1 INTRODUCTION

The question of whether the Sun has a companion was first raised about a hundred years ago [1]. As candidates for a companion of the Sun, both known objects (46 Tau [1]) and hypothetical ones, whose presence could be confirmed by indirect signatures, were considered: a relativistic object [2], a black dwarf [3], and a brown dwarf [4]. These works are also related to studies of the question of whether the Sun has an unknown companion with a planetary mass [5‒7].

Among the possible indicators of the presence of a companion near the Sun, the following were considered:

• (i) acceleration of the barycenter of the Solar System, leading to anomalous changes in the period of pulsars located in a certain direction in the sky [8, 9];

• (ii) an abundance of comets with hyperbolic orbits caused by the gravitational influence of a compact massive object (neutron star or black hole) [10] or even a high speed solar mass object [11]. Note that at present, only one comet with a hyperbolic orbit has been 100% recorded, namely, comet 2I/Borisov [12, 13] (also known is the interstellar asteroid 1I/’Oumuamua [14] and moving, with a certainty of 99.999%, along the perbolic orbit CNEOS meteor 2014-01-08 [15]);

• (iii) the presence of a cycle duration of 26 × 106 years in biological mass extinctions, which are supposed to be caused by intense comet showers [3] (the expected boundary of the Oort cloud—a reservoir of comets that can be perturbed by a hypothetical companion of the Sun—is about \({{10}^{5}}\) AU, see discussion in Section 3); and

• (iv) features of the orbits of Sedna and other trans-Neptunian objects caused by the presence of the Sun’s companion [5, 6, 16].

In a number of papers, attempts were made to indicate the boundaries for the possible values of the companion and orbit parameters or even to estimate them [3, 6, 16–18]. Assumptions were made about the presence of the desired object in modern large infrared photometric surveys (IRAS, 2MASS, and WISE) [6, 16, 19].

With the advent of preliminary results from the Gaia astrometric space observatory, it became possible to refine the characteristics of this hypothetical object. The fact that the Gaia observatory has not (yet) discovered the Sun’s companion places certain restrictions on its nature and location.

This paper investigates the possibility of the existence and detection of a brown dwarf, a possible companion of the Sun. Its limiting parameters, the possible distance from the Sun and the magnitude of its proper motion are estimated. A comparison is made both with real wide binary systems and with the estimates of other authors investigating this issue.

2 OBSERVED WIDE BINARIES

First of all, we determine what values of the orbital parameters and parameters of the components have the widest of the observed binary systems with known orbital elements. The most authoritative source of information about such systems is the catalog of orbital binaries ORB6 [20]. About a dozen of the 3310 stars contained in ORB6 can be called very wide pairs (having a period \(P{{ > 10}^{5}}\) years and/or semi-major axis \(a>100'' \)).

Not all of these objects will be considered in this study. As shown in works on determining the masses of the components of orbital binaries [21, 22], the WDS 23100+3651 system can be suspected of optical binarity, while others (WDS 14396–6050, WDS 07204–5219) show a discrepancy between the dynamic mass values determined by Kepler’s third law, and photometric masses determined from the brightness of the components, the distance to the system, and the mass–luminosity ratio. For the WDS 19464+3344 and WDS 00057+4549 systems, ORB6 provides three rather different orbital solutions each.

For our purposes, we can consider double WDS 00524 6930 and WDS 01522 5220. Their parameters are given in Table 1. Orbital elements (\(P\), \(A\), and \(e\)) are taken from ORB6, spectral classification from WDS [23] and SIMBAD, parallaxes from Gaia DR3. The semi-major axes in linear units and the maximum distance between the components \({{d}_{{\max }}} \approx a(e + 1)\) are estimated here.

Table 1.   Orbital and physical parameters of wide binary systems
Full size table

3 ABSOLUTE MAGNITUDES OF BROWN DWARFS AND THE POSSIBILITY OF THEIR DETECTION

This section lists (both published and obtained in this paper) the absolute magnitudes of brown dwarfs.

To estimate the brightness of the companion at different distances from the main component (the Sun), it is necessary to know the absolute magnitude of the object \(M\). Table 2 contains the absolute magnitudes in  several photometric bands of the Gaia [24, 25], 2MASS [26], WISE [27] surveys for the case when the companion is a cold red dwarf (M9.5), a hot brown dwarf (L5), a warm brown dwarf (T4.5) and a cool brown dwarf (Y2).

Table 2.   Absolute magnitudes of cold dwarfs
Full size table

The source of the values M for the M9.5 and L5 stars was Mamajek tablesFootnote 1 published in [28]. Data on \({{M}_{H}}\) and \({{M}_{{Ks}}}\) for brown dwarfs T4.5 and Y2 are taken from the same tables.

To estimate the values \({{M}_{G}}\) and \({{M}_{{W1}}}\) for the last two spectral types, we used the catalog of brown dwarfs closest (up to 20 pc) to the Sun 525 L, T, and Y [29]. It contains data for the WISE photometric system, and to estimate the \({{M}_{G}}{\kern 1pt} \)-values, we identified the cataloged sources with the Gaia DR3 archive. The identification was carried out with the identification radius \(1'' \). We managed to identify 132 objects out of 496. Two objects, 2MASS 0915+0422 and 2MASS 1520–4422, each received two analogues in Gaia DR3, in both cases the one located closer to the source object was recognized as suitable. Further analysis included a comparison of the proper motions and parallaxes found in [29] with the Gaia data, as well as the construction of various photometric relationships between the G-value and the photometry presented in [29]. As a result, five objects (2MASS 0355+1133, WISE 0715–1145, WISE 0720–0846B, DENIS 1253–5709, and Gaia 1831–0732) were found to be incorrectly identified and discarded.

The identification results are shown in Fig. 1. It can be seen that objects of spectral types no colder than T6 were found in the Gaia archive, and that although the limiting magnitude of Gaia objects reaches \(G \approx {{21.25}^{m}}\), however, the limiting brightness required to determine the parallax is at the \(G \approx {{20.75}^{m}}\) level.

Fig. 1.
figure 1

Identification of brown dwarfs from the catalog [29] with the Gaia DR3 archive. Left panel: distribution of brown dwarfs by spectral type (red—all stars and green—identified with Gaia). The spectral type is defined as follows: 0—L0, …, 5—L5, …, 10—T0, …, 20—Y0, … . Right panel: distribution of identified objects by brightness \(G\) (red—all, blue—no parallax value in Gaia).

Full size image

The results obtained can be used to obtain at least a rough relationship between the G-value of an object and its brightness in the infrared bands \(H\) (2MASS) and \(J\) (2MASS). Figure 2 shows all identified objects with \(G\), \(H\), and \(J\) photometry, as well as the result of linear approximation,

$$G = 1.363{\kern 1pt} H + 1.358,$$
(1)
$$G = 1.235{\kern 1pt} J + 1.918,$$
(2)

are the correlation coefficients in formulas (1) and (2)—0.97 and 0.96, respectively. The object 2MASS 0746+2000 (the brightest in the diagrams) has the attribute VARIABLE in the Gaia DR3 archive and did not participate in the approximation.

Fig. 2.
figure 2

Identified brown dwarfs in the \(H{-} G\) (left) and \(J{-} G\) (right) diagrams. Straight lines are the results of linear approximation according to formulas (1) and (2).

Full size image

Extrapolating these ratios to the region of low temperatures, it is possible to estimate the absolute G-values of T and Y type brown dwarfs (in the Mamajek table, these values are given only for L8 and hotter). This estimate is shown in Fig. 3, left panel. It is noteworthy that in the region of hot brown dwarfs, our data are in good agreement with Mamajek data.

Fig. 3.
figure 3

Left panel - absolute magnitudes of brown dwarfs. The red curve is an approximation of values \({{M}_{J}}\) (2MASS) from the Mamajek table, the green curve is the values \({{M}_{G}}\) (Gaia) calculated by Eq. (2), and the blue dots are the values \({{M}_{G}}\) (Gaia) from the Mamajek table. The right panel shows the limiting distance for the band \(J\) 2MASS (red curve) and the band \(G\) Gaia DR3 (green curve), where brown dwarfs can be observed. Objects located below these curves should fall into the corresponding views. The horizontal stripes are the approximate positions of the inner (yellow) and outer (grey) boundaries of the Oort cloud. The principle of encoding the spectral types of brown dwarfs is similar to that given in the caption to Fig. 1.

Full size image

The data obtained make it possible to estimate the limiting distance at which a brown dwarf of one type or another will be detected by the Gaia observatory (Fig. 3, right panel). \(G{{.75}^{m}}\) was used as the limiting magnitude for Gaia observation, and interstellar extinction was assumed to be negligible. It can be seen that a brown dwarf later than Y3 will not be detected by the Gaia observatory, even if it is inside the Oort cloud. Here it is considered that the inner boundary of the Oort cloud is at a distance of 2–5 thousand AU from the Sun, and the outer one is 50–100 thousand AU (see, for example, [30]; see also the siteFootnote 2 on the NASA portal). At the same time, this hypothetical brown dwarf will definitely be included in the 2MASS catalog (red curve; here we consider that 2MASS is full up to \(J = {{16.5}^{m}}\)) and even more so in WISE. Hotter objects (L and T dwarfs) need to be located outside the Oort cloud in order not to fall into the Gaia observational archive.

4 PROPER MOTION

It seems appropriate to estimate the magnitude of the proper motion of a cold brown dwarf, a satellite of the Sun, located in the region of the inner boundary of the Oort cloud. The orbital velocity at any instant of time can be written as

$${v} = \sqrt {G{\kern 1pt} ({{m}_{1}} + {{m}_{2}}){\kern 1pt} \left( {\frac{2}{r} - \frac{1}{a}} \right)} {\kern 1pt} .$$
(3)

Here, the gravitational constant G = 6.6743 × 10‒11 m3 s–2 kg–1, \({{m}_{1}},\;{{m}_{2}}\) are the masses of the components: \({{m}_{1}} = {{m}_{ \odot }} = 1.989 \times {{10}^{{30}}}\) kg, and the mass of the component \({{m}_{2}}\) can be neglected. Let’s take the current distance to the companion \(r\) as 2000 AU (= \(3 \times \) 1014 m), and for the semi-major axis \(a\) consider options \(r = a\) (circular orbit) and \(r = 1.9{\kern 1pt} a\) (companion is located in the vicinity of the aphelion of a very eccentric orbit). Then the orbital velocities estimated from (3) for these two cases will be 665.2 and 210.4 m/s. Proper motion,

$$\mu = \frac{{v}}{{4.74{\kern 1pt} r}}{\kern 1pt} ,$$
(4)

in these cases it will be equal to 14.5 and 4.6\('' \)/year, respectively (compare with the proper motion of Barnard’s star, 10.4\('' \)/year). In Eq. (4), the values \(\mu \), \({v}\), and \(r\) are expressed in units of \('' \)/year, km/s, and pc, respectively.

Such a noticeable proper motion would immediately highlight the object in the astrometric survey, but at the same time it would make it extremely difficult to cross-identify it between photometric surveys, even with a small difference in observation epochs.

5 BLACK DWARFS

The term “black dwarf” does not have a clear quantitative definition. Moreover, in the literature, this concept refers to two completely different types of objects.

1. Stars of very low mass (\(m < 0.08{\kern 1pt} {{m}_{ \odot }}\)), which do not pass through the stage of the thermonuclear reaction of burning hydrogen, become completely degenerate objects or black dwarfs in a time much shorter than the age of the Galaxy [31].

2. Old white dwarfs that have cooled their outer layers to several thousand degrees and have a luminosity of the order of \({{10}^{{ - 5}}}{\kern 1pt} \;{{L}_{ \odot }}\) or less are also called black dwarfs [32]. This process is much longer; formulas for estimating the parameters of these objects can be found in [33].

Obviously, if the Sun’s companion is a black dwarf, then the probability of its detection is greatly reduced. Note, however, that a black dwarf of the “second type” (former white dwarf), in contrast to a black dwarf of the “first type” (former brown dwarf), has a mass that should not be neglected when estimating the proper motion of such an object. Assuming that the companion has a mass equal to the Sun, the values given in Section 4 should be increased by 40%.

6 COMPARATIVE ANALYSIS

Let us compare the estimates made here with those observed and predicted by other authors.

Note that the binary system “Sun–brown dwarf Y3" discussed at the end of Section 3 would be very similar to the systems WDS 01522–5220 and WDS 00524–6930 described in Section 2 and would be even more compact.

The authors of [3] suggest that the Sun has a companion, a black dwarf with a mass from \(2 \times {{10}^{{ - 4}}}\) to \(7 \times {{10}^{{ - 2}}}{\kern 1pt} {{m}_{ \odot }}\), in a very eccentric (\(e \geqslant 0.9\)) orbit with semi-major axis \(a \approx 8.8 \times {{10}^{4}}\) AU. As follows from Fig. 3 (right panel), it is unlikely that such an object could be included in the Gaia archive and existing infrared surveys.

In [17], in order to determine the boundaries of the Solar System, hypothetical satellites of the Sun were considered in a wide mass range, from 0.005 to 0.3 \({{m}_{ \odot }}\) in orbits with semi-major axis \(a \approx 9 \times {{10}^{4}}\) AU. In this case, obviously, the probability of detection depends on the mass of the object.

Finally, in [16], a relationship was proposed between the mass of a hypothetical object and the major semiaxis of its orbit: \(m{\text{/}}{{m}_{J}} \approx 5\sqrt {9000\;{\text{AU/}}a} \).

It can be seen that in these studies, the emphasis was mainly on high (above the Oort cloud) orbits for the hypothetical companion of the Sun. In [19], however, lower orbits were also considered, but they dealt with objects of planetary masses, (1–5) \({{m}_{J}}\). However, as the analysis performed in this paper shows, under certain conditions, a companion, a brown dwarf, can also exist in low orbit (below the Oort cloud) and remain unnoticed by the Gaia observatory and unrecognized (although possibly present) in modern infrared photometric surveys.

7 CONCLUSIONS

The paper considers the restrictions on the nature and characteristics of the hypothetical component of the Sun, imposed by the current results of the work of the Gaia space observatory. In particular, it is shown that even being in an inner orbit with respect to the Oort cloud, a brown dwarf of spectral type Y3 and colder will go unnoticed by the Gaia observatory, although it may well be already included in the 2MASS and WISE infrared survey catalogs. It is shown that such an object should demonstrate a noticeable proper motion—units or tens of arc seconds per year. A decrease in the temperature of a hypothetical companion and/or its location in a higher orbit makes its detection by modern astrometric and photometric methods practically impossible. The paper also obtained an empirical dependence \({{M}_{G}}\)(SpT) for brown dwarfs.