An Astrometric Planetary Companion Candidate to the M9 Dwarf TVLM 513–46546

We use a periodogram code to search for astrometric signatures that indicate the possible presence of one or more companions to the main source. The periodogram of the astrometric data is obtained using a modified version of the classic least-squares periodogram method described by Curiel et al. (2019). The new version of the code, which we call a recursive least-squares periodogram with a circular orbit (RLSCP), takes into account the possibility of fitting the Keplerian orbits of several companions (see also, e.g., Anglada-Escudé & Tuomi 2012). This recursive periodogram consists of fitting all the parameters of the already detected signals together with the signal of a new companion, which is under investigation. We start assuming circular orbits, but we can include a fixed eccentricity for the signals already found. When no previous planets have been detected, the initial periodogram is obtained by comparing the least-squares fits of the basic model (proper motions and parallax only) and a one-companion model (proper motions, parallax, and Keplerian orbit of a single companion). When a signal has already been detected, the recursive periodogram compares the least-squares fits of a one- and a two-companion model (proper motions, parallax, and Keplerian orbits of two companions), and so on.

The weighted least-squares solution is obtained by fitting all of the free parameters in the model for a given period. The sum of the weighted residuals divided by N_obs is the so-called χ² statistic, where N_obs is the number of data points. Then, each ${\chi }_{P}^{2}$ of a given model with k_P-free parameters can be compared to the ${\chi }_{0}^{2}$ of the null hypothesis with k₀-free parameters by computing the power, z, as (e.g., Anglada-Escudé & Tuomi 2012; Curiel et al. 2019)

$$\begin{eqnarray}&&z(P)=\displaystyle \frac{({\chi }_{k}^{2}-{\chi }_{P}^{2})/({N}_{k+1}-{N}_{k})}{{\chi }_{P}^{2}/({N}_{\mathrm{obs}}-{N}_{k+1})},\end{eqnarray} \tag{ 2 }$$

where ${\chi }_{k}^{2}$ is the χ² statistic for the model with k planets (the null hypothesis), ${\chi }_{P}^{2}$ is the χ² statistic for the model including one more planet with an orbital period P, N_k is the number of free parameters in the model with k planets, and ${N}_{k+1}$ is the number of free parameters in the model including one more candidate in a circular orbit with an orbital period P. In this model, a large z is interpreted as a very significant solution. The values of z follow a Fisher F-distribution with ${N}_{k+1}-{N}_{k}$ and ${N}_{\mathrm{obs}}-{N}_{k+1}$ degrees of freedom (Scargle 1982; Cumming 2004). Even if only noise is present, a periodogram will contain several peaks (see Scargle 1982, as an example) whose existence has to be considered in obtaining the probability that a peak in the periodogram has a power higher than z(P) by chance, which is the so-called false-alarm probability (FAP),

$$\begin{eqnarray}&&\mathrm{FAP}=1-{\left(1-\mathrm{Prob}[z\gt z(P)]\right)}^{O},\end{eqnarray} \tag{ 3 }$$

where O is the number of independent frequencies. In the case of uneven sampling, O can be quite large and is roughly the number of periodogram peaks one could expect from a data set with only Gaussian noise and the same cadence as the real observations. We adopt the recipe O ∼ ΔT/P_min given in Cumming (2004, Section 2.2), where ΔT is the time span of the observations and P_min is the minimum period searched. For instance, assuming that ΔT = 560 days and P_min = 20 days, the astrometric data is expected to have O ∼ 28 peaks.

Here we use the least-squares algorithm and the asexual genetic algorithm (AGA) presented by Curiel et al. (2019). In short, we use the source barycentric two-dimensional position described as a function of time (α(t), δ(t)), accounting for the (secular) effects of proper motions (μ_α and μ_δ), the (periodic) effect of the parallax Π, and the (Keplerian) gravitational perturbation induced on the host star by one or more companions, such as low-mass stars, substellar companions, or planets (mutual interactions between companions are not taken into account). Given a discrete set of N_obs data points (α(i), δ(i)) with associated measurement errors σ_i, one seeks the best possible model (in other words, the closest fit) for these data using a specific form of the fitting function, (α(t), δ(t)). This function has, in general, several adjustable parameters, whose values are obtained by minimizing a "merit function," which measures the agreement between the data (α(i), δ(i)) and the model function (α(t), δ(t)). The maximum-likelihood estimate of the model parameters (c_i, ..., c_k) is obtained by minimizing the χ² function (e.g., Cantó et al. 2009; Curiel et al. 2019),

$$\begin{eqnarray}\begin{array}{rcl}{\chi }_{\min }^{2} & = & \displaystyle \sum _{i=1}^{N}{\left(\displaystyle \frac{{\alpha }_{i}-\alpha ({t}_{i};{c}_{1},...,{c}_{k})}{{\sigma }_{i}}\right)}^{2}\\ & & +\ \displaystyle \sum _{i=1}^{N}{\left(\displaystyle \frac{{\delta }_{i}-\delta ({t}_{i};{c}_{1},...,{c}_{k})}{{\sigma }_{i}}\right)}^{2},\end{array}\end{eqnarray} \tag{ 4 }$$

where each data point (α_i, δ_i) has a measurement error that is independently random and distributed as a normal distribution about the "true" model with standard deviation σ_i.

First, we used both the least-squares and AGA algorithms (Curiel et al. 2019) to fit the proper motions and parallax to the 17 new VLBA astrometric observations without taking into account any possible companion (single-source solution). Then, we fitted all of the VLBA astrometric data, including six previous VLBA detections of this source, obtained by Forbrich et al. (2013; see Table 1). The results of a single-source solution are shown in Table 2 and Figure 2. We find that the fitted parameters (proper motions and parallax) are very similar in both cases. However, the residuals are large and show a temporal trend that suggests the presence of at least one companion with a possible orbital period of a few hundred days (see Figure 2).

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Table 2.  Single-source Astrometry Fits^a

Parameter	VLBA_new	VLBA_combined
	Parameters Fitted
Epochs	17	23
μα (mas yr−1)	−43.21 ± 0.12	−43.158 ± 0.012
μδ (mas yr−1)	−65.37 ± 0.13	−65.532 ± 0.013
Π (mas)	93.450 ± 0.057	93.423 ± 0.053
	Other Parameters
D (pc)	10.7009 ± 0.0065	10.7040 ± 0.0061
Δα (mas)b	0.079	0.099
Δδ (mas)b	0.123	0.138
χ2, ${\chi }_{\mathrm{red}}^{2}$	38.14, 1.36	66.18, 1.65

Notes.

^aThe parameters presented here were obtained with AGA. Very similar results were obtained with the least-squares fitting method. The second column contains the astrometric fit of the new VLBA data. The third column corresponds to the astrometric fit of the combined VLBA data. ^bThe rms dispersion of the residual.

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We also fitted the astrometric data with acceleration terms, which take into account an astrometric signature due to a possible companion with a large orbital period. We find that the fits do not improve substantially when including acceleration terms (see Table 3). The fitted acceleration terms are small in the case of the combined VLBA data (a_α = −0.0144 ± 0.0045 and a_δ = 0.0332 ± 0.0049 mas yr⁻²) and somewhat larger using only the new VLBA data, but they are consistent with zero within the errors. In this case, the acceleration terms are a_α = −0.34 ± 0.30 and a_δ = 0.41 ± 0.33 mas yr⁻², which suggests that this source might have a companion with an orbital period larger than about 1.5 yr (the time span of the new astrometric VLBA data) and smaller than 9.8 yr (the time span of the combined astrometric VLBA data).

Table 3.  Single-source Astrometry Fits^a

Parameter	VLBA_new	VLBA_combined
	Parameters Fitted
Epochs	17	23
μα (mas yr−1)	−43.08 ± 0.12	−43.187 ± 0.012
μδ (mas yr−1)	−65.49 ± 0.13	−65.460 ± 0.013
aα (mas yr−2)	−0.34 ± 0.30	−0.0144 ± 0.0045
aδ (mas yr−2)	0.41 ± 0.33	0.0332 ± 0.0049
Π (mas)	93.407 ± 0.057	93.451 ± 0.053
	Other Parameters
D (pc)	10.7058 ± 0.0065	10.7008 ± 0.0061
Δα (mas)b	0.081	0.089
Δδ (mas)b	0.111	0.130
χ2, ${\chi }_{\mathrm{red}}^{2}$	33.57, 1.29	53.87, 1.42

Notes.

^aThe parameters presented here were obtained with AGA. Very similar results were obtained with the least-squares fitting method. The astrometric fit includes acceleration terms. The second column contains the astrometric fit of the new VLBA data. The third column corresponds to the astrometric fit of the combined VLBA data. ^bThe rms dispersion of the residual.

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In what follows, we obtain the astrometric fit of the data without taking into account possible acceleration terms.

The RLSCP of the new astrometric VLBA data (see Figure 3) does not show a narrow prominent peak. However, it shows a somewhat "broad signal" with an orbital period between 200 and 300 days. The periodogram also shows that this broad signal is part of a large plateau-like structure that extends beyond the orbital periods considered in the plot (1000 days). This plateau has a drop in the periodogram power around 297 days, suggesting that there may be two broad signals in the periodogram, one at about 241 days and one with an orbital period larger than the time span of the new VLBA observations (about 1.5 yr). The main broad signal is not well constrained but seems to have a relatively weak peak at about 241 days. The FAP of this main peak is 1.43%, suggesting that this signal is real and could be due to a companion.

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The RLSCP of the combined (old and new) VLBA data also shows a broad signal between 200 and 300 days, nearly coinciding with the broad signal observed in the periodogram of the new data. In this case, the periodogram appears somewhat noisier, especially at orbital periods larger than 100 days. The main peak of the combined data is located at 220 days and has an FAP of 5.39%, which, although it is somewhat above the 1% limit usually used to consider a signal as possibly real, also suggests that the signal is real and due to the presence of a companion.

To further investigate the possibility that the peak that appears in the periodograms is real, we computed the recursive periodogram of the two data sets including the signal of this possible companion (two-companion solution). We now simultaneously fit the parameters of the already detected signal together with the signal under investigation (a second possible companion). To obtain an improved global solution (with two possible candidates), we include in the fitting the orbital period of the first companion using, as an initial guess, the orbital period of the peak in the initial periodogram. The RLSCP algorithm includes the possibility that the orbital period of the first companion adjusts during the simultaneous fit of both possible companions. The resultant periodogram is shown in Figure 3. The new periodograms show that the signal of the initial candidate disappears, leaving some residual noise. In addition, the new periodograms show no significant signals, indicating that there is only one significant signal in the periodogram. The new periodogram of the combined data shows two very narrow and relatively strong signals between 3 and 5 days that do not appear in the periodogram of the new VLBA data. These signals are most likely spurious signals or artifacts. The new periodograms also show a slow rising signal close to the end of the plot. This suggest that there may be a second companion that, if real, produces a small astrometric signal and that its orbital period is larger than the time span of the new VLBA data (>558 days). Further observations will be needed to confirm this putative second companion.

We then used both the least-squares and AGA algorithms to fit the astrometric observations of this source, including a possible single companion (single-companion solution). First, we used both methods to fit the new VLBA astrometric observations to obtain proper motions and parallax, taking into account a single companion. Table 4 summarizes the best fit and the ${\chi }_{\mathrm{red}}^{2}$ per degree of freedom (${\chi }_{\mathrm{red}}^{2}={\chi }^{2}/({N}_{\mathrm{data}}-{N}_{\mathrm{par}}-1)$, where N_data = 2 × N_points and N_par is the number of fitted parameters). The fits of the parallax, proper motions, and orbital motions of the candidate are presented in Figure 4. The fit of the astrometric data clearly improves when including a companion, as seen by the ${\chi }_{\mathrm{red}}^{2}$. The ${\chi }_{\mathrm{red}}^{2}$ is now about a factor of 2 smaller, compared with the single-source solution. Tables 2 and 4 and Figures 2 and 4 show that the residuals of the single-companion solution (rms ∼0.10 mas) are a factor of 1.4 smaller than in the case of the single-source solution (rms ∼0.14 mas). The residuals are now comparable to the mean noise present in the data (rms ∼0.08 and 0.13 mas for both R.A. and decl.) and the astrometric precision expected with the VLBA (<96 μas). The astrometric signal in the source due to the companion is 0.17 ± 0.10 mas, i.e., significant at 1.7σ. Although the astrometric signal is small with a relatively large error, the same signal appears when we analyze both data sets using three different algorithms (periodogram, least-squares, and AGA). This indicates that this astrometric signal is real.

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Table 4.  Single-companion Astrometry Fits^a

Parameter	VLBA_new	VLBA_combined
	Parameters Fitted
Epochs	17	23
μα (mas yr−1)	−43.05 ± 0.17	−43.165 ± 0.017
μδ (mas yr−1)	−65.43 ± 0.18	−65.529 ± 0.019
Π (mas)	93.326 ± 0.079	93.405 ± 0.074
P (days)	241 ± 20	220 ± 5
T0 (days)	2,458,263 ± 21	2,457,631 ± 22
eb	0.0	0.0
ω (deg)b	0.0	0.0
Ω (deg)	122 ± 35	139 ± 39
a1 (mas)	0.17 ± 0.10	0.128 ± 0.088
i (deg)	88 ± 36	71 ± 38
	Other Parameters
D (pc)	10.7151 ± 0.0091	10.7060 ± 0.0085
m (M⊙)c	0.08, 0.06	0.08, 0.06
m2 (M⊙)	0.00044 ± 0.00023, 0.00036 ± 0.00019	0.00036 ± 0.00024, 0.00030 ± 0.00020
m2 (MJ)	0.46 ± 0.25, 0.38 ± 0.20	0.38 ± 0.24, 0.31 ± 0.21
a1 (au)	0.0018 ± 0.0011, 0.0018 ± 0.0011	0.00138 ± 0.00094, 0.00138 ± 0.00094
a2 (au)	0.325 ± 0.016, 0.295 ± 0.015	0.3063 ± 0.0036, 0.2782 ± 0.0032
Δα (mas)d	0.063	0.070
Δδ (mas)d	0.075	0.110
χ2, ${\chi }_{\mathrm{red}}^{2}$	17.74, 0.77	38.02, 1.09

Notes.

^aThe parameters presented here were obtained with AGA. Very similar results were obtained with the least-squares fitting method. The astrometric fit includes the orbital motions of a companion. The second column contains the astrometric fit of the new VLBA data. The third column corresponds to the astrometric fit of the combined VLBA data. ^bFixed eccentricity. ^cFixed mass of the star. ^dThe rms dispersion of the residual.

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Table 4 summarizes the best fit of the new VLBA astrometric data, including a companion. The orbit of the companion has an orbital period P ∼ 241 days, a position angle of the line of nodes Ω ∼ 122°, and an inclination angle i ∼ 88°, which indicates that the orbit of the companion is prograde (i < 90°). However, the large error in the inclination angle (∼36°) suggests that the orbit could also be retrograde (i > 90°). In addition, the astrometric fit of the data indicates that the eccentricity of the orbit is not well constrained; thus, we use a fixed eccentricity, e = 0. The orbital period obtained with the astrometric fit is consistent with that obtained with the periodogram. The orbit of the companion is relatively well fitted; however, the errors of the orbital parameters are large. This is consistent with the relatively broad signal observed in the periodogram (see Figure 3). Further observations are needed to better constrain the orbital parameters of this companion. With this astrometric fit, we cannot estimate the dynamical mass of the system; thus, to estimate the mass of the companion, we use the lower and upper limits of the best estimated mass for this source, M_* = 0.06−0.08 M_⊙ (Martín et al. 1994; Reid et al. 2002; Hallinan et al. 2008), as a fixed mass. Table 4 summarizes the estimated parameters of the companion, hereafter TVLM 513b. We find that the mass of the companion is between 0.00036 (M_* = 0.06 M_⊙) and 0.00044 M_⊙ (M_* = 0.08 M_⊙), which is consistent with a planetary companion with a mass between 0.38 and 0.46 M_J. The semimajor axis of the orbit of this planetary companion is between 0.295 and 0.325 au.

We also fitted the combined VLBA data, including the Keplerian fit of a single companion (single-companion solution). Table 4 summarizes the best fit and ${\chi }_{\mathrm{red}}^{2}$ per degree of freedom. The fit of the combined data also improves when including a companion. The residuals of the single-companion solution (rms ∼0.13 mas) improve by a factor of 1.3 compared to the case of a single-source solution (rms ∼0.17 mas). Similarly, the ${\chi }_{\mathrm{red}}^{2}$ is smaller by a factor of 1.7 compared to the single-source solution. The orbital fit to the combined data is in general similar to that obtained in the case of the fit of the new VLBA astrometric data (see Table 4 and Figure 4). The orbital parameters and their estimated errors are similar, with relatively small differences from those obtained using only the new VLBA data (see Table 4). The estimated mass and semimajor axis of the orbit of the companion are also similar to those obtained with the new VLBA data. These results further support the detection of a planetary companion.

These results indicate that the best fit of the orbit of the companion is obtained with the new VLBA data. This is not surprising because these observations are in general deeper and with smaller error bars than previous VLBA observations. However, it is important to point out that the astrometric signal appears in both the new VLBA data and the combined data. Furthermore, the same astrometric signal is found using the two different algorithms (least-squares and AGA) that we have used here, which is also consistent with the astrometric signal found in the least-squares periodogram. Figure 4 suggests a reasonably good astrometric fit. However, the residuals, although small (rms ∼0.1 mas), are comparable to the astrometric signal (0.17 mas).

Figure 4 shows that the data are well fitted when considering a single planetary companion; however, Table 4 shows that the orbital parameters obtained with the fit have large error bars, which indicates that the orbital motion of the companion is not well constrained. We find that this is the result of several contributions, which combined increase the astrometric errors and produce a larger error in the orbital fit. The astrometric signal of the companion is small (0.17 mas), just larger than both the residuals of the fit (rms ∼0.1 mas) and the mean noise present in the data (rms ∼0.08 and 0.13 mas for R.A. and decl.). In addition, the periodogram of the data (see Figure 3) indicates that the orbit of the companion is not completely constrained, and that there may be a second companion with a larger orbital period. The presence of a second companion would appear in the residuals as an additional source of noise, and it would worsen the astrometric fit of the detected companion. Thus, all of these contributions preclude a better estimate of the orbital parameters. However, the fact that the astrometric signal appears in the periodogram and the fits obtained with two different algorithms (least-squares and AGA) supports the detection of the planetary companion. Further observations are needed to better constrain the orbital solution and possibly to confirm the presence of the putative second companion.

Table 4 shows that the estimated parallax to TVLM 513 does not change substantially when fitting only the new VLBA data or combining the new data with the previous VLBA data. Taking into account that the different fits give slightly different values for the parallax, we have calculated the weighted average of the estimated parallax as follows:

$$\begin{eqnarray}&&\langle {\rm{\Pi }}\rangle =\displaystyle \frac{{\displaystyle \sum }_{i}^{N}{\pi }_{i}/{\sigma }_{i}^{2}}{{\displaystyle \sum }_{i}^{N}1/{\sigma }_{i}^{2}},\end{eqnarray} \tag{ 5 }$$

and the uncertainty is

$$\begin{eqnarray}&&\sigma (\langle {\rm{\Pi }}\rangle )=\sqrt{\tfrac{{\displaystyle \sum }_{i}^{N}(1/{\sigma }_{i}^{2}){\left({\pi }_{i}-\langle {\rm{\Pi }}\rangle \right)}^{2}}{{\displaystyle \sum }_{i}^{N}1/{\sigma }_{i}^{2}}},\end{eqnarray} \tag{ 6 }$$

where π_i and σ_i are the estimated parallax of each fit and its uncertainty, respectively.

We obtain that the weighted parallax is 93.368 ± 0.039 mas, which corresponds to a weighted distance d = 10.7102 ± 0.0045 pc. The estimated error corresponds to the standard deviation of the fitted values, which better reflects the dispersion seen in the different astrometric fits. This estimate is an improvement on the distance to this source of 10.762 ± 0.027 pc, previously obtained with VLBI observations (Forbrich et al. 2013; Gawroński et al. 2017). The estimated error that we obtain here is about a factor of 10 smaller than those obtained previously. This is mainly due to the larger number of observations used for the present astrometric fit, the accuracy of the observations we present here (see Table 1), and better coverage of the parallax.

Table 4 also shows that the estimated proper motions of TVLM 513 do not change substantially when fitting the new VLBA data and the combined data. The fit of the combined VLBA data gives slightly better estimates for the proper motions because they cover a larger time span. We obtain that the weighted average proper motions are μ_α = −43.164 ± 0.011 and ${\mu }_{\delta }=-65.528\pm 0.010$ mas yr⁻¹.

The solution that we obtain for the single-companion astrometry can be used to estimate an expected induced RV of the star due to the gravitational pull of the companion as follows (e.g., Cantó et al. 2009; Curiel et al. 2019):

$$\begin{eqnarray}&&K={\left(\displaystyle \frac{2\pi G}{T}\right)}^{1/3}\displaystyle \frac{{m}_{p}\sin (i)}{{\left({M}_{* }+{m}_{p}\right)}^{2/3}}\displaystyle \frac{1}{\sqrt{1-{e}^{2}}},\end{eqnarray} \tag{ 7 }$$

where G is the gravitational constant, and T, M_*, m_p, and e are the orbital period, star and companion masses, and eccentricity of the orbit of the companion. Using the solutions of the astrometric fit given in Table 4, we obtain that the maximum RV of TVLM 513 induced by TVLM 513b is K ∼ 64−81 m s⁻¹ (for the combined and new VLBA data, respectively). This RV could, in principle, be observed with high spectral resolution spectrographs. Future short- and long-term high-resolution spectroscopic observations of TVLM 513 may be able to detect the RV signal that we find that TVLM 513b induces on TVLM 513. Furthermore, these kinds of observations may also be able to confirm the putative second companion suggested by the periodogram.

The radio continuum flux density of TVLM 513 is clearly variable in time. Figure 5 shows that the flux density of this source has short-term, and probably long-term, variability. The timescale of the short-term variability is a few days, where the flux changes by about a factor of 2 or so. This flux variability is observed with both the VLBA observations obtained at 8.4 GHz and the European VLBI Network (EVN) observations obtained at 5 GHz by Gawroński et al. (2017). This source is well known to be variable on a timescale of ∼1.96 hr (when polarized bursts of emission occur), which has been inferred to be the rotation period of this UCD (e.g., Osten et al. 2006; Hallinan et al. 2007, 2008; Berger et al. 2008). These variability timescales may be unrelated, since the integration time of the VLBI observations is several hours (∼4.0 and 7.5 hr); thus, a single VLBI observation integrates over several of the very short time span flux variations.

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In the long term, the flux density of the source has in general decreased as function of time in the past 10 yr. The source was in general stronger in the first VLBA-observed epochs and weaker in the last VLBA-observed epochs. This suggests that the flux density of the source may have a general tendency to decrease as function of time, at least in the past 10 yr. To investigate the long-term variability, and just for description purposes, we have fitted the data with a function of the type

$$\begin{eqnarray}&&\mathrm{Flux}={f}_{0}+{f}_{m}\ {e}^{-\tfrac{{\left(t-{t}_{0}\right)}^{2}}{2{\sigma }^{2}}}.\end{eqnarray} \tag{ 8 }$$

This function fits a single Gaussian function plus a flux base to all of the VLBA epochs. Here f₀ is a constant flux density (in mJy), f_m is the maximum increment in the flux density (in mJy) during the outburst, t₀ is the time of the maximum flux of the source during the outburst (in days from the first observed epoch), and σ is the FWHM (in days) of the Gaussian function. In this fit, the single outburst observed about 10 yr ago is fitted with a single Gaussian function. The fit of the observed flux density data gives f₀ = 0.14 mJy, f_m = 0.71 mJy, t₀ = 23.21 days, and σ = 9.60 days. This fit suggests that the source had a maximum flux outburst of about 0.85 mJy centered at the epoch JD = 2,455,297.139, which is very close to the third observed epoch (BF100C; see Table 1), and that the outburst had an FWHM of 9.60 days and lasted for about 70 days. For the fit, we have only used the observations obtained with the VLBA because they were obtained at 8.4 GHz, while the EVN observations were obtained at 5 GHz. In Figure 5, we plot the observed flux densities, the fit that we obtain, and the residuals of the fit. The residuals of the fit show that the previous VLBA observations can be well fitted with a single Gaussian function with an amplitude of about 0.71 mJy and an FWHM of about 9.60 days, and that the new VLBA observations do not show a similar outburst. The fit also shows that the source is generally weak, having a mean flux density around 0.14 mJy with a small flux fluctuation of about 0.1 mJy in short periods of time, probably a few days, or even shorter. The source may also have strong outbursts, such as the one observed about 10 yr ago that lasted for about 70 days (see Figure 5). The large temporal gap of about 7 yr in the VLBA data precludes the possibility of finding whether these outbursts may be periodic or not. Our recent VLBA observations, which were obtained in a time span of about 560 days, do not show any strong outbursts, suggesting that if the source undergoes periodic outbursts, they are probably at intervals longer than this timescale. Thus, we find that the source seems to undergo flux fluctuations with at least three different time spans: (a) a short-period variation with a time span of about 1.96 hr, observed with the VLA and correlating with the rotation period of this UCD; (b) an intermediate-period variation with a time span of a few days (not well established), observed with the VLBA; and (c) a possible longer-period variation, observed with the VLBA as a single outburst about 10 yr ago. Future observations will tell whether the source undergoes periodic outbursts, as that observed about 10 yr ago, and what their origin may be.

There is only one exoplanet that has been found using astrometry (HD 176051b; Muterspaugh et al. 2010). It was found using optical differential astrometry. This planetary companion is associated with a relatively nearby (14.99 ± 0.13 pc) binary system (1.07 and 0.71 M_⊙) and has an estimated mass of 1.5 ± 0.3 M_J, assuming that it is associated with the low-mass star. The mass of the planetary companion is expected to be higher if it is associated with the higher-mass star.

The best fit of the astrometric data of the M9 dwarf TVLM 513 indicates that this UCD has at least one substellar companion, TVLM 513b. Furthermore, the estimated weighted average mass and semimajor axis of TVLM 513b are 0.347 ± 0.035 M_J and 0.2789 ± 0.0034 au, respectively, when assuming the lower mass limit of TVLM 513 (0.06 M_⊙), or 0.418 ± 0.040 M_J and 0.3072 ± 0.0040 au, respectively, when assuming the upper mass limit of TVLM 513 (0.08 M_⊙). The estimated weighted average period and inclination angle of the orbit are 221 ± 5 days and 80° ± 9°, respectively. The estimated mass is consistent with this planetary companion being a Saturn-mass planet (0.30 M_J). Figure 6 shows all the confirmed planets that have been found up to now for which the planetary mass has been estimated (either m_p or m_p × sin(i)). We include TVLM 513b in this figure. This figure shows that TVLM 513b is located in a region in the M_* − m_p and M_* − a_p phase space where very few planets have been found. TVLM 513 is one of the lowest-mass stars with known Jovian-mass planetary companions.

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The estimated weighted average astrometric signal of TVLM 513 is 0.145 ± 0.019 mas. Although this astrometric signal is relatively small, it is consistent with a planetary companion associated with this M9 UCD. However, this astrometric signal could be contaminated by the expected astrophysical "jitter" added to the true source position due to stellar activity. It is estimated that M9 UCDs have a stellar radius of ∼0.1 R_⊙ (e.g., Dahn et al. 2002; Hallinan et al. 2006; Chabrier et al. 2000). Thus, assuming that the radio emission originated within ∼one stellar radius of the photosphere (e.g., White et al. 1994), the expected radius of TVLM 513 at a distance of 10.7 pc is about 0.05 mas. Thus, the expected jitter is about a factor of 3 smaller than the astrometric signal observed in TVLM 513. This result supports the detection of the planetary companion TVLM 513b.

As we have mentioned before, in recent years, it has been found that giant-mass planets, such as the one we have found orbiting TVLM 513, have a very low occurrence around UCDs, which is consistent with predictions of planetary formation theories. The core-accretion theory predicts that the formation of giant-mass planets scales with the mass of the central star; thus, it is expected that very few Jovian-mass planets are formed around UCDs (e.g., Laughlin et al. 2004; Kennedy & Kenyon 2008). The core-accretion theory indicates that these planets would be formed in orbits far from the star, at several au. On the other hand, it is expected that disk instability may also be able to form giant-mass planets around UCDs (e.g., Boss 2006). In this case, the orbit of the planet is expected to be relatively closer to the star, from a few to several au. The semimajor axis of the orbit of TVL M513b, a ∼ 0.3 au, is smaller than expected from core-accretion and disk instability theoretical models (e.g., Laughlin et al. 2004; Boss 2006). It may be that TVLM 513b was formed by the same collisional accumulation process that led to the formation of the terrestrial planets in our solar system. Alternatively, TVLM 513b may have formed with a wider orbit at several au from the star and then migrated inward to its current orbit. However, it is not clear what would stop the migration of the planet at 0.3 au. Further theoretical models will be required to understand the formation of giant-mass planets, such as the one we find associated with the M9 UCD TVLM 513.

Finally, to our knowledge, this is the second exoplanet found using astrometry and the first exoplanet found using absolute astrometry. In addition, this is also the first exoplanet found using radio astrometric observations. This result suggests that radio observations with the VLBA can be used to search for giant-mass planets around very low mass stars, such as M dwarfs, and in particular around UCDs.