the photometric data of main sequence stars from Pecaut and Mamajek (2013), using the G-band photometry of the companion, the extinction in this band, the given extinction relations G ∕ V = 0.789 and Ks ∕ V = 0.078 from Wang and Chen (2019), and the parallax of TOI 4301. Therefore, if the companion were a main sequence star, it should be clearly visible in the 2MASS image, similar like 2MASS 11050267-4602149. In contrast, if the companion were a white dwarf, we could infer its expected Ks-band photometry as above, but using the evolutionary DA white dwarf models of Bergeron et al. instead. For an assumed companion mass of 0.6 ⊙ we obtain ∼ 18.3 mag, which is clearly below the 2MASS detection limit. We therefore conclude that TOI 4301 B is a white dwarf, due to its intrinsic faintness in the optical and near-infrared spectral range. With the assumed mass of this degenerated companion and the white dwarf evolutionary models mentioned above, the absolute G-band photometry of the companion yields its effective temperature, which is ef f = 7447 +211 −158 K. Figure 7 shows the histograms of the properties of all companions detected in this project. The companions have angular separations to the (C)TOIs, ranging from about 0.4 to 167 arcsec, corresponding to projected separations from 51 up to 9736 au. According to the underlying cumulative distribution function, the frequency of the companions is highest and constant between about 300 and 750 au, decreasing for larger projected separations. Half of all companions have projected separations of less than 1500 au. In total, 5 stellar systems (four binaries, and one hierarchical triples) with projected separations below 200 au are detected, namely: TOI 3073 AB, TOI 3634 AB, TOI 4175 AB+C, TOI 4349 AB, and CTOI 248960573 AB, i.e. the most challenging environments for planet formation identified in this study.
The companion masses range from about 0.11 to 2 ⊙ (the mean mass is ∼ 0.6 ⊙ ). The highest companion frequency in the cumulative distribution function is found in the mass range between 0.2 and 0.5 ⊙ , corresponding to mid to early M dwarfs, according to the relation between mass and spectral type (SpT) from Pecaut and Mamajek (2013). For higher masses, the companion frequency is lower but constant between about 0.7 and 1.1 ⊙ , from where it decreases continuously towards higher masses. This peak in the companion population is also seen in the distribution of their effective temperature, where the companion frequency is highest at temperatures between about 3200 and 3700 K. There is also a second but weaker clustering of companions between about 4600 and 6100 K, corresponding to mid K to late F-type stars, according to the ef f -SpT-relation from Pecaut and Mamajek (2013).
As can be seen from the separation-mass-diagram in Figure 8, of the 149 companions presented here, 18 are the primary, 124 are the secondary, 5 the tertiary, and 2 the quaternary component of their stellar systems.

FIGURE 8
The separation-mass-diagram of the companions detected in this study. Companions that are the primary components of their stellar systems are shown as star symbols, secondaries as grey circles, tertiary components as black triangles, and quaternary components as black squares, respectively. Detected white dwarf companions, assumed to have a mass of 0.6 ⊙ , are plotted as white crossed circles (note that the symbols of the white dwarf companions TOI 4301 B, and CTOI 13073396 B overlap at a separation of about 5000 au).
To characterize the detection limit reached in this project, we plot the magnitude difference of all detected companions over their angular separation to the associated (C)TOIs, as shown in Figure 9. For comparison, we plot the Gaia detection limit determined by Mugrauer et al. (2022), which agrees well with the detection limit presented here. Only at small angular separations below about 0.4 arcsec does it seem to be too conservative. In this angular separation range one companion is detected that is about 0.4 mag fainter than the associated TOI. The magnitude difference of all detected companions plotted against their angular separation from the associated (C)TOI. The Gaia detection limit, found by Mugrauer et al. (2022), is shown as a dotted line for comparison. The expected average magnitude differences for companions with 0.1 or 0.6 ⊙ are shown as grey dashed horizontal lines.
The expected brightness difference between the targets of this study and low-mass main sequence companions (shown as grey dashed lines in Figure 9) is estimated using the expected absolute G-band magnitude of these stars, as listed in Pecaut and Mamajek (2013), and the average absolute G-band magnitude of our targets ( G ∼ 4.2 mag). As shown in Figure 9, a magnitude difference of about 3.7 mag is reached at an angular separation of about 0.9 arcsec around the targets of this project. This allows the detection of companions with masses down to about 0.6 ⊙ (mean mass of all detected companions) that are separated from the (C)TOIs by more than 250 au. In addition, companions with masses down to about 0.1 ⊙ are detectable beyond 4 arcsec, corresponding to a projected separation of 1100 au at the average target distance of 280 pc.

SUMMARY AND OUTLOOK
In recent years, several surveys have been carried out using ground-based observations to explore the multiplicity of exoplanet host stars (see e.g. Ginski, Mugrauer, Adam, Vogt, & van Holstein, 2021;Mugrauer & Ginski, 2015). Here we present the latest results of our survey based on data of the ESA-Gaia mission, which was initiated at the University Observatory Jena in early 2020 with the aim of detecting and characterizing stellar companions of (C)TOIs, i.e. potential exoplanet host stars. In the study presented in this paper we search for companions of 1106 (C)TOIs announced in the FIGURE 10 (RGB)-color-composite images of the hierarchical quadruple system TOI 3106 AC+BD, and of the triple system TOI 3584 AB+C from z-(or y-), i-, and g-band Pan-STARRS images. A detailed y-band Pan-STARRS image is shown for TOI 3548 AB+C, in which the primary and secondary component of this triple star system are well resolved. The (RGB)-colorcomposite images of the binary systems TOI 4001 AB, and TOI 4079 BA, are composed of R-, V-, and B-band images taken with the CTK-II at the University Observatory Jena.
(C)TOI release of the ExoFOP-TESS, whose multiplicity has not yet been investigated in the course of our survey, using Gaia DR3 data.
In total, about 33000 sources with accurate astrometric solutions are detected in the Gaia DR3 around 891 targets, while around the remaining 215 targets of this survey no companioncandidates are identified within the applied search radius. In total, new co-moving companions are detected around 142 of all targets whose multiplicity is studied here. In addition, companions around a further 23 (C)TOIs are found in the Gaia DR3, but these had already been detected in the Gaia DR2 by Mugrauer (2019), Mugrauer and Michel (2020), and Michel and . Thus, the multiplicity rate of the investigated (C)TOIs is at least 14.9±1.1 %, which is lower than the rate found in our survey so far (e.g. 20.1 ± 0.9 %, Mugrauer et al., 2022). The reason for this difference is the low multiplicity rate of the CTOIs examined in this study, which is only about 5 %. In contrast, the multiplicity rate of the TOIs is 19.9±1.5 %, which is well in line with the multiplicity rates found earlier in our survey. As described above, the CTOIs whose multiplicity is studied here are closer to the Sun than the TOIs. A smaller distance usually leads to higher proper motion, brightness and angular separation of potential companions, which should facilitate their detection. Therefore, the surprisingly low multiplicity rate of the CTOIs found in this study must be an effect of the target selection by the ExoFOP-TESS.
Color-composite images of some of the detected stellar systems are shown in Figure 10, taken with Pan-STARRS or with the Cassegrain-Teleskop-Kamera (CTK-II, Mugrauer, 2016) at the University Observatory Jena. In addition to 134 binaries, two quadruple star systems are revealed in which the (C)TOIs have both a close companion and a distant binary companion. Furthermore, three hierarchical triple star systems are detected, whose individual components are listed in the Gaia DR3. Moreover, significant ( ( ) = 20) astrometric non-single-star solutions are listed in the Gaia DR3 for the co-moving companions of TOI 3875, TOI 4128, and TOI 4420 (indicated with the NSS flag in Table 5). For the companions of TOI 3875 and TOI 4128, a Keplerian orbital motion of their photocenter is detected in the Gaia astrometry ( = 502.5 ± 5.0 days, = 0.99 ± 0.05 mas, = 0.36 ± 0.10, = 86.9 ± 2.5 • for TOI 3875 B, and = 802.7 ± 11.2 days, = 1.86 ± 0.03 mas, = 0.61 ± 0.02, = 129.1 ± 1.2 • for TOI 4128 B, respectively). So these companions are themselves close binary systems. With the masses of TOI 3875 B and TOI 4128 B, and their Gaia DR3 parallaxes redetermined in the non-single star solutions ( = 2.482 ± 0.026 mas for TOI 3875 B, and = 3.210 ± 0.017 mas for TOI 4128 B, respectively), the masses of the unseen tertiary components and the semi-major axis of these close binary systems can be determined. We obtain 0.37 ± 0.03 ⊙ for the mass of TOI 3875 C, and 0.38 ± 0.02 ⊙ for TOI 4128 C, respectively. The semi-major axis of the TOI 3875 BC system is = 1.33 ± 0.03 au, which corresponds to about 3 mas at the distance of the system, and = 1.78 ± 0.05 au (∼ 6 mas) for TOI 4128 BC, respectively. The companion of TOI 4420 is identified as a single-lined spectroscopic binary in the Gaia DR3. According to its significant ( = 12) orbital solution, the companion has a radial velocity (RV) semi-amplitude of = 20.9 ± 1.8 km/s, a period of = 4.7257 ± 0.0009 days, and an eccentricity of = 0.24 ± 0.09. Together with the mass of TOI 4420 B this gives the minimum-mass of the unresolved companion TOI 4420 C (∼ 0.16 ⊙ ), and the minimum of the semi-major axis of this close binary system (∼ 0.05 au). Thus, TOI 3875, TOI 4128, and TOI 4420 are three more hierarchical triples composed of the TOIs and their distant binary companions. Hence, a total of 6 hierarchical triple star systems are detected in the multiplicity study presented here.
Furthermore, the companion TOI 3956 A, is listed in the Gaia DR3 as having a significant acceleration of its proper motion (̇= 1.22 ± 0.04 mas/yr 2 ), i.e. this star probably has an additional close stellar companion with an orbital period longer than the 34-months period on which the Gaia DR3 is based on. We therefore classify TOI 3956 as a potential hierarchical triple star system, whose triple nature will have to be confirmed by follow-up observations.
As expected for components of stellar systems the (C)TOIs and the detected companions are equidistant and share a common proper motion, as verified by their accurate Gaia DR3 parallax and proper motion. In particular, the direct proof of the equidistance of the individual components of the stellar systems, as done in this study by comparing their parallax, was not possible in previous multiplicity surveys prior to the release of the precise Gaia data, because in particular for most of the faint companions their parallax could not be measured by the ESA-Hipparcos mission (Perryman et al., 1997).
However, 23 companions identified in this project are already listed in the WDS, either as co-moving companions or as companion-candidates of the (C)TOIs, requiring confirmation of their companionship, which is finally provided by this study. Although the WDS is currently the most complete catalog of multiple star systems available, containing relative astrometric measurements of multiple star systems over a period of more than 300 years, 126 (i.e. about 84 % of all) companions not listed in the WDS are detected in this project and are marked with the ★ flag in Table 4. This demonstrates the great potential of the ESA-Gaia mission for the study of stellar multiplicity, especially for the detection of wide companions, as shown by the derived detection limit of this study in Figure 9. On average, all stellar companions with masses down to about 0.1 ⊙ are detectable around the targets in this study beyond about 4 arcsec (or 1100 au of projected separation), and more than half of all detected companions have such separations. Overall, companions with projected separations between about 50 and 9700 au are detected, and the frequency of companions is constant and highest for separations between about 300 and 750 au, while it decreases significantly for larger projected separations. The companions found in this project have masses ranging from about 0.11 to 2 ⊙ and are most abundant in the mass range between 0.2 and 0.5 ⊙ . In addition to main sequence stars, four white dwarfs are identified as co-moving companions of the (C)TOIs, whose true nature is revealed based on their photometric properties.
A significant (rel ≥ 3) differential proper motion rel relative to the associated (C)TOIs is detected for 122 (i.e. about 80 % of all) companions presented here. We derive the escape velocity esc of all these companions using the approximation described in Mugrauer (2019). The differential proper motion of most of these companions is consistent with orbital motion. In contrast, the differential proper motion of 19 companions significantly exceeds the expected escape velocity. Since these companions all have a high degree of common proper motion ( -≥ 10), this may indicates a higher degree of multiplicity as described in Mugrauer (2019). In fact, five of the companions are members of already confirmed hierarchical triple or quadruple star systems. For the companion TOI 3956 A, a significant acceleration of its proper motion is listed in the Gaia DR3, indicating that this star is itself a close binary, i.e. TOI 3956 is a potential hierarchical triple star system. Follow-up (high contrast imaging) observations are needed to further investigate the multiple status of all these particular systems and their companions, which are summarized in Table 2.

TABLE 2
List of all detected companions (sorted by their identifier) whose differential proper motion rel relative to the (C)TOIs significantly exceeds the expected escape velocity esc . Companions already known to be members of hierarchical triple or quadruple star systems are labelled with ★★★ or ★★★★, those in potential triple star systems with (★★★), respectively. In addition to astrometry, also RV data can be used to check the companionship of the components of stellar systems. For common proper motion pairs, we do not expect significant differences between the RVs of their components, since their orbital motion (especially for wide ones) around their barycenter is typically much smaller than the systematic velocity. For 59 companions detected in this study (i.e. less than 40 % of all) a comparison between their RV and that of the associated (C)TOI is possible in the Gaia DR3. As expected, the RVs of most of these companions and the (C)TOIs do not significantly deviate from each other. Only four companions have significant RV deviations, namely: TOI 3956 A (8.35 ± 2.70 km/s), TOI 4175 B (1.98 ± 0.60 km/s), TOI 4179 A (2.34 ± 0.36 km/s), and TOI 4420 B (14.99 ± 4.01 km/s). This may indicate the presence of additional close stellar companions in these systems that induce faster orbital motions. This is particulary the case for companions, belonging to stellar systems whose components have a high degree of common proper motion, as is the case for the four companions mentioned above ( -≥ 16). The companions TOI 3956 A, and TOI 4420 B both have non-single star solutions in the Gaia DR3, so are themselves close binary systems. TOI 4175 B is a member of a hierarchical triple star system whose components are resolved by Gaia, and TOI 4179 A is a member of a potential triple star system according to its Gaia DR3 astrometry.
As demonstrated with the target sample studied here, even in the latest data release of the ESA-Gaia mission, RVs are available for only a minority of the detected companions. For this reason, we do not use RV data for companionship verification in this survey so far, but will include it as soon as it is available for most of the detected companions, which is expected for the upcoming Gaia data releases.
The survey, whose latest results are presented here, is an ongoing project whose target list is continually growing as a result of the ongoing analysis of photometric data collected by the TESS mission. The multiplicity of all these newly detected (C)TOIs will be explored in the course of this survey, and the detected companions and their determined properties will be reported regularly in this journal and also published online in the VizieR database (Ochsenbein, Bauer, & Marcout, 2000) 11 , and on the website of this survey 12 . The results of this survey, combined with those of high-contrast imaging observations of the (C)TOIs, which can detect close companions with projected separations down to only a few au, will complete our knowledge of the multiplicity of all these potential exoplanet host stars.

ACKNOWLEDGMENTS
We used data from: (1) the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/ consortium). The DPAC is funded by national institutions, in particular those participating in the Gaia Multilateral Agreement.
(2) the Exoplanet Follow-up Observing Program website, operated by the California Institute of Technology, on behalf of the National Aeronautics and Space Administration under the Exoplanet Exploration Program.
(3) the Simbad and VizieR databases operated at the CDS in Strasbourg, France.
(5) the Pan-STARRS1 surveys, made possible by contributions from the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, and the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation Grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), and the Los Alamos National Laboratory. The Pan-STARRS1 Surveys are archived at the Space Telescope Science Institute (STScI) and are available through MAST, the Mikulski Archive for Space Telescopes. Additional support for the Pan-STARRS1 public science archive is provided by the Gordon and Betty Moore Foundation.
(6) the Second Palomar Observatory Sky Survey (POSS-II), made by the California Institute of Technology with funds from the National Science Foundation, the National Geographic Society, the Sloan Foundation, the Samuel Oschin Foundation, and the Eastman Kodak Corporation. The UK Schmidt Telescope was operated by the Royal Observatory Edinburgh, with funding from the UK Science and Engineering Research Council (later the UK Particle Physics and Astronomy Research Council), until 1988 June, and thereafter by the Anglo-Australian Observatory. The blue plates of the southern Sky Atlas and its Equatorial Extension (together known as the SERC-J), as well as the Equatorial Red (ER), and the Second Epoch [red] Survey (SES) were all taken with the UK Schmidt. Supplemental funding for sky-survey work at the ST ScI is provided by the European Southern Observatory. The Digitized Sky Surveys were produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166. The images of these surveys are based on photographic data obtained using the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope. The plates were processed into the present compressed digital form with the permission of these institutions.
(7) the Two Micron All Sky Survey, a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.
(8) the University Observatory Jena, which is operated by the Astrophysical Institute of the Friedrich-Schiller-University.       4 This table lists for each detected companion (sorted by its identifier) the angular separation and position angle to the associated (C)TOI, the difference between its parallax and that of the (C)TOI Δ with its significance (in brackets calculated also by taking into account the Gaia astrometric excess noise), the differential proper motion rel of the companion relative to the (C)TOI with its significance, as well as its -. The last column indicates (⋆) if the detected companion is not listed in the WDS as companion(-candidate) of the (C)TOI.