Minimum Energies and Magnetic Field Strengths of Edge-brightened Compact Symmetric Objects

Compact symmetric objects (CSOs) are a class of compact radio-emitting active galactic nuclei (AGN), similar in morphology to the traditional Fanaroff–Riley type I and type II objects (B. L. Fanaroff & J. M. Riley 1974), but 10–10⁴ × smaller and shorter-lived. CSOs have long been studied using very long baseline interferometry (VLBI), providing the necessary resolution to resolve their symmetric radio lobes. Through an extensive study carried out by S. Kiehlmann et al. (2024a, 2024b; hereafter K24a and K24b), their definition has recently been redefined. In particular, the authors have defined CSOs to (i) exhibit two-sided radio structure symmetrically distributed about the center of activity (which need not be visible), (ii) have a largest projected length <1 kpc, (iii) not have strongly beamed emission toward the observer, and (iv) not have apparent super-luminal motion β_app > 2.5. Criterion (iii) is often concretized by the source exhibiting a lack of variability (<20% yr⁻¹) in conjunction with criterion (iv), which would suggest order unity lobal Doppler factors. An extensive literature search by K24a resulted in the identification of 79 observed bona fide CSOs, and since then, work done by E. E. Sheldahl et al. (2025) has identified an additional 65 CSOs meeting the classification criteria. Defining the spectral index α through F_ν ∝ ν^α, P. V. de la Parra et al. (2024) have shown that CSOs exhibit steep (α < − 0.5) and flat (α ≥ −0.5) spectra between 5 and 8 GHz; CSOs are typically GHz-peaked spectrum sources, but as surveyed by C. P. O’Dea (1998), not all GHz-peaked spectrum sources belong in the CSO class.

S. E. Tremblay et al. (2016) showed that, like Fanaroff–Riley sources, CSOs typically fall into an edge-dimmed class (designated as CSO-1s) and an edge-brightened class (designated as CSO-2s). Using flux-limited complete samples of radio sources, K24b argue that CSO-2s may constitute a distinct class of AGN, as suggested by a tentative sharp cutoff in the size distribution at around 500 pc and a distinct redshift distribution compared to the jetted-AGN population. The former, in particular, suggests that CSO-2s are short-lived sources at surface brightness levels accessible by the Very Long Baseline Array (VLBA). CSO-1s were not analyzed due to their small occurrence rate in the complete samples. CSO-2s are further divided into three classes (CSO-2.0–2.2), where R24 argue that CSO-2s evolve from 2.0s (younger, higher luminosity) to 2.1s and 2.2s (older, lower luminosities), as modeled by A. G. Sullivan et al. (2024) in a (P, D), or luminosity–size diagram. However, these simulations suggest a pile-up of CSO-2s at large sizes and luminosities, which are not consistent with VLBA observations of CSOs (see Figure 1 in A. C. S. Readhead et al. 2024, hereafter R24).

Consistencies with CSO-2 birth rates, lifetimes, and energies led R24 to suggest that CSO-2s could be the result of tidal disruption events (TDEs) of a post-main-sequence star of a few solar masses, as had first been suggested by A. C. S. Readhead et al. (1994). Scaling from known CSO equipartition energies derived previously, R24 estimated a range of equipartition energies from 10⁻⁴ to 7 M_⊙c² among the CSO-2 population. R24 suggest that the low-energy CSO-2s can be triggered by the stellar capture of evolved ∼1 M_⊙ stars, whereas the high-energy CSO-2s can be triggered by more massive stars, despite being rarer.

In this paper, we conduct a more rigorous synchrotron minimum energy analysis of the CSO-2s identified in K24a, K24b, and R24 to derive the characteristic component magnetic field strengths and minimum energies to constrain proposed models of their formation and evolution. We present two different ways of estimating these quantities, the first using individual components of CSO-2s and the second using the full component sum of the radio spectrum. We then model the broadband AGN spectrum of coreless CSO-2s to evaluate whether the synchrotron and external photons from the AGN and host galaxy can well explain their observed X-ray emission without deviating significantly from minimum energy.

Throughout this paper, we assume a flat ΛCDM cosmology with h = 67.66 and Ω_M,0 = 0.3111 (Planck Collaboration et al. 2020). In addition, we define the spectral index α so that F_ν ∝ ν^α.

In this paper, we focus on conducting a synchrotron and inverse Compton emission analysis of CSO-2s to derive magnetic field strengths, minimum energies, and departures from thereof. This requires careful modeling of the volume of the emitting regions of CSO-2s and their optically thin synchrotron spectrum to obtain their bolometric synchrotron luminosities. The data collected to perform this analysis are motivated by two different methods we considered for estimating the CSO-2 lobal minimum energy conditions, each with its own merits and drawbacks. Because the data quality does not allow us to reliably distinguish them, we call any resolved island of flux (jet, knot, lobe, core, or hotspot) a component in this paper.

1. 1.  

   We use spatially resolved VLBA maps at two different frequencies in the optically thin part of the synchrotron spectrum and derive minimum energies and magnetic field strengths for each identifiable component that is not a core, using the volumes and luminosities of each component.

2. 2.  

   We use lower-resolution radio surveys that generally do not allow for the separation of components, and use them to fit a broadband radio spectrum to estimate the component sum of the synchrotron luminosity of the source. We use this spectrum, along with the sum of the VLBA-derived component volumes, to find the total minimum energy and averaged magnetic field strength for the source as a whole.

While method 1 enables us to infer minimum energies and magnetic field strengths on a per-component basis, it can only be applied to CSO-2s that have two high-quality VLBI maps available for which components can be matched by eye. Furthermore, due to missing u-v coverage at the shortest baselines, VLBA maps may underestimate the flux density of components and produce steeper spectral indices when only taking the difference between two maps. Finally, without a full spectral decomposition, it is difficult to know whether the two frequencies are optically thin or partially absorbed for a given component. These concerns justify method 2, for which fitting to a full spectrum can better estimate the overall synchrotron luminosity of the source at the expense of losing the distinction of individual components. However, as discussed later in Section 4, method 2 can suppress the energy contribution from large-volume, low surface brightness components that do not contribute significantly to the synchrotron luminosity.

For method 1, we collected calibrated images from the Radio Fundamental Catalog (RFC; L. Y. Petrov & Y. Y. Kovalev 2025). Out of the R24 sample of 43 sources with spectroscopic redshifts, we identified only 25 to have VLBA maps in the RFC showing unambiguous symmetric emission at two different VLBA frequencies that could be matched by eye. An additional nine sources were identified to have symmetric components at only one frequency and were thus used for component volume estimation in method 2. The remaining nine sources belong in one of the following categories and thus were not considered in the following analyses: (i) they have no available RFC data; (ii) at least two symmetric components cannot be detected in any available RFC map; or (iii) elliptical Gaussians are not a good approximation to the component morphology. Case (iii) occurs when there is a significant amount of diffuse or resolved-out emission. Altogether, we analyze 34 CSO-2s of which 16 are 2.0s, 10 are 2.1s, and 8 are 2.2s based on the R24 classification.

To select VLBA maps, we first prioritize using those that are of good quality and within which symmetric components can be identified by eye. We then prioritize using maps that are taken as close as possible in time to more easily identify cores that dominate intrinsic source variability. For most sources, we use the frequently available S band (2.3 GHz) as our lower-frequency VLBA map and the C (5.0 GHz) or X band (8.4 GHz) as our higher-frequency VLBA map. For sources that peak at higher frequencies or do not have resolved components at lower frequencies, we opt to use the U-band (15 GHz) MOJAVE observations (M. L. Lister et al. 2018) as the higher-frequency map. Seven sources have maps that are separated by at least 2 yr, but the inclusion or exclusion of these sources does not change our conclusions, and spectral indices of components are generally consistent with what is found in the literature. In Table 4 of Appendix A, we list the frequencies and observation dates of the selected RFC maps per source.

For method 2, with the advent of all-sky radio surveys, we are able to measure the shape of the synchrotron spectra. We constructed radio spectra by collecting flux density measurements from literature catalogs, of which the central frequencies span the 54 MHz–442.5 GHz range. We note that the catalogs span multiple epochs.

Sorted by central frequency from low to high, these catalogs derive from the LOFAR LBA Sky Survey (LoLSS; F. de Gasperin et al. 2023), the VLA Low-Frequency Sky Survey redux (VLSSr; W. M. Lane et al. 2012), the LOFAR Two-Metre Sky Survey (LoTSS; e.g., T. W. Shimwell et al. 2017, 2022), the TIFR GMRT Sky Survey Alternative Data Release (TGSS ADR; H. T. Intema et al. 2017), the Galactic and Extragalactic All-Sky MWA Survey (GLEAM; N. Hurley-Walker et al. 2017), the Westerbork Northern Sky Survey (WENSS; R. B. Rengelink et al. 1997), the VLASS Commensal Sky Survey (VCSS; E. Polisensky et al. 2016; T. E. Clarke et al. 2016), the Rapid ASKAP Continuum Survey Low (RACS-low; C. L. Hale et al. 2021), the Rapid ASKAP Continuum Survey Mid (RACS-mid; S. W. Duchesne et al. 2024), the NRAO VLA Sky Survey (NVSS; J. J. Condon et al. 1998), the Faint Images of the Radio Sky at Twenty Centimeters (FIRST; R. H. Becker et al. 1995), the Rapid ASKAP Continuum Survey High (RACS-high; S. W. Duchesne et al. 2025), the VLA Sky Survey (VLASS; Y. A. Gordon et al. 2021), the Combined Radio All-Sky Targeted Eight GHz Survey (CRATES; S. E. Healey et al. 2007), the VLA Calibrator Catalogue (VLA CC)⁴
, the Planck Multi-frequency Catalogue of Non-thermal Sources (PCNT; Planck Collaboration et al. 2018), and the ALMA Calibrator Catalogue (ACC; M. Bonato et al. 2019).

For each survey, we list the central frequency and the point-spread function (PSF) full width at half maximum in Table 5. Uncertainties on the VLA calibrator flux densities are assumed to be 10% for the L and X bands, 20% for the U and K bands, and conservatively 30% for Q, following the broad guidelines on the calibrator site. If there is a quality flag “?” at any configuration or the quality flag is “X” across all configurations, the measurement is ignored. We assume a nominal 10% uncertainty on the CRATES flux densities. We do not account for angular resolution differences across instruments. CSOs are generally compact (≲100 mas in size), and the chosen surveys are either single dish or arrays with short u-v spacing. Therefore, CSOs will generally be unresolved, and if not, any extended diffuse emission would be accounted for. While the Atacama Large Millimeter/submillimeter Array (ALMA) can achieve resolutions where CSOs become resolved in the most extended configuration, image plane model fitting was done to model the flux of potentially resolved sources (M. Bonato et al. 2018). Similarly, for the VLA CC, the quality cuts ensure that flux measurements were selected for sources at frequencies where the structure is known and can be used at a minimum as a suitable phase calibrator. Additionally, by nature of the high VLBA surface brightness limits (∼1 mJy beam⁻¹ at GHz frequencies), the selected CSOs are among the brightest at each frequency and are thus less likely to be affected by source confusion.

2.1. Notes on Individual Sources

2.2. High-energy Observations of CSO-2s

For both methods, we apply the same formalism to estimate the synchrotron minimum energies and magnetic fields. If limits on the synchrotron spectrum are to be set by the minimum and maximum energies of the underlying particle energy distribution assumed to be a power law, then the characteristic synchrotron frequency (in SI),

which has been averaged over isotropic pitch angles, will set the lower and upper frequency cutoffs. In this paper, we assume a fiducial and for first-order Fermi acceleration (as done for FR IIs in M. J. Hardcastle & D. M. Worrall 2000), so for typical CSO magnetic field strengths on the order of 10 mG,  the synchrotron spectrum will have cutoffs at and The bounds on the synchrotron spectrum are set by the magnetic field strength of the source and and . We assume that the CSO-2s are continuously refreshing the electron population in their components. Even if the high-energy electrons cool faster than they can get replenished, leading to a cooling break, their contribution to the overall energetics will be minimal. In Section 4, we also discuss the effect of the electron γ range on the minimum energy quantities, and a reduction in can act as fixing a cooling cutoff. Given the different formalisms often used to calculate the minimum energy conditions for CSOs, we clarify our procedure in Appendix C. Equations (C8) and (C9) give the explicit formulae for the magnetic field strength and minimum energy, respectively. The main observables required to use these equations are (i) the volume of the emitting region and (ii) the synchrotron luminosity at a reference optically thin frequency, which are explained in Sections 3.1 and 3.2 for each method. Similar derivations of synchrotron minimum energies and magnetic field strengths can be found in G. R. Burbidge (1956), R. Beck & M. Krause (2005), and M. S. Longair (2011).

Our methodology primarily differs from R24 in that we explicitly model the volumes of the emitting regions from the VLBA maps, and use the intrinsic bolometric synchrotron luminosity determined by integrating an extrapolation of the optically thin part of the spectrum (between the low- and high-frequency synchrotron cutoffs) as a proxy for the total particle content. In contrast, R24 use the scaling relation to estimate the maximum CSO-2 equipartition energy, assuming a peak observed radio luminosity L_ν,peak ≃ 3 × 10²⁷ W Hz⁻¹ and mean lobe size D_lobe ≃ 50 pc. We argue that using the intrinsic synchrotron luminosity best represents the total particle content within the CSO components; the peak radio luminosity, instead, is sensitive to synchrotron self-absorption, and thereby discounts particles in the (frequency-dependent) optically thick interiors of CSO components. Furthermore, our spectral integration domain is the characteristic synchrotron frequency range of electrons with Lorentz factors between some fixed minimum and maximum , rather than the frequency range presently accessible to radio surveys (i.e., 54 MHz–442.5 GHz).

We note that the proton (or more generally, hadronic) contribution to the particle energy density K and the volume filling factor η remain unconstrained and are assumed to be 0 and 1, respectively. We also note that the assumption K = 0 is consistent with that of R24. Additionally, VLBA maps, especially those at higher frequencies, may underestimate volumes V due to lower sensitivity to extended emission and lack of short-spacing u–v coverage. In a minimum energy analysis, uncertainties in K, η, and V are suppressed in the magnetic field calculation given that , where typically α < −0.5. Therefore, order of magnitude uncertainties in (1 + K)/ηV will correspond to uncertainties of order unity on . However, the minimum energy is dependent on larger powers of the unconstrained variables and thus more affected by systematic assumptions on them. For example, an order of magnitude uncertainty on ηV alone can produce uncertainties of a factor of ∼3 on

3.1. Individual Component Analysis

For each of the 25 sources identified to have high-quality VLBA maps at two frequencies, we use the radio source finding tool AegeanTools (P. J. Hancock et al. 2012, 2018) to identify all component (and core) locations. We use a nominal “seeding” index of 10σ and “flooding” index of 4σ above the map rms (which determines the boundary of the ellipse); consequently, all the components considered in this analysis have a signal-to-noise ratio (SNR) > 10. Background rms values are estimated using the program BANE from AegeanTools. We match components across both the VLBA maps for spectral index computation. To improve component matching, we convolve the higher-frequency map to the resolution of the lower-frequency map (if possible and if components do not get blurred together).

Because AegeanTools does not reliably return uncertainties on all fits, we use a Markov Chain Monte Carlo (MCMC) approach via the emcee package (D. Foreman-Mackey et al. 2013b) to obtain uncertainties on the component volumes and flux densities. For each component or group of components that overlap in the image plane identified by AegeanTools, we use the output of AegeanTools as a starting point to fit elliptical Gaussians with freely varying major and minor FWHMs, position angles, and amplitudes; the positions remain fixed. The MCMC samples the intrinsic major and minor FWHMs first and then convolves them by the beam to match the VLBA map, so that uncertainties on the true component size can be obtained. We use a Gaussian likelihood with the derived background rms being the per-pixel intensity standard deviation. We use these samples to propagate uncertainty into the flux density, spectral index, and volume calculations. Due to the higher sensitivity to extended flux, we use the intrinsic Gaussian fit to the lower-frequency VLBA maps to estimate the component volume. We assume that the components are ellipsoids, that the intrinsic projected major and minor FWHMs equal the intrinsic semimajor and semiminor axes, and that the third axis is the average of the two projected axes. For all sources, we fit to the same maps used for AegeanTools.

Figure 1 shows maps at the low frequency of all CSO-2s in the sample (including those with single VLBA maps), with the fitted components labeled. For morphologically simple sources with two elliptical components like J1734+0926, the Gaussians can well explain the emission. However, for more morphologically complex sources like J1159+5820, where there is faint diffuse emission, and J0029+3456, whose central core was not detected in the higher-frequency X-band map and thus not covered by the model, the full emission decomposition is harder to capture with resolved Gaussians using AegeanTools. For such sources, we ignore the contribution by the diffuse emission, assuming that the detected components at the nominal SNR > 10 level at both frequencies dominate the energetics, given that the uncaptured emission is much lower in surface brightness and in filling factor. Furthermore, such emission is not observed in the corresponding higher-frequency map, and for sources with two VLBA maps, we only consider components identified in both maps. There are also sources like J0405+3803 (which has a second core as identified by H. L. Maness et al. 2004) and J0741+2706, where at the lower S-band frequencies, cores are merged into brighter components (C1 for both sources). The spectral indices of these components, however, suggest that these cores are likely faint at the frequencies imaged.

For each noncore component, we use its reference luminosity L₀ corresponding to the rest-frame frequency of the low-frequency map, modeling the intrinsic synchrotron spectrum as a power law (Equation (C4)) and the corresponding component volume from the lower-frequency map to calculate the minimum energy magnetic field strength using Equation (C8) and the minimum energy using Equation (C9). We find typical component magnetic field strengths of ∼20 mG and a rather broad range of minimum energies on the order of 10⁻²–10² M_⊙c².

Cores are filtered out in our calculations by their generally flat or inverted spectral indices α > −0.5; we also confirm by comparing to existing literature on these sources. The following sources had a detected SNR > 10 core at two frequencies that were filtered out: J0943+1702, J1644+2536, J1234+4753, J1244+4048 (S. E. Tremblay et al. 2016), J1511+0518 (T. An et al. 2012), J1158+2450 (X. Yang et al. 2017), J1945+7055 (A. B. Peck et al. 1999), and J2022+6136 (W. Tschager et al. 2000).

3.2. Spectral Analysis

A significant concern with using VLBA map-fitted component fluxes is the possibility of resolving out flux compared to single-dish or short-baseline interferometric observations. We therefore consider using the broadband spectrum to constrain our estimates of the overall synchrotron luminosity of each source, avoiding VLBA total flux densities. We use a double power-law model that enables a fit of the optically thin and optically thick spectral index:

Here, F₀ and ν₀ correspond to a reference flux density and frequency, α₁ is the optically thick spectral index (which can be < 0 if not fully absorbed) and α₂ < 0 is the optically thin spectral index (I. A. G. Snellen et al. 1998). We note that F₀ and ν₀ are not exactly at the break of the spectrum, as stated by I. A. G. Snellen et al. (1998), but will be close in practice; the break frequency and flux density are not analytically found. An alternative parameterization is to do a parabola in log-space, allowing for an estimate of the observed luminosity without knowledge of , , and the magnetic field. However, we find that it tends to underestimate the highest (lowest) frequency measurements, which are more consistent with having minimal spectral curvature with respect to other observations in the optically thin (thick) portions of the spectrum closer to the turnover.

Sample posterior draws along with the maximum a posteriori (MAP) are shown in Figure 2. Fits are done using an MCMC in order to obtain more robust uncertainties on the model parameters, using wide uniform priors on all parameters and requiring the reference flux and frequency to be within the range of the data, with the exception of J0405+3803, which does not appear to have any break frequency over the observed range (see Figure 2). In this case, we set the upper bound on the prior to the break frequency to be the lowest frequency measurement. We allow the optically thick spectral index to be negative for cases like J1434+4236, which are more weakly absorbed (see Figure 2) and may only see a turnover well below the frequencies at which it has been observed. For J0713+4349, we narrow the range of the break frequency to be above 2 GHz to obtain a reasonable fit to the optically thin regime (also in Figure 2). Trace plots of the MCMC chains were visually inspected to confirm convergence in the fit parameters. The ACC 100 GHz measurements are able to strongly constrain our estimate of the optically thin spectral index, and their general consistency with lower-frequency measurements suggests that the global synchrotron cutoff may occur above 100 GHz. However, such measurements are not available in all cases. Significant deviations from the double power-law fit may be due to the intrinsic variability of the sources (particularly the cores), given that data were taken from surveys taken at different times (>20 yr time range), and calibration errors. In particular, we find that for J0713+4349, J1511+0518, and J2022+6136, which are also X-ray sources, more than one radio flux density measurement exists that may be inconsistent with a strict double power law. We show the radio spectra of these CSO-2s in Figure 2. The high-frequency data of J0713+4349 and J2022+6136 are consistent with a constant optically thin spectral index, so we do not expect the fit to bias the energy and magnetic field calculations. For J1511+0518, the high-frequency measurements from the ACC also help solidify the high-frequency spectral index despite the potentially discrepant VLA CC measurements at 22 and 43 GHz. We confirm the result of P. V. de la Parra et al. (2024) that shows that known CSO-2s exhibit a broad range of spectral types, although the majority have a steep high-frequency spectrum: α₂ < −0.5.

For the volumes, we use the sum of the lower-frequency map components (and core, if detected) derived in Section 3.1 from the RFC VLBA map. Since we only require one VLBA map, we include the nine additional sources that did not have two VLBA maps at two frequencies into this analysis, applying the same method to fit the component volumes (see Figure 1 and Table 4 for the specific RFC maps used). We use the reference flux density at 50 GHz, well into the optically thin regime, and the optically thin spectral index to parameterize the intrinsic luminosity density profile with Equation (C4), correcting for redshift. We again use Equations (C8) and (C9) to calculate the magnetic field strengths and minimum energies. For sources with identifiable cores, we do not consider subtracting the core as its contribution to the radio spectrum is not well understood from the typical two RFC VLBA maps at different frequencies. We note that the resulting magnetic fields are then an average over all components of the source (including the core). Including the core, nevertheless, provides a better estimate of the energy lower bound for tests of the CSO-TDE scenario.

In Table 6, we report the intrinsic component sizes, fluxes, spectral indices, and minimum energy quantities from the individual component analysis. In Table 7, we list the spectral indices and minimum energy quantities from the spectral analysis (statistical errors only). R24 note that, owing to their asymmetry, the sources J1159+5820, J1227+3635, J1244+4048, and J1400+6120 are likely mildly relativistically boosted and were thus excluded from their analysis. Indeed, we find that all of these sources have minimum energies generally larger than the rest of the sample, with , and are flagged in the tables. However, for mildly relativistic lobe speeds of β ≈ 0.4 observed in CSOs and L_ν ∝ f_ν ∝ D^3−α for Doppler factor D, given that , jet viewing angles of 0^∘ ≤ θ ≤ 45^∘ produce 1.28 ≤ D ≤ 1.53, which can lead to ≤2.3× larger energies than estimated, comparable to the level of systematics that exist in the modeling as discussed later in this section. We consider the minimum energy and magnetic field strength statistics of the CSO-2 sample by ignoring these four sources. However, we also provide a separate statement of the statistics if these sources were included in Section 4.1.

Ignoring the potential four beamed sources, we show a histogram distribution of posterior median magnetic fields and total minimum energies for both methods in Figure 3. By resampling the individual posterior distributions, we can report uncertainties on the distribution median. Under our fiducial assumptions, the median magnetic field strengths resulting from the individual component analysis and the spectral analysis are and mG, respectively. The median total minimum energies are and , respectively. However, the means of the energy distributions are more consistent than the medians across the two methods when taken in log-space, with and , respectively.

In Figure 4, we compare the minimum energy quantities for sources without cores across the two methods, colored by CSO-2 subclasses. In about two-thirds of the cases, the magnetic field strengths derived from these two approaches are in reasonable agreement, while in the remaining one-third, the spectral analysis underestimates the magnetic field strength relative to the individual component analysis. The minimum energies appear to be more consistent to within a factor of a few. We identify 18/23 from the individual component analysis and 22/30 CSOs from the spectral analysis to have minimum energies larger than M_⊙c². We note that 16/18 of those from the individual component analysis with energies >1 M_⊙c² are in common with the spectral analysis, while the remaining two have minimum energies above and below M_⊙c² but consistent to within a factor of ∼2. No clear trend appears in the energetics of the CSO-2 subclasses, but we find that the CSO-2.2s and most of the 2.1s tend to have weaker magnetic field strengths than the 2.0s, consistent with 2.2s being more evolved. In Figure 5, we plot the distribution of magnetic field strengths from the spectral analysis for each CSO-2 subtype, generally supporting this observation when extending the larger sample of CSO-2s with cores. We find no distinction in the fraction of sources that exhibit minimum energies >1 M_⊙c² between the CSO-2 subtypes. Namely, the fraction of CSO 2.0s, 2.1s, and 2.2s in the sample of 30 used for the spectral analysis is consistent with the fraction of those that are >1 M_⊙c².

4.1. Potential Systematics

The discrepancies between methods are likely due to the following systematics: (i) the resolving out of flux in the VLBA maps, and (ii) the averaging of the spectral index across individual components in the spectral analysis. (i) can lead to systematically higher component-wise magnetic field strengths and energies due to steeper spectral indices, particularly for those that are low surface brightness and large volume. Several individual components in Table 6 have spectral indices α < −1.5, which make up most of the component-wise magnetic field strength measurements ≳75 mG. These magnetic field estimates and corresponding energies are likely overestimated and should thus be interpreted with caution. (ii) can have the opposite effect, dampening out steep spectral indices from large volume, low surface brightness components that have large energies or magnetic field strengths. In the spectral analysis, the brightest components (which may include the core) will set the observed spectral index, potentially leading to milder spectral indices compared to the individual components. While flatter spectral indices typically lead to lower magnetic field strengths, better coverage of the overall synchrotron luminosity constrained by the broadband radio spectrum typically estimates larger energies.

Another potential source of systematic error is the choice of electron and , particularly if the high-energy electrons cool rapidly at the edges of the hotspots. We find that decreasing to 10⁴ or 10³ does not significantly impact the individual component analysis results; all distribution medians are generally consistent within the error bars. However, for the spectral analysis, we find a small reduction in the magnetic field strengths and minimum energies by ∼0.9×  in the magnetic field strength and ∼0.7× going down to . The highest frequencies contribute the least to the total synchrotron luminosity, and fewer electrons have high enough γ to emit at these frequencies. Consequently, increasing will also not significantly impact the minimum energy quantities. We conclude that spectral aging of high-energy electrons would not significantly change our conclusions.

On the other side, by decreasing to 1, we find the median magnetic field strength increases by ∼1.7× and ∼1.3× from the individual component analysis and spectral analysis, respectively. The median minimum energy also increases by a factor of ∼3.3× and ∼1.4×, respectively. This is because there would be more electrons emitting at a lower energy (or γ), and the lowest frequencies dominate the intrinsic synchrotron luminosity density under an extrapolation of the optically thin spectral index. Finally, considering the observable range of frequencies (54 MHz–442.5 GHz), for typical magnetic field strengths on the order of 10 mG, we restrict the electron energy range from to . The median magnetic field strengths become and in the individual component and spectral analyses, respectively. The minimum energies drop to and , respectively, but these frequency bounds are likely not representative of the intrinsic synchrotron spectrum of the CSO-2s.

We investigate the impact on the median magnetic field strengths and energies relative to the fiducial case caused by the inclusion or exclusion of sources that may introduce systematic effects. We still assume the fiducial 10 ≤ γ ≤ 10⁵. Removing the X-ray sources that show evidence of broad X-ray iron emission (three of which show potential deviations from the radio double power-law fits) changes the magnetic field strengths by ∼0.95× and ∼0.7× and minimum energies by ∼1.1× and ∼0.9× across the individual component and spectral analyses, respectively. Alternatively, removing the sources that contain at least one steep spectrum component with α < −1.5 in the individual component analysis and global spectrum with average α < −1 in the spectral analysis, we find changes of ∼1× and ∼0.55× in the magnetic field strengths and ∼1.2× and ∼0.97× in the minimum energies across the two methods, respectively. Ignoring the seven sources with VLBA maps separated by more than 2 yr, we find changes of ∼1.05× in the magnetic field strengths and an ∼0.8× change in the minimum energy for the individual component analysis. We conclude that the median minimum energies are generally robust to the systematics created by the inclusion of sources with unusually steep spectrum components and potential core variability. Finally, with the inclusion of the four asymmetric sources, we find that the median minimum energies are consistent within the errors of the fiducial case. These various systematics tests are summarized in Table 1. In any case, our conclusion that the majority of the VLBA-detected CSO-2s are consistent with having minimum energies >1 M_⊙c² is unchanged.

Table 1. Summary of Systematics Tests on the Median Minimum Energy Magnetic Field Strengths B_ME and Minimum Energies on the Individual Component Analysis (Subscript 1) and Spectral Analysis (Subscript 2)

Note. The last four rows assume the fiducial 10 ≤ γ ≤ 10⁵. All tests ignore the four CSO-2s that are potentially mildly relativistically boosted, with the exception of the last one, where they are included.

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ASCIITypeset image

Finally, we note that these minimum energies and magnetic field strengths are determined for the brightest CSO-2s in the sky and are therefore expected to be the largest among the CSO-2s for any given physical size, as shown in Figure 6. While we expect that in the typical size regime of CSOs, the radio luminosity should gradually rise (M. J. Hardcastle 2018), the radio source size will likely grow faster, resulting in an overall decrease in radio surface brightness. Hence, conservation of magnetic flux requires that at large physical sizes, magnetic field strengths should decrease despite the surface brightness selection effect, which is also observed in Figure 6. We expect a large population of CSO-2s at intermediate sizes not detectable within VLBA surface brightness limits, with significantly lower luminosities and weaker magnetic field strengths.

In order to better estimate the energetics and cooling times of CSO-2s, we seek estimates of the true magnetic field strengths of the CSO-2s in the sample, rather than the minimum energy magnetic field strengths. There is strong evidence that CSOs and GHz-peaked spectrum sources are close to minimum energy (A. C. S. Readhead 1994; A. G. Polatidis & J. E. Conway 2003; M. Orienti & D. Dallacasa 2008). Methods for determining how close radio sources are to minimum energy come in two forms: (i) under the assumption that the source is synchrotron self-absorbed, the spectral turnover can directly constrain estimates of the magnetic field (M. A. Scott & A. C. S. Readhead 1977; M. Orienti & D. Dallacasa 2008), and (ii) modeling the inverse Compton X-ray emission from cosmic microwave background (CMB), AGN, or synchrotron photons (synchrotron self-Compton) generated by the same electrons emitting synchrotron (J. H. Croston et al. 2005; Ł. Stawarz et al. 2008; L. Ostorero et al. 2010). In this section, we focus on method (ii) to examine whether comptonization by the electron population in the components can explain the observed X-ray emission identified in a subset of CSO-2s. We note that method (i) may be less reliable given that it is often difficult to disentangle free–free and synchrotron self-absorption, and both may generally contribute to the break in the spectrum (S. J. Tingay & M. de Kool 2003; J. M. Marr et al. 2013; P. V. de la Parra et al. 2024). Furthermore, if self-absorption is the case, an upper bound on the magnetic field can only be obtained if the angular size at spectral turnover is known (see updated formula in Appendix B of A. C. S. Readhead et al. 2021), which is only approximately known.

With the ability of Chandra to resolve X-ray emission confined to the radio lobes of FR IIs, J. H. Croston et al. (2005) showed that Comptonization of CMB photons could explain their X-ray fluxes, yielding magnetic field strengths that are within order unity of equipartition and thus minimum energy. We note that inverse Compton of non-CMB photon fields has also been considered to account for X-ray fluxes in such radio galaxies (e.g., see G. Brunetti et al. 2001; M. C. Erlund et al. 2008; M. Brienza et al. 2023, and references within). A similar analysis for CSO-2s is particularly challenging; given the largest separation of ∼500 pc (corresponding to subarcsecond scales), the radio lobes are unresolved in Chandra, and Comptonization is expected to occur of host galaxy, AGN, and synchrotron emission rather than primarily of CMB photons, given the former’s larger energy density. Ł. Stawarz et al. (2008) and L. Ostorero et al. (2010) argue that the X-ray emission of GHz-peaked spectrum sources can be well explained by inverse Compton scattering of non-CMB radiation fields, thus motivating this analysis.

Using recent X-ray analyses of CSOs done by S. Swain et al. (2025) and Y.-Y. Gan et al. (2025), we focus on a sample of eight CSO-2s that do not have a detected Fe Kα line nor a strong radio core emission that would otherwise suggest a significant coronal contribution to the X-ray emission. Both independent analyses suggest that the jet/component radiation likely dominates the X-ray emission of the CSOs, given the characteristic photon spectral indices. Previous inverse Compton analyses of CSOs (L. Ostorero et al. 2010; D. Ł. Król et al. 2024) use blackbodies to model the broadband spectral energy distribution (SED). Here, we use AGNfitter (G. Calistro Rivera et al. 2016; L. N. Martínez-Ramírez et al. 2024) to more robustly separate out the host-AGN, dust, and starlight contributions.

We collect IR-UV flux densities from the Sloan Digital Sky Survey (SDSS) DR16 (R. Ahumada et al. 2020), Pan-STARRS1 (PS1) DR2 (E. A. Magnier et al. 2020), Two Micron All Sky Survey (2MASS; R. M. Cutri et al. 2003), and Wide-field Infrared Survey Explorer (R. M. Cutri et al. 2021) and additionally from UKIDSS DR9 (A. Lawrence et al. 2007) for those without 2MASS measurements, correcting for galactic foreground extinction (E. F. Schlafly & D. P. Finkbeiner 2011); extinction in W3/W4 is assumed to be negligible. No available FIR emission is found in existing catalog searches for any selected source. Due to possible source confusion, we set the W3/W4 flux for J1326+3154 and the W4 flux for J1609+2641 to be an upper limit (despite not being given as an upper limit). We use the SDSS model magnitudes and PS1 Kron magnitudes to collect the extended emission of the host galaxy. We ignore radio and X-ray emission, considering only fits in the IR-UV range. We use the S17 cold dust, CAT3D dusty torus, BC03 stellar population, and THB21 accretion disk model templates for SED fitting (see Section 3 of L. N. Martínez-Ramírez et al. 2024, for more details). These measurements are not enough to strongly constrain estimates of the cold dust emission in the mid-IR–far-IR, but are sufficient to constrain models of the stellar, torus, and/or disk component. Figure 7 shows the SED fits for all sources considered, showing that in any case, at least one of the stellar, disk, or torus emission is well constrained. However, the stellar and disk contributions tend to be strongly degenerate.

Given the lack of knowledge about precise source geometry with respect to the accretion disk and that the components are unresolved in X-ray emission, we make the assumption that the external photon fields are isotropic, and their energy densities are averaged across the maximum linear extent of the source. We ignore the contribution to the inverse Compton luminosity by cold dust, given that it will be more widely distributed throughout the host galaxy. We approximate the spectral energy density at the location of the components from the AGN emission as

where l_p is the overall projected length of the source, defined by the separation of the symmetric components. Values are taken from Table 3 of K24a.

The spectral energy density due to synchrotron emission is taken to be

where V is the total component volume. In general, the synchrotron energy density will exceed the contribution by external photon fields, given the smaller volume that the components occupy compared to the overall linear extent (V^1/3 < l_p/2).

Finally, we consider the starlight spectral energy density. Given that the PS1 photometry suggests these CSO-2 host galaxies are extended sources, we use the package imfit (P. Erwin 2015) to fit a PSF-convolved (∼1″ FWHM) Sérsic profile to the i-band PS1 cutout images to estimate the Sérsic index and circularized half-light radius, masking any nearby sources that would bias the fits. We also incorporate a point source for the AGN. PSFs are built using at least two isolated stars near the galaxy with the ePSF of photutils (L. Bradley et al. 2016). Because the PS1 cutouts are already background-subtracted, we estimate the sky level as 3× the rms of the current background above the minimum pixel. This provides half-light radii qualitatively consistent with the images. We approximate the starlight’s spectral energy density to be

where L_ν,⋆(r < l_p/2) = fL_ν,⋆ and f is the fraction of the luminosity within l_p/2, derived using the Sérsic profile. We note that for typical starlight luminosities of 10⁴⁴ erg s⁻¹, the median torus or disk energy density exceeds the starlight energy density by at least 100×, so our conclusions would remain unchanged if the starlight component were not included. However, starlight is still considered for completeness because a few combinations of the SED posterior can result in a nonnegligible starlight spectral energy density.

We calculate the inverse Compton volume emissivity using Equation (7.28) in G. B. Rybicki & A. P. Lightman (1979), which holds in the Thomson regime under the assumption that γ ≫ 1, emission is isotropic, and there are negligible secondary scatterings. We bootstrap the synchrotron, external photon, and unabsorbed X-ray 2–10 keV rest-frame luminosities to get a distribution over the required magnetic field strength to match the sampled luminosity; given the expensive minimization required to solve the nonanalytic equation for the magnetic field, we use only 2500 samples. We have symmetrized the uncertainties on the observed 2–10 keV luminosity and assumed it to be a normal distribution. In Table 2, we list the host galaxy, torus, and accretion disk (dereddened) luminosities outputted from AGNfitter as well as the observed 2–10 keV luminosity and the ratios B/B_ME and , where B_ME is the minimum energy magnetic field, and is the minimum energy obtained from the spectral analysis. Averaging over the sources, we find a typical B/B_ME ∼ 1.9 and a typical , but we note that among individual sources, the statistical uncertainty in the energy ratio is large, with distributions typically having long tails toward large , and model-dependent uncertainties are not included. Even ignoring J1035+5628 and J1944+5448, which have fractional uncertainties in B/B_ME as large as 50% and in larger than 100%, the typical deviations from minimum are unchanged. Three sources are consistent with minimum energy to ∼1σ, and the rest are consistent to within order unity. These results differ from J. H. Croston et al. (2005), who find a typical B/B_eq ∼ 0.7 for FR IIs.

We note that the predicted spectral indices from the inverse Compton analysis are not always consistent with the observed X-ray spectral shape, but a direct comparison is likely fraught with uncertainty for some sources with low signal-to-noise X-ray spectra. This requires the photon index to be fixed or often results in large uncertainties in power-law fits to observations. Furthermore, photon indices tend to differ significantly between epochs or instruments (e.g., see Table 6 of Y.-Y. Gan et al. 2025). In Table 3, we compare the modeled photon index from observations in the literature and predicted from the inverse Compton analysis. For sources where the spectral index is not fixed, the agreement is to within 2σ. We additionally note that changes in the magnetic field by a factor of a few can change the predicted X-ray luminosity by an order of magnitude, which already discourages strong deviations from minimum energy. Therefore, while we urge the reader to be cautious about these comparisons, there is evidence that CSO-2s are close to minimum energy or equipartition under the assumption that the 2–10 keV emission is primarily inverse Compton.

For characteristic synchrotron frequencies of 1 GHz for CSO-2s, and typical minimum energy magnetic field strengths of ∼20 mG, the characteristic single-electron Lorentz factor is γ ∼ 100. Then, the typical synchrotron cooling lifetime at VLBA frequencies is given by

which would be ∼4× smaller for the average deviation from minimum energy we found in the previous section. The shortest kinematic timescale for the largest CSO-2s is approximately t_kin ≥ l_p/2c ∼ 10³ yr. Therefore, if CSO-2s are to abruptly shut off at l_p ∼ 500 pc in size as suggested by the models of A. G. Sullivan et al. (2024) and observations of R24, then CSO-2s will fade on timescales at VLBA frequencies faster than the kinematic timescale. This provides an alternative explanation to prevent the build-up in a population of high-luminosity CSO-2s around 500 pc at VLBA frequencies to those mentioned in A. G. Sullivan et al. (2024). Furthermore, the smaller magnetic field strengths in CSO-2.1s and CSO-2.2s corroborate the evolutionary models suggested by R24 and A. G. Sullivan et al. (2024) that they are more evolved than the CSO-2.0 counterparts.

Among our sample of CSO-2s, which is selected from surveys of the brightest radio sources in the sky, careful modeling of the volume of the emitting regions and the total intrinsic synchrotron luminosity shows that the typical minimum energy is close to the maximum 7 M_⊙c² inferred by R24. The total energy of the radio-emitting components is likely several factors higher than the minimum energy, accounting for departures from minimum energy, kinetic energy of the bulk motion of the lobes, and work done on the ISM as the CSO-2s expand. Furthermore, additional energy can be accounted for in mass that is yet to be accreted and non–radio accretion disk emission. If energy extraction efficiencies of <100% are assumed during a TDE, then the CSO-TDE scenario can hold if massive (>10 M_⊙) evolved stars (>500 R_⊙) are disrupted by massive black holes (see Figure 14 and Equation (4) of R24) to explain the combination of the fallback timescale, kinematic timescale, and the energies. This may be unlikely given their short lifetimes in the giant phase and typical stellar initial mass functions (IMFs). Top-heavy IMFs have been observed in the Milky Way (H. Bartko et al. 2010) and inferred in high redshift starforming and local starburst galaxies (e.g., see C. Weidner et al. 2011; Z. Guo et al. 2024 and references within), but may not be realistic for the elliptical host galaxies of CSO-2s. If the CSO-TDE connection is real, then CSO-2s could provide a window into the IMFs of the cores of elliptical host galaxies. On the other hand, a several solar mass star might require additional energy extraction from the black hole spin to produce these observed energies, which has been invoked previously (R. D. Blandford & R. L. Znajek 1977; R. Blandford & N. Globus 2022).

Certainly, the CSO-2s observed thus far are likely to be the most luminous and energetic due to selection effects in surveys. On the low-energy end, we find that minimum energies of CSO-2s can go down to ∼10⁻² M_⊙c², which can more easily be explained by the disruption of less massive stars without invoking high efficiencies. And deeper complete surveys of CSOs might yield a substantial population of lower luminosity CSO-2s compared to those observed thus far with energies <1 M_⊙c². But if the high-luminosity CSO-2s do turn off after ∼10³–10⁴ yr as the complete surveys suggest, then their origin remains quite uncertain.

Whether these sources can be triggered by other means remains subject to future work. A. G. Sullivan et al. (2024) note that O(10 M_⊙) gas accretion can result in accretion timescales comparable to the lifetimes of CSO-2s, although magnetic pressure or turbulent motion might suspend these gas clouds against collapse. Furthermore, detailed forward modeling is required to check whether the occurrence rates/birth rates of CSO-2s can be commensurate with jetted TDE rates if triggered by more massive, evolved stars than previously thought.

We have presented two different methods of estimating the minimum energy conditions of edge-brightened compact radio sources (CSO-2s). We fit components to RFC VLBA maps to determine their volumes and spectral indices, thus enabling us to estimate the energies and magnetic field strengths of individual components for which two high-quality VLBA maps are available. To address the possibility of resolved-out flux in VLBA maps, we also estimate the synchrotron flux from the overall radio spectrum from a combination of different single-dish and interferometric surveys. With this method, we find typical magnetic field strengths of ∼10–20 mG and minimum energies of ∼5–10 M_⊙c², with CSO-2s exhibiting a large range of minimum energies from 10⁻² to 10² M_⊙c². Modeling of the X-ray inverse Compton emission by the same electron population generating the synchrotron emission shows that CSO-2s are likely within a factor of a few from minimum energy.

A number of sources of systematic error that may significantly affect the minimum energy quantities of the CSO-2 population exist, including uncertainties in the component volume modeling (due to resolved-out flux or blended components), cutoffs of the injection electron distribution, volume filling factors, and baryonic contributions to the particle content. Furthermore, improvements to this analysis to reduce systematic errors in modeling require performing a full decomposition of components across ≥3 VLBI frequencies to improve the determination of the individual component synchrotron spectrum. This would also allow us to place constraints on the existence of cooling breaks in each component and improve the synchrotron emission modeling.

If the cutoff in the CSO-2 size distribution at 500 pc is real, then the typical magnetic field strength of CSO-2s is able to explain their disappearance once the jet turns off. Deeper, lower-frequency interferometric surveys should be able to identify an even larger sample of relic emission of once actively accreting CSO-2s in the Local Universe. In addition, the identification of the smallest CSO-2s (≤10 pc) will provide strong constraints on theories of their origin and early time evolution.

On the other hand, the typical CSO-2 minimum energies found by our analysis exceed those of R24. If CSOs are to be explained by the disruption of an evolved, several solar mass star, then it must tap into a significant amount of the spin energy of the SMBH. In order to determine whether CSO-2s are truly triggered by discrete mass injections, it will be necessary to verify whether the size cutoff at ∼500 pc is real through deeper, complete surveys. As more CSOs are identified (e.g., E. E. Sheldahl et al. 2025), we plan to achieve a greater census of their components’ magnetic field strengths and internal energy distributions.

We thank the referee for comments that helped clarify the presented methodology and encouraged a deeper discussion of the systematics present within the analyses.

This work is supported by NSF grant AST2407603. T.D.S. acknowledges support from the Caltech Doug and Betty Nickerson Fellowship. AGS acknowledges the support of the National Science Foundation Graduate Research Fellowship and a Giddings Fellowship at the Kavli Institute for Particle Astrophysics and Cosmology at Stanford.

This paper depended on VLBA data obtained from the RFC (DOI:10.25966/dhrk-zh08). The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.

Facilities: VLBA – Very Long Baseline Array, VLA, ALMA – Atacama Large Millimeter Array, LOFAR – , GMRT, MWA, WSRT, ASKAP, Planck,OVRO:40 m – , WISE, FLWO:2MASS, UKIRT, PS1, Sloan, XMM – Newton X-Ray Multimirror Mission satellite, CXO – Chandra X-ray Observatory satellite.

Software: AegeanTools (P. J. Hancock et al. 2012, 2018), astropy (Astropy Collaboration et al. 2013, 2018; Astropy Collaboration et al. 2022), Xspec (K. Arnaud et al. 1999), AGNfitter (G. Calistro Rivera et al. 2016), Imfit (P. Erwin 2015), Photutils (L. Bradley et al. 2016), emcee (D. Foreman-Mackey et al. 2013a).

T.D.S. was responsible for implementing the component fitting, spectral fitting, minimum energy calculations, and inverse Compton modeling, and writing the manuscript. M.S.S.L.O. contributed to the derivations and methodological ideas for the analysis and collected the data from the radio surveys. M.S.S.L.O., A.C.S.R., and A.G.S. provided useful discussions and checks on the analysis procedure and comments on the manuscript.

Table 4 shows the selected VLBA maps used from the Radio Fundamental Catalog necessary for decomposing the CSO-2s into individual components to compute volumes and computing component-wise spectral indices when a low and high frequency map are available.

Table 5 lists the primary catalogues from which the broadband synchrotron radio spectrum was constructed for each CSO-2, which was used to perform double power law fits in the spectral analysis.

In this appendix, we derive the “minimum energy” expressions used in this paper’s CSO component modeling. We assume that the component lepton (i.e., electron and positron) population generating the synchrotron photons follows a truncated power-law distribution in their Lorentz factors (or, equivalently, in their energies): f_Γ(γ) = C_eγ^p, where p  <  0 and . It is assumed that p is related to the synchrotron spectral index α by p = 2α − 1. Here, C_e is a normalization constant that ensures that f_Γ(γ) integrates to 1 over the full range of Lorentz factors. In this paper, we have assumed and The total energy density comprises contributions from particles (electrons and positrons on the one hand, and protons on the other) and from magnetic fields:

Here, the protons contribute an energy density Ku_e; the magnetic field energy density u_B = B²/2μ₀. In this paper, we take the jets to be leptonic and ignore entrainment, so that the proton contribution to the total energy is negligible (K ≈ 0). We can express the lepton energy density as the total (i.e., rest plus kinetic) energy of the N_e leptons, which have some mean Lorentz factor , divided by the product of the plasma’s volume filling fraction η and the lobe volume V:

In this paper, we assume η = 1. Neglecting synchrotron radiation from protons, the total number of electrons and positrons, N_e, is given by the ratio of the total synchrotron luminosity to the pitch angle-averaged mean synchrotron power per lepton:

We compute L_s by assuming that the lobe produces, at least internally, a synchrotron spectrum with a power-law profile; the exponent is the optically thin spectral index α:

Here, L₀ is the internally generated (rather than outwardly radiated) luminosity density at reference rest-frame frequency ν₀. We integrate L_ν(ν) between the characteristic frequencies (Equation (1)) of the minimum and maximum leptonic Lorentz factors and . For α ≠ −1, we obtain

We rewrite the total energy density as

with C₂ being the lobe’s rest-frame particulate energy density per lepton, and C₃ encompassing all other factors:

One obtains the minimum energy magnetic field strength by solving . The result is

for α ≠ −1. The minimum energy in a synchrotron-emitting region can then be found through

Table 6 lists the individual components fitted to each CSO-2 from which volumes were computed. Minimum energies and magnetic field strengths obtained from the individual component analysis are also provided for CSO-2s where two VLBA maps were available. Table 7 lists the total minimum energy and average magnetic field strengths derived from the spectral analysis, along with relevant information describing the shape and normalization of the optically thin part of the broadband radio spectrum.