Binarity Enhances the Occurrence Rate of Radiation Belt Emissions in Ultracool Dwarfs

Despite a burgeoning set of ultracool dwarf ($\leq$M7) radio detections, their radio emissions remain enigmatic. Open questions include the plasma source and acceleration mechanisms for the non-auroral"quiescent"component of these objects' radio emissions, which can trace Jovian synchrotron radiation belt analogs. Ultracool dwarf binary systems can provide test beds for examining the underlying physics for these plasma processes. We extend a recently developed occurrence rate calculation framework to compare the quiescent radio occurrence rate of binary systems to single objects. This generalized and semi-analytical framework can be applied to any set of astrophysical objects conceptualized as unresolved binary systems with approximately steady-state emission or absorption. We combine data available in the literature to create samples of 179 single ultracool dwarfs (82 M dwarfs, 74 L dwarfs, and 23 T/Y dwarfs) and 27 binary ultracool dwarf systems. Using these samples, we show that quiescent radio emissions occur in $54^{+11}_{-11}$ - $65^{+11}_{-12}$ per cent of binaries where both components are ultracool dwarfs, depending on priors. We also show that binarity enhances the ultracool dwarf quiescent radio occurrence rate relative to their single counterparts. Finally, we discuss potential implications for the underlying drivers of ultracool dwarf quiescent radio emissions, including possible plasma sources.


INTRODUCTION
Magnetic activity undergoes a fundamental shift as ultracool dwarfs (M7 and later spectral types) drop in effective temperature and become more analogous to gas giant planets.Ultracool dwarfs at late as ∼T8 (Route & Wolszczan 2012;Kao et al. 2016;Williams et al. 2017;Vedantham et al. 2020Vedantham et al. , 2023;;Rose et al. 2023) can produce bursting radio emissions that are orders of magnitude stronger than predicted by coronal X-ray emissions (e.g.Berger et al. 2001;Williams et al. 2014;Pineda et al. 2017;Callingham et al. 2021).The trio of papers establishing the auroral paradigm of magnetic activity in the ultracool dwarf regime (Hallinan et al. 2015;Kao et al. 2016;Pineda et al. 2017) argued that their coherent and periodically bursting radio emissions, emitted via the electron cyclotron maser instability (Hallinan et al. 2007(Hallinan et al. , 2008)), trace the radio component of aurorae in these objects.Searches for such coherent emissions from ultracool dwarfs have enabled the only direct measurements of magnetic field strengths on planetary-mass objects outside of the Solar System (Kao et al. 2016(Kao et al. , 2018)), yielded the first direct discovery of a brown dwarf using radio observations (Vedantham et al. 2020), confirmed that late T dwarfs can host kiloGauss magnetic fields (Route & Wolszczan 2012, 2016a;Williams & Berger 2015;Kao et al. 2016Kao et al. , 2018;;Rose et al. 2023), and demonstrated that our understanding of dynamos in the mass and temperature regime bridging low mass stars and planets remains limited (Kao et al. 2016(Kao et al. , 2018)).
Ultracool dwarfs can also produce long-lived incoherent and quasi-steady nonthermal radio emissions (e.g.Hallinan et al. 2007;Kao et al. 2016Kao et al. , 2023;;Guirado et al. 2018;Climent et al. 2022Climent et al. , 2023;;Kao & Shkolnik 2024, and references therein), which extends to at least 95 GHz (Williams et al. 2015b;Hughes et al. 2021) and which numerous studies attribute to optically thin gyrosynchrotron emissions (e.g.Berger et al. 2005;Osten et al. 2006;Williams et al. 2015b;Lynch et al. 2016).Williams et al. (2015b) proposed that a blanket of low-level flaring could be one possible explanation, and indeed strong white-light flares can occur on objects as late as early to mid-L dwarfs (e.g.Gizis et al. 2013;Paudel et al. 2018a;Jackman et al. 2019;Paudel et al. 2020;Medina et al. 2020).At present, the possibility of flare behavior extending to cooler objects remains unknown, as is also the case regarding the existence and rate of weaker radio flares in the coldest T-and Y-spectral type brown dwarfs.
However, existing studies show that flare activity may not correlate with radio activity on ultracool dwarfs.Flares become less frequent with later spectral types (Paudel et al. 2018a;Medina et al. 2020), and emission strengths from the chromospheric activity marker H also decrease (Schmidt et al. 2015;Pineda et al. 2017).If flares indeed correlate with quiescent radio emissions, we expect the latest type ultracool dwarfs to similarly exhibit a lower radio occurrence rate when all other factors are equal.Instead, T/Y dwarfs do not exhibit a lower quiescent radio occurrence rate than M dwarfs (Kao & Shkolnik 2024).Though these results do not control for existing biases in science that prioritize publishing detections over non-detections, nor for other possible confounding factors such as age or rotation rate, they are consistent with other studies finding that ultracool dwarf quiescent radio emissions misalign with flare behaviors.For instance, quiescent radio luminosities from ultracool dwarfs depart (Williams et al. 2014;Pineda et al. 2017) from well-established X-ray vs. radio correlations for stellar flare activity (Guedel & Benz 1993).Instead, for auroral emitters, they correlate with H luminosities (Pineda et al. 2017;Richey-Yowell et al. 2020) attributed to aurorae (Hallinan et al. 2015;Kao et al. 2016;Pineda et al. 2017).Together, these findings suggest that flare activity cannot account for quiescent ultracool dwarf radio emissions.Hallinan et al. (2006) postulated an alternative interpretation in which these objects' quiescent radio emissions trace dense regions of energetic particles trapped in their large-scale magnetospheres, similar to the radiation belts possessed by magnetized Solar System planets (Sault et al. 1997;Bolton et al. 2004;Clarke et al. 2004;Horne et al. 2008;Mauk & Fox 2010).In later work, Pineda et al. (2017) and Kao et al. (2019) developed this idea into a coherent framework tying together the auroral and quiescent components of ultracool dwarf radio emissions.Using multi-epoch imaging at 8.4 GHz, Kao et al. (2023) demonstrated that quiescent radio emissions from the canonical auroral ultracool dwarf LSR J1835+3259 traces synchrotron instead of gyrosynchrotron emissions from ∼15 MeV electrons in Jovian radiation belt (or Earth van Allen belt) analogs.Notably, they showed that the observed radiation belts cannot trace electrons accelerated solely by flare processes.A later epoch of resolved imaging at 4.5 GHz confirm their result (Climent et al. 2023).Together, these images show persistent equatorial belts of high energy plasma captured in the dipole magnetic field of LSR J1835+3259 and extending further than nine times the object's radius despite the absence of stellar wind from a host star to seed its magnetospheric plasma.All confirmed auroral ultracool dwarfs exhibit similar quiescent radio emissions (Pineda et al. 2017;Kao et al. 2016;Richey-Yowell et al. 2020), though whether such emissions can occur around objects without aurorae remains an open question.
Recently, Kao & Shkolnik (2024) argued for quantifying ultracool dwarf magnetic activity using their quiescent emissions in addition to their periodically bursting aurorae.They noted that the rarity of detected ultracool dwarf radio emissions -between ∼5-10 per cent detection rates for M, L, and T ultracool dwarfs (Antonova et al. 2013;Lynch et al. 2016;Route & Wolszczan 2016b) in volume-limited surveystogether with the wealth of data available in the literature underscored the value of moving toward studies of radio occurrence rates as a function of object characteristics, such as  eff , mass, rotation rate, and age.Occurrence rate studies also lend themselves well to all-sky surveys from current and upcoming instruments such as the LOw Frequency Array (LOFAR1 ; van Haarlem et al. 2013), the Square Kilometre Array (SKA2 ; Braun et al. 2019), and Deep Synoptic Array 2000 (DSA-2000 3 ; Hallinan et al. 2019).Accordingly, Kao & Shkolnik (2024) developed a generalized semi-analytical framework for calculating occurrence rates for approximately steady-state emission or absorption in populations of single objects.By applying this framework to observations available in the literature, they show that quiescent radio occurrence rates are between 15 +4 −4 -20 +6 −5 per cent for isolated ultracool dwarfs and that T/Y and M dwarfs exhibit similar radio occurrence rates.To date, no comprehensive study of ultracool dwarf radio emissions has focused on binaries, yet comparing the population of single objects to ultracool dwarf binaries provides a complementary means of identifying characteristics that influence magnetic activity at the stellar-planetary boundary.For instance, in stars, tidal spin-up may enhance the rotation rates and thus the magnetic activity of the binary population (Zahn 1977;Morgan et al. 2016).Here, we assess whether binarity enhances the quiescent radio occurrence rate of ultracool dwarfs in binary systems compared to their single counterparts and discuss implications.

EXTENDING AN OCCURRENCE RATE FRAMEWORK FOR SINGLE OBJECTS TO BINARY SYSTEMS
Does binarity affect the magnetic activity of ultracool dwarfs?To answer this question, we compare quiescent radio occurrence rates for ultracool dwarfs in single-object versus binary systems by adapting the occurrence rate framework developed by Kao & Shkolnik (2024).We use their notation here, where P denotes probability and P refers to a probability density distribution function.
This Bayesian occurrence rate framework for a sample of single objects considers the probability  single that an object is emitting between an assumed luminosity range  ∈ [ min ,  max ] for a dataset  consisting of individual observations where the  th observation   = {detect  ,  rms i ,   ,  d i ,   }.They treat each observation property as a random variable, where detect  ∈ {0 if undetected, 1 if detected},  rms i is the reported rms noise for each object,   is the distance to each object,  d i is the error for the measured   , and   is the assumed specific luminosity for quiescent radio emission.They show that for single-object systems, the probability P( |  single ) of observing a dataset given a fixed occurrence rate  single = Θ depends on the probability of drawing the particular observational and object properties captured in : where P(  ≥  thr,i |  rms i ,  d i ,   ,  = 1) is the probability that the object has a detectable luminosity  thr,i , and  ∈ {0 = not emitting, 1 = emitting} denotes whether or not an object is emitting within the luminosity range of interest.Here, we follow Kao & Shkolnik (2024) and use a slight abuse of notation, where P( rms i ), P( d i ), and P(  ) refer to the probabilities that an object possesses the given observations/properties within a representative physical interval of interest.We also use distance and parallax interchangeably, and Kao & Shkolnik (2024) provide the algebra for using either.In the analysis that we present in §4.3, we use parallaxes when available and distance estimates otherwise.
To adapt the single-object framework to binary systems, we treat each system as an unresolved binary, as is the case for most observations reported in the literature.This approach allows us to account for the fact that we often do not know which component in the binary is emitting for detected systems.Thus,  single becomes the occurrence rate for quiescent radio emissions in binary systems  binary .Kao & Shkolnik (2024) describe how to calculate P( rms i ), P( d i ), and P(  ), which remain unchanged for unresolved binary systems.They additionally describe how to calculate P(  ≥  thr,i |  rms i ,  d i ,   ,  = 1) for single objects.To calculate this probability that an emitting object is also detectable, they integrate over the set of possible object distances.They also integrate over detectable object luminosities from an assumed luminosity distribution for emitting objects P ( single |  = 1) within a luminosity range of interest  single ∈ [ min ,  max ]: ( The only change required for applications to unresolved binary systems is replacing P ( single |  = 1) with the luminosity distribution of unresolved binary systems given that the system is emitting, P ( binary |  = 1), between an assumed luminosity range of interest  binary ∈ [ binary, min ,  binary, max ].Below, we define these minimum and maximum luminosities for binary systems.
For the remainder of this section, we address P ( binary |  = 1).To compare binary and single-object samples, we construct P ( binary |  = 1) within the context of the luminosity distribution for single objects P ( single ).This is because unresolved binaries emit when one or both of their components are emitting.Furthermore, they can satisfy the emission threshold even if both components are individually too faint to meet it ( §2.1).In this framework, we account for all cases.
The luminosity for an unresolved binary is equal to the summed luminosities of the  and  components such that  binary =   +   .We assume that the luminosities for the individual components each follow the distribution P ( single ) , which gives the convolution P ( binary ) = P ( single ) * P ( single ) , 0 ≤  binary ≤ 2 max 0 , otherwise . ( Notice that this extends possible luminosities for P ( binary ) to higher luminosities.Accordingly, the binary luminosity range of interest for where we remind the reader that  max is the maximum luminosity for a single object.
Here, the probability mass for P ( single ) is distributed such that the occurrence rate of quiescent radio emissions for a single object,  single , is equal to the integrated probability mass between the minimum and maximum possible luminosities for a single object: The remaining probability mass 1 −  single is distributed over  single ∉ [ min ,  max ].Notice that the combined luminosity of an unresolved binary depends on how probability mass is distributed both inside and outside the luminosity range of interest for single objects (see §3.2).
When we consider the case that both components are emitting, the distribution of probability mass 1 −  single outside of  single ∉ [ min ,  max ] becomes relevant.For instance, an individual object that is emitting at  single = 0.8 min is not considered "emitting" in the single-object framework.In contrast, if both components in a binary are emitting at that luminosity, the binary luminosity  binary = 1.6 min is within the binary luminosity range of interest.In §3, we discuss our choice for distributing the non-emitting probability mass 1 −  single .
We can now re-write equation 1 to: (5) Here, we have also added the multiplicative term P(B), or the probability that an object is in a given population , where B ∈ [single, binary].This allows us to condition on the appropriate luminosity distribution for each population.The probability that an emitting object's radio luminosity is detectable now depends on the assumed binary radio occurrence rate, and by implication, the single-object radio occurrence rate.
2.1 Constructing a predicted binary occurrence rate  predict from the single-object occurrence rate  single Our experiment design is straightforward: We calculate the actual quiescent emissions occurrence rate for binaries  binary from equation 5 and compare it to a "control" occurrence rate.We design the control to reflect the scenario where binarity does not impact the occurrence rate of quiescent emissions in ultracool dwarfs.If the actual occurrence rate differs from the control occurrence rate, we conclude that binarity does in fact impact the occurrence rate for quiescent emissions in binaries.
To construct the control occurrence rate, we assume that binarity does not affect the radio occurrence rate of individual components or their luminosities.In §3, we examine this latter assumption.With these assumptions in hand, we can now calculate the occurrence rate of quiescent emissions predicted for binaries  predict from the single-object occurrence rate  single by considering the appropriate change of variables.
The sum of both components in a binary fall into the binary luminosity range of interest such that We choose an appropriate  max such that the single-object luminosity prior takes the form where we discuss  max in §3.2, and F 1 ( single ) and F 2 ( single ) are normalized density functions that depend on  single .This generalized form meets the condition defined in equation 4 and gives us the freedom to examine how distributions with different shapes impact the final calculated binary occurrence rate.For instance, a situation where objects are either "on" or "off" can be described by a luminosity prior where the probability mass for  single <  min is concentrated as a delta function at  single = 0.While this is not the case for thermal emission, it may be the case for non-thermal magnetospheric radio emission: not all planets in the Solar System host radiation belts, and not all ultracool dwarfs may flare.By selecting an appropriate  max ( §3.2), equation 6 simplifies to  binary = P(  +   ≥  min ).
We then account for all combinations of   +   by invoking the law of total probability and integrating where P (  = ) is the probability density distribution that the  component has a luminosity equal to .This distribution is our chosen single-object luminosity distribution, such that P (  = ) = P ( single = ), and it gives the relationship between  single and  predict .Note that integrating equation 3 for  binary >  min equivalently arrives at equation 8.In Appendix A, we show that we can write the predicted binary quiescent emissions occurrence rate as where C( single ) = C(1 −  single ) is a correction function and C is a constant that depends on how the probability mass for single-object luminosities is distributed for  single <  min .In the case where objects are either "on" or "off", the non-emitting probability mass is a delta function at zero and C( single ) = 0. Intuitively, this is because the summed luminosities of an emitting and non-emitting component still fall below  min .The occurrence rate of binaries with quiescent luminosities within [ min ,  max ] is then one minus the probability that neither component is emitting.When the mass is not distributed as a delta function, C( single ) is positive and accounts for the additional pairs of objects that may be too faint individually to fall above  single ≥  min but together are luminous enough to fall above  binary ≥  min .From equation 9, we see that  predict >  single and expect that unresolved binaries are more likely to emit radio emissions than their single counterparts when all else is equal.

Do individual binary components follow the same luminosity distribution as single objects?
Self-consistent comparisons between single and binary systems require a binary luminosity prior constructed from that of single objects ( §2).
For our analysis, we do not condition the underlying single-object luminosity distribution P ( single ) on binarity.One consequence of our treatment is that we expect unresolved binary systems to be more luminous on average than single objects when all else is equal.Indeed, the data seem to support this assumption, with mean quiescent radio luminosities for detected binaries brighter than that of detected single objects (Kao & Pineda 2022).
Figure 1 further assesses our assumption by examining empirical cumulative distribution functions (ECDFs) using the Kaplan-Meier estimator for observed single objects, binaries, and a hypothetical set of binaries constructed with our single-object sample.§4.1 describes our data.Briefly, the Kaplan-Meier estimator is a non-parametric statistic used in survival analysis (Kaplan & Meier 1958;Lee & Wang 2003;Feigelson & Jogesh 2012), which can incorporate non-detections and limits within cumulative distribution functions.In our case, it estimates the probability that the observed object's luminosity is less than some value.
We use the full set of compiled observations from the literature for objects that are within 35 pc, including non-detections.This distance cutoff is consistent with all detected objects, and it includes 154 and 25 single and binary systems, respectively, or 86 per cent and 93 per cent of observed systems.Figure 1 shows a histogram of 4 sensitivity limits for all undetected single object systems with radio observations that have 3 distances within 35 pc.For objects with repeated observations, we use the median measured luminosity.We treat binaries as unresolved systems by combining the reported luminosities of each binary component for resolved systems such as 2MASS J07464256+2000321AB Figure 1.Left: Empirical cumulative distribution function for all observed single objects that meet the data inclusion criteria described in §4.1 (teal), a hypothetical set of binaries that have   = 2 single (blue), and detected binary systems (purple) within 35 pc.Shaded regions are 95 per cent confidence intervals, calculated with the exponential Greenwood's formula, a standard survival analysis error estimator (Feigelson & Jogesh 2012).Detected objects with more than one measurement are represented with their median value.Distributions and confidence intervals are the mean distributions from 5000 trials in which we randomly draw single objects to match the spectral type distribution of the binary systems.The low luminosity end of the binary distribution is consistent with the single-object distribution, and the high luminosity end is consistent with the hypothetical distribution.Right: Histogram of 4 luminosity limits for undetected single objects with 3 distances within 35 pc.85 per cent of single-object observations had luminosity limits that would have been sufficient to detect the brightest quiescent radio emissions observed in single objects (maroon dashed line).The blue dashed line shows our selected  max , which we discuss in §3.2.(Zhang et al. 2020).Finally, each undetected object is represented only once as an upper limit corresponding to the most sensitive observation that is available in the literature for that object.
We account for upper limits by treating non-detections as left-censored data, allowing for the possibility that non-detections are too faint to be detected.The Kaplan-Meier estimator makes the fundamental assumption that all objects in the given data set are emitting quiescent emissions, even if they are not detectable.It is important to note that this assumption may not be true, since some undetected objects may not be emitting non-thermal radio emissions at all.For instance, Kao & Shkolnik (2024) find that ∼15-20 per cent of objects may not be emitting non-thermal quiescent radio emissions brighter than 10 11.7 erg s −1 Hz −1 , and future observations at higher sensitivities may yield a significant number of upper limits at lower luminosities.This would shift the probability mass in the ECDFs for single and hypothetical populations leftward to lower luminosities that better align with "on"/"off" cases.
For our hypothetical set of binaries, we randomly draw objects from the single-object sample to match the spectral type distribution of the individual components in the binary sample.We repeat this trial 5000 times and show the mean distribution and 95 per cent confidence interval in Figure 1.We find that the low luminosity end of the binary distribution is consistent with the single-object distribution and the high luminosity end is consistent with the hypothetical binary distribution.Even though Kao & Pineda (2022) observed that the highest luminosities observed in single ultracool dwarfs could not account for the most luminous detected binaries, controlling for spectral type distributions seems to show that single and binary objects do not have statistically different underlying luminosity functions.This further grounds our choice not to condition the underlying single-object luminosity distribution P ( single ) on binarity.

Choosing a self-consistent luminosity distribution
To assess if binaries have elevated radio occurrence rates relative to analogous single objects, we must use a luminosity distribution that is consistent for both the single and binary populations.Kao & Shkolnik (2024) adopted a luminosity prior with minimum and maximum single-object luminosities that correspond to those observed for the known population of radio-bright ultracool dwarfs at our frequencies of interest.Although we show in §3.1 that the underlying luminosity distribution for the individual components of binary systems is statistically indistinguishable from that of observed objects, the actual set of detected binaries includes systems that can be more luminous than summed luminosities from the most luminous detected single objects (Kao & Pineda 2022).Thus, we cannot adopt the same minimum and maximum observed single-object luminosities that Kao & Shkolnik (2024) used.Instead, we must revise the luminosity range.
First, we choose  max such that all empirical detections of single objects have  single ≤  max and it reasonably accounts for detection limits.Twice the brightest reported quiescent radio luminosity for single objects corresponds to   ∼ 10 13.9 erg s −1 Hz −1 , but the M7+M7 binary 2MASS J13142039+1320011 (also known as NLTT 33370 AB) has luminosities up to   ∼ 10 14.6 erg s −1 Hz −1 (McLean et al. 2011).Surprisingly, resolved observations demonstrate that only one component of this system is emitting (Forbrich et al. 2016).As Kao & Pineda (2022) note, this suggests the possibility that individual components in binary systems may be overluminous compared to single-object systems.We obtain a probability density distribution calculated by differentiating this cumulative distribution function and fit a piece-wise linear prior P ( single |  = 1) (magenta) as well as a uniform prior (black).We run calculations for both of these fitted distributions.
However, radio luminosities for this system can differ by greater than a factor of two (Williams et al. 2015a;Forbrich et al. 2016), and some portion of its radio emissions have been attributed to gyrosynchrotron flares (Williams et al. 2015a).
Since the radio luminosity of 2MASS J13142039+1320011 may not trace solely quiescent radio emission, we turn to the next most luminous binary system, 2MASS J13153094-2649513AB.This L3.5+T7 binary has   ∼ 10 14.2 erg s −1 Hz −1 (Burgasser et al. 2013a), which is still overly luminous compared to the brightest single objects.
This corresponds to a single-object maximum luminosity  max = 10 13.9 erg s −1 Hz −1 , which we choose for this calculation.This upper limit, revised from 10 13.6 erg s −1 Hz −1 , is consistent with reported binary luminosities and it includes 97% of luminosity limits for single objects within 35 pc.Finally, it corresponds to two times the maximum quiescent radio luminosity of detected single ultracool dwarfs, which is consistent with observed levels of variability in ultracool dwarf quiescent radio emissions (Kao & Shkolnik 2024, and references therein).The one outlier is the L2.5 dwarf 2MASS J05233822-1403022, which Antonova et al. (2007) report can vary by a factor of ∼5 from ≤45 to 230±17 Jy with no evidence of short-duration flares during 2-hr observing blocks.Although the low circular polarization of this object rules out coherent aurorae, non-auroral flares at both radio and optical frequencies can persist for at least several hours (e.g.Villadsen & Hallinan 2019;Paudel et al. 2018b).This outlier object has been excluded from the single-object sample by Kao & Shkolnik (2024) on the basis of the uncertain nature of its radio emission, similar to our reasoning for excluding the binary 2MASS J09522188-1924319 from the binary sample (see §4.1).
Following Kao & Shkolnik (2024), we assume two luminosity distributions for our calculations, which we show in Figure 2: • Uniform =1 : Emitting single objects have radio luminosities that are uniformly distributed between  single ∈ [ min = 10 11.7 ,  max = 10 13.9 ] erg s −1 Hz −1 .Here,  min corresponds to the minimum detected radio luminosity for single ultracool dwarfs, and our chosen  max provides an upper bound for existing detection measurements.By including repeat observations of the same object, we can account for the intrinsic variability of individual objects.Calculations that use this luminosity distribution assume that our only prior knowledge about the luminosity distribution are the minimum and maximum values, as is the case with the existing limited data.
• KM =1 : Alternatively, we can assume that emitting objects follow the same luminosity distribution of all measured detections.We construct this prior by fitting the Kaplan-Meier ECDF of emitting single objects with a piece-wise linear function (Figure 2) and differentiating the fitted function.This prior also accounts for intrinsic variability in individual objects, but it may be skewed from the true underlying luminosity distribution due to the non-uniformity of repeated observations and small sample size (33 measurements).The KM probability density distribution initially extends only to   = 10 13.6 erg s −1 Hz −1 , but it is characterized by two concatenated uniform distributions (Figure 2).Accordingly, we extend the luminosity distribution to our chosen maximum limit   = 10 13.9 erg s −1 Hz −1 and re-normalize.
Figure 2 shows our chosen luminosity priors for P ( single |  = 1).Using the same luminosity priors for both the single and binary populations controls for luminosity differences between single and binary populations and allows us to ask whether binaries have elevated occurrence rates relative to analogous single objects.
As discussed in §2, the probability mass outside of the luminosity prior for emitting single objects ( = 1) impacts the probability distribution for the binary luminosities.The single-object radio occurrence rate  single corresponds to the probability that an object is emitting within the assumed luminosity bounds  single ∈ [10 11.7 , 10 13.9 ] erg s −1 Hz −1 .Thus, we attribute  single of the probability mass for P ( single ) within these assumed bounds.Finally, we must decide how to distribute the rest of the probability mass (1 −  single ) between  single ∈ [0, 10 11.7 ) erg s −1 Hz −1 .For this work, we examine the case where (1 − ) of the mass is located at  single = 0.This simplest case equivalently assumes that single ultracool dwarfs only emit radio emissions between  single ∈ [10 11.7 , 10 13.9 ] erg s −1 Hz −1 .
Our assumption is motivated by the fact that a number of factors can contribute to a lack of quiescent radio emissions from a given ultracool dwarf.Such factors include but are not limited to the absence of a plasma source, magnetic fields that may not be sufficient in size or strength to confine the plasma, or the absence of a mechanism that can accelerate magnetospheric plasma to the mildly relativistic energies required for gyrosynchrotron emissions.Indeed, planets demonstrate that not all objects meet the necessary conditions for producing quiescent radio emissions.For example, the magnetized planets in our solar system (Mercury, Earth, Jupiter, Saturn, Uranus, Neptune) are "on", whereas the unmagnetized solar system planets (Mars, Venus) are "off" (Khurana et al. 2004;Bolton et al. 2004;Girard et al. 2016;Mauk & Fox 2010;Kellermann 1970;Basharinov et al. 1974;Ganushkina et al. 2011).Similarly, deep observations of the nearest brown dwarf binary system Luhman 16AB (2.02±0.15pc, Luhman 2013) and the Y dwarf WISE J085510.83-071442.5 (2.23±0.16pc, Tinney et al. 2014) rule out radio emissions down to   ≤ 10 11.0 and ≤ 10 11.2 erg s −1 Hz −1 , respectively, at 4 significance (Osten et al. 2015;Kao et al. 2019).
Assuming that ultracool dwarfs emit only within the bounds  single ∈ [10 11.7 , 10 13.9 ] erg s −1 Hz −1 captures the observed range of detected ultracool dwarf quiescent radio luminosities.However, the existing set of detected objects may not fully capture the faint end of the ultracool dwarf luminosity distribution.Thus, we also include calculations for a hypothetical P ( single ) with  single ∈ [10 9.8 , 10 13.9 ] erg s −1 Hz −1 .Existing quiescent radio luminosities span 1.9 dex, and the revised lower limit reflects an additional 1.9 dex.

Data
To compare the radio occurrence rates of binary systems to single objects, we use the compilations of single objects in Kao & Shkolnik (2024).This set includes 82 ultracool M dwarfs, 74 L dwarfs, and 23 T/Y dwarfs.To define the luminosity priors that we assume for our occurrence rate calculations in §3.2, we use the combined dataset of all detected quiescent radio emissions measurements for single objects and binaries, the latter of which are compiled in Table 4 of Kao & Pineda (2022).
Finally, we performed a literature search to compile a list of all radio observations, including non-detections, of ultracool dwarf binaries with individual components that have spectral type ≥M7 (Table 1).For objects with multiple recorded observations, we follow the data inclusion policy described in Kao & Shkolnik (2024) to produce a data set that represents observations that are independent from each other: In this data set, each object is represented only once.For each non-detected object, we select its most sensitive observation.For each detected object, we select the detection with the lowest rms noise.Furthermore, the flux densities that we list correspond to quiescent emissions and exclude contributions from identified flares.This latter dataset serves as the input dataset for our occurrence rate calculations in §4.
Our binary sample contains 27 systems, summarized in Table 1.For binaries with reported radio emissions for separate components, we choose the higher rms value to be conservative and designate the binary as detected if at least one component was detected.We include only binary systems for which both components are ultracool dwarfs and exclude triples except for hierarchical systems in which observations resolve the binary and where we can reasonably treat the binary as a self-contained system for our science case.
The triple system 2MASS J12560183-1257276 consists of an L7.0 secondary that is widely separated at ≥ 102 ± 9 AU from its primary component, an M7.5+M7.5 close-in binary system (Gauza et al. 2015;Stone et al. 2016).We include the primary component because it is easily resolved from the secondary in all observations.In units of host star stellar radii, this large separation is ≳300 times that between the Sun and Neptune (Pineda et al. 2021), the latter of which has a faint radiation belt populated by the Solar wind (Mauk & Fox 2010).By mass conservation, stellar wind densities approximately drop off as 1/ 2 at far separations , so we include the secondary in the single-object sample.
We also include the binary systems 2MASS J00043484-4044058 BC (GJ 1001 BC) and 2MASS J22041052-5646577 ( Ind Bab) using similar reasoning.These systems are part of widely-separated triple systems in which the primary components are separated from the secondary binary systems by ∼180 AU (18 ′′ , Golimowski et al. 2004) and ∼1459 AU (402 ′′ , Scholz et al. 2003).We do not include the primaries for these systems in the single sample because they are not ultracool dwarfs.Instead, they are M4 and K5V stars, respectively.
We exclude from our binary sample 2MASS J00244419-2708242, LP 415-20, GJ 564 BC, GJ 569 Bab, and 2MASS J09522188-1924319.For 2MASS J00244419-2708242, the ATCA 6A configuration cannot resolve this triple system.For LP 415-20, compelling evidence suggests that the primary component of this system, which radio observations cannot resolve from its secondary, may in fact itself be an unresolved ultracool dwarf binary system (Dupuy & Liu 2017).GJ 564 BC resides in a triple system that is unresolved by the VLA.The primary component is a solar analog at a distance of ∼47 AU, or ∼1.6 times the separation of Neptune from the Sun (Potter et al. 2002;Dupuy et al. 2009).Although solar radio emissions from 4-8 GHz during solar maximum at this object's distance is not detectable by the VLA (Villadsen et al. 2014), we cannot rule out the possibility that the solar wind from the primary may populate radiation belts around the secondary binary, so we cannot confidently treat the secondary component as a self-contained binary system.GJ 569 Bab is also in a triple and separated from its M3 primary by ∼50 AU (5 ′′ , Forrest et al. 1988).Although Berger (2006) do not report the observation date, VLA configuration, or observing frequency for  2023) report only bursting emission.Distances calculated from parallaxes are provided for the reader's convenience but truncated to three significant figures.We report upper limits or flux densities as presented in the literature but define detected = 1 if the signal-to-noise ratio ≥ 4.0 to remain consistent with our occurrence rate calculation.† Excluded from the binary sample (see §4.1). a The primary component of this system may be an unresolved ultracool dwarf binary (Dupuy & Liu 2017).b Observed with ATCA in 6A configuration, which cannot resolve this triple system.c McLean et al. (2012) list 2MASS J00275592+2219328 as LP 349-25 B, but the VLA cannot resolve this binary.d Reid et al. (2002) report that 2MASS J09522188-1924319 is an unresolved double-lined spectroscopic binary, but Guenther & Wuchterl (2003) observed only a single line.Its detected radio emission may be flaring rather than quiescent emission (McLean et al. 2012).e The discovery that the primary in the 2MASS J12560183-1257276 AB system is actually an equal-mass binary (Stone et al. 2016) suggests that this system may be as far as 17.1 ± 2.5 pc rather than the 12.7 ± 1.0 pc reported by Gauza et al. (2015).The wide ≥ 102 ± 9 AU separation between the primary and secondary components (Gauza et al. 2015) is easily resolved in all observations.f GJ 569 Bab is a companion to an M3 primary that is separated by ∼53 AU.This corresponds to ∼5" (Forrest et al. 1988) and is resolved by the VLA.g GJ 564 BC resides in a triple system, where the primary component is a solar analog at a distance of ∼47 AU (Potter et al. 2002;Dupuy et al. 2009).This separation is not resolvable by the VLA for the reported observations.h 2MASS J00043484-4044058 (GJ 1001) is a triple system, in which the M4 primary is separated by ∼180 pc from the L5+L5 binary secondary (Golimowski et al. 2004).The observation reported here is for the binary component of the triple.This distance is easily resolved, though Golimowski et al. (2004) do not give error estimates for their measured angular separation.i Dupuy & Liu (2017) note that the orbit fit for 2MASS J12281523-1547342 is of questionable quality.j 2MASS J22041052-5646577 is also known as  Ind Bab.
predicted: Figure 3. Calculations comparing binary to single-object systems using the full set of data available in the literature and assuming a uniform luminosity prior between  ,single ∈ [10 11.7 , 10 13.9 ] erg s −1 HZ −1 .Left: Occurrence rate distributions of quiescent radio emissions in ultracool dwarf binaries (mauve) and the predicted occurrence rate distribution for binaries from single objects, where  predicted = 1 − (1 −  single ) 2 (blue).Shaded regions correspond to the 68.3% credible intervals.Individual components in binary systems are more likely to emit radio emissions than their single counterparts.Right: Probability density distribution for the difference in occurrence rates Δ =  binary −  predict between binaries and predictions from single objects.Shaded regions correspond to 68.3%, 95.5% and 99.7% credible intervals.Existing data rule out the null hypothesis that binarity does not affect quiescent radio occurrence rates.
this target, we surmise from an archive search that this target was observed during VLA B configuration at C band (Project Code AB1179, PI Bower).The separation between GJ 569 Bab and its primary is resolvable by the VLA for this observation.However, ∼50 AU is only ∼4.5 times the stellar radii separation between Neptune and the Sun, so we cannot be confident that GJ 569 Bab remains largely uninfluenced by its primary star.Finally, we exclude the M7+M7 binary 2MASS J09522188-1924319 because the radio emissions detected from this object may be flaring rather than quiescent emissions.McLean et al. (2012) detected 233 ± 15 Jy emissions from this object, which corresponds to   = 10 14.4 erg s −1 Hz −1 .However, they reported that followup observations at 4.96 and 8.46 GHz after the initial detection did not yield a detection to a limit of 69 Jy, or a factor of 2.4 below the original detection.They concluded that the initial detection was likely a flare or that 2MASS J09522188-1924319 exhibits long-term variability.Excluding this object gives a more conservative binary quiescent radio occurrence rate.

Analysis
To assess if binaries have an observed quiescent radio occurrence rate that is consistent with that of analogous single-objects, we compare occurrence rate distributions for two populations in Figure 3: the set of observed binaries, and a control set of constructed binaries.
For the control set, we randomly draw single-object systems without replacement to match the M, L, and T dwarf distribution of the individual components in the binary sample.The number of observed T dwarf systems limits the sizes of our drawn samples to 108 objects.We account for different possible combinations of single-object systems by repeating this procedure 5000 times and calculating the probability density distribution of the occurrence rate for each trial using the single-object framework defined in Kao & Shkolnik (2024).In other words, this occurrence rate reflects the occurrence rate that one would observe if each binary were split into its individual components.
We then take the mean distribution for single objects P ( single ) and transform it to a predicted distribution for our hypothetical binaries P ( predict ) by ensuring that which is the standard transformation for random variables.Equation 9gives  predict = 1 − (1 −  single ) 2 .We compare P ( predicted ) to the actual calculated binary radio occurrence rate distribution P ( binary ) in Figure 3. Finally, we also calculate the probability density distribution for differences in occurrence rates Δ between predicted and actual binary  Kao et al. (2016Kao et al. ( , 2018) ) to reduce possible bias from their selection effects.
systems, where We can then calculate the probability that the binary radio occurrence rate exceeds the predicted occurrence rate by integrating

Results: Occurrence rate of binary systems compared to predictions from single objects
In Figure 3, we show that binary ultracool dwarf systems have an elevated radio occurrence rate compared to the rate predicted from their single counterparts.This figure uses the full set of objects that have been observed at radio frequencies and assumes that  ,single ∈ [10 11.7 , 10 13.9 ] erg s −1 Hz −1 .We find a quiescent occurrence rate of 56 +11 −11 per cent for the binary sample, where uncertainties correspond to 68.3 per cent credible intervals.In comparison, the predicted binary occurrence rate from single-object systems with the same spectral type distribution as our binary systems is 23 +8 −7 per cent.The radio occurrence rate of binaries exceeds the predicted rate from single objects with probability P( binary >  predict ) = 100 per cent, so we conclusively rule out the null hypothesis and further explore implications for the observed binary enhancement of radio occurrence rates in §5.
We also find that binarity may enhance the ultracool dwarf quiescent radio occurrence rate even when we exclude objects from Kao et al. (2016).This targeted study selected objects with likely markers of auroral magnetic activity at other wavelengths such as H emissions and infrared variability.These selection criteria resulted in a very high detection rate (80 per cent) compared to the 5-10 per cent detection rate in the literature (e.g.Route & Wolszczan 2016b;Richey-Yowell et al. 2020) that is consistent with that of volume-limited surveys (Antonova et al. 2013;Lynch et al. 2016, e.g.).While Richey-Yowell et al. (2020) showed that infrared photometric variability at 0.5-4 m does not trace quiescent radio emissions, but they reaffirm an earlier finding by Pineda et al. (2017) that H emissions are correlated with the quiescent radio of ultracool dwarfs that have detected radio aurorae.Excluding these objects removes any obvious bias for magnetic activity in our samples.With the exception of the Kao et al. (2016) study, other survey attempts to introduce a positive detection bias have not yielded detection rates that are distinct from volume-limited studies (e.g.Richey-Yowell et al. 2020).Similarly, attempts to target individual objects that may be promising have yielded mixed results (e.g.Audard et al. 2005;Pineda & Hallinan 2018).We therefore elect to include all remaining available data to allow the hypothetical effects of these various selection attempts to average out.
For both datasets, we repeat this calculation for four luminosity distribution cases.We summarize these calculations in Table 2.In all cases, binary ultracool dwarfs have an elevated quiescent radio occurrence rate.
The detection rate for our sample is 37 +11 −11 per cent (with bootstrapped 68.3 per cent confidence intervals), or ∼57-69 per cent of the calculated binary quiescent radio occurrence rate depending on the luminosity prior.This is consistent with correction factors between ∼51-66 per cent calculated for single-object systems using simulated data that followed literature distributions for relevant properties (Kao & Shkolnik 2024).These correction factors can be attributed largely to noise floors, such that detection rates begin to converge with occurrence rates as rms noise for samples decrease (Kao & Shkolnik 2024).This calls for re-observing undetected systems with high noise floors, which may yield new detections.

DISCUSSION
Comparing binary versus single-object systems gives strong evidence that binarity may enhance the occurrence rate of quiescent radio activity in ultracool dwarfs.The radio occurrence rates that we report are specifically for non-flaring non-thermal radio emissions, suggesting that radiation belts are more likely to develop around ultracool dwarfs in binary systems than in their single-object counterparts.We additionally rule out excess flaring induced by magnetic interactions (e.g.Morgan et al. 2012;Lanza 2012).The smallest separation binaries in our sample (∼ 0.6 AU) exclude significant magnetospheric interactions.The strongest ultracool dwarf dipole fields (∼5 kG; Kao et al. 2018) will decay as 1/ 3 to ≲ 20 nG at the half-way point between binary components, which is less than ISM field strengths (Sofue et al. 2019).
Since the quiescent emissions that we focus on does not directly trace flares (e.g.Kao et al. 2023, see also §1 and §4.1 in this paper), our finding of a binary enhancement effect raises interesting questions about the plasma sources for this emissions.
One possibility is that volcanic activity from satellites may not be the dominant source of emitting magnetospheric plasma for ultracool dwarfs.Planet occurrence rates are suppressed for close-in binary stellar systems compared to wide-separation binaries and single-star systems (Moe & Kratter 2019).Theory suggests that this suppression occurs because binaries can truncate the mass and radius of the circumprimary disk, increase disk turbulence, and/or clear out disk material on timescales faster than planet formation (e.g., Artymowicz & Lubow 1994;Haghighipour & Raymond 2007;Rafikov & Silsbee 2015).Like stars, ultracool dwarfs are known to host planets (e.g. the TRAPPIST-1 system, Gillon et al. 2017;Hardegree-Ullman et al. 2019) and theory predicts multiple rocky satellites around brown dwarfs or gas giant planets (Canup & Ward 2006;He et al. 2017;Cilibrasi et al. 2021).Thus, similar mechanisms may suppress planet formation around close-in brown dwarf binaries.In this scenario, we would also expect a suppressed quiescent radio occurrence rate if volcanic satellites are a key electron source.
We emphasize, however, that our results do not rule out plasma contributions from satellites.As an example, massive companions on non-circular orbits may lead to higher occurrence rates of tidally-driven heating in close-in satellites orbiting the radio-bright components of binaries in our sample (e.g.Mardling 2007).Alternatively, other mechanisms may on average outweigh the contribution of volcanic activity from satellites.For instance, if binarity enhances the rotation rates of ultracool dwarfs, this could in turn increase the occurrence rate of luminous radiation belts by both enriching ultracool dwarf magnetospheric plasma environments as well as supplying a larger reservoir of energy to power radiation belt electron acceleration mechanisms.
For the first case, increased rotation-driven flaring on ultracool dwarfs may provide a significant source of plasma for their radiation belts.Though Kao et al. (2023) showed that quiescent radio emissions are unlikely to directly trace flare emissions, they also proposed that flares may seed the magnetospheres of the warmest ultracool dwarfs with electrons that then undergo several acceleration stages to reach the MeV energies found in their radiation belts.This hypothesis was motivated by recent studies demonstrating that flares can occur on objects that are at least as late as L5 (Paudel et al. 2018a;Jackman et al. 2019;Paudel et al. 2020;Medina et al. 2020).Ultracool dwarfs appear to flare less frequently and with less energy as they decrease in spectral type (Paudel et al. 2018a;Medina et al. 2020), so flare contributions to ultracool dwarf magnetopsheric plasma populations may only be significant for the warmest objects.In this scenario, ultracool dwarf binaries may experience rotation-enhanced flaring similar to the increased magnetic activity from binary stars attributed to enhanced rotation rates from tidal spin-up (Zahn 1977;Morgan et al. 2016).Similar phenomena occurring for binary ultracool dwarfs would enrich their magnetospheric plasma environments compared to isolated objects.How rotation drives increased ultracool dwarf flare activity is an open question given key differences in atmospheric ionization and internal structure relative to more massive stars.Nevertheless, rotation increases flare activity even in stars with fully convective dynamo regions (Medina et al. 2020) similar to those possessed by ultracool dwarfs.
For the second case, radiation belt activity itself is likely also enhanced by increased rotation rates when all else is equal.For Jupiter's radiation belts, rotation contributes to various stages of electron acceleration (Krupp et al. 2007;Cowley & Bunce 2001;Kollmann et al. 2018).Modeling suggests that rotation is an important energy reservoir for extrasolar radiation belt emissions (Leto et al. 2021;Owocki et al. 2022) and ultracool dwarf radio detection rates indeed increase with increasing projected rotational velocity  sin  (Pineda et al. 2017).In fact, isolated T/Y dwarfs may show tentative evidence of a higher quiescent radio occurrence rate compared to L dwarfs, which Kao & Shkolnik (2024) suggested could be consistent with age-related spin-up (Scholz et al. 2018) contributing to the acceleration of electrons in ultracool dwarf radiation belts.In this picture, binarity-related enhancements in the rotation rates of ultracool dwarfs could also contribute to the acceleration of radiation belt electrons and therefore the production of luminous radiation belts.
Finally, we must acknowledge that one possible contributor to the high binary radio occurrence rate may be that some of the included binary systems are in fact unresolved triples.With higher multiplicity, a greater fraction of combinations of radio-bright individual components can result in an unresolved system appearing radio-bright as a whole.However, ultracool dwarf triple systems are very rare (Bardalez Gagliuffi et al. 2014), and we have excluded all targets that have evidence of higher-order multiplicity.Our results imply that binaries are excellent targets for radio studies that aim to increase the number of known radio-bright ultracool dwarfs because they may have an elevated radio occurrence rate compared to single objects.
Gaining further insight into flare contributions to the plasma reservoir traced by ultracool dwarf quiescent radio emissions will require studies that (1) examine how rotation rates correlate with ultracool dwarf flare rates and (2) compare the quiescent radio occurrence rate for flaring versus non-flaring ultracool dwarfs.If such studies find a higher occurrence rate of quiescent radio emissions in flaring versus non-flaring objects, this may imply that flares provide a significant source of plasma in the magnetospheres of ultracool dwarfs.

CONCLUSIONS
We present the first detailed statistical comparison of non-flaring quiescent radio emissions in ultracool dwarf binary systems compared to their single-object counterparts.Prior to this work, detection rate studies did not account for possible effects of binarity.
After compiling all published radio observations of binary ultracool dwarfs and compare the radio luminosities of binary systems to single objects, we show that although binarity previously appeared to possibly enhance the quiescent radio luminosities of detected ultracool dwarf binary systems relative to single objects, controlling for spectral types may account for this observed phenomenon.
Finally, we show how to apply a generalized analytical Bayesian framework for calculating the occurrence rate of steady emissions in astrophysical populations introduced by Kao & Shkolnik (2024) to comparisons between binary versus single-object systems in a rigorous and self-consistent manner.We then compare the quiescent radio occurrence rate of ultracool dwarf binary systems to single objects with a similar spectral type distribution as for our binary sample and find that binarity enhances the the quiescent radio occurrence rate in ultracool dwarfs compared to their single counterparts.This enhancement effect is consistent with existing evidence that ultracool dwarf quiescent emissions may largely trace radiation belt emissions (Kao et al. 2023;Climent et al. 2023).

Figure 2 .
Figure2.The empirical cumulative distribution function for   for all radio detections of single ultracool dwarfs, calculated using the Kaplan-Meier estimator (teal).We obtain a probability density distribution calculated by differentiating this cumulative distribution function and fit a piece-wise linear prior P ( single |  = 1) (magenta) as well as a uniform prior (black).We run calculations for both of these fitted distributions.

Table 1 .
Radio Observations of Multi-object Ultracool Dwarf Systems from the Literature Note -We do not include the radio-bright T dwarf binary WISEP J101905.63+652954.2 in this table sinceVedantham et al. (

Table 2 .
Calculated occurrence rates For the single sample, single objects were randomly drawn from a samples of 82 ultracool M dwarfs, 74 L dwarfs, and 23 T/Y dwarfs to match the spectral type distribution of the individual components in the binary sample.The number of observed T dwarf systems limits sample sizes for constructed single-object samples.Listed values are calculated from the mean distribution of 5000 trials.Reported uncertainties correspond to 68.3% credible intervals.
Excluding observations from