Paper Title: Two-Component gamma-ray Emission Spectrum and X-Ray Polarization of the Radio Galaxy Pictor A.





Jin Zhang & Hai-Ming Zhang
j.zhang@bit.edu.cn; hmzhang@nju.edu.cn

0009-0003-9471-4724]Jia-Xuan Li
School of Physics, Beijing Institute of Technology, Beijing 100081, People's Republic of China; j.zhang@bit.edu.cn


School of Physics, Beijing Institute of Technology, Beijing 100081, People's Republic of China; j.zhang@bit.edu.cn

0000-0003-2547-1469]Ji-Shun Lian
School of Physics, Beijing Institute of Technology, Beijing 100081, People's Republic of China; j.zhang@bit.edu.cn

0009-0000-6577-1488]Yu-Wei Yu
School of Physics, Beijing Institute of Technology, Beijing 100081, People's Republic of China; j.zhang@bit.edu.cn


Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical Science and Technology, Guangxi University, Nanning 530004, People's Republic of China


Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical Science and Technology, Guangxi University, Nanning 530004, People's Republic of China

0000-0001-6863-5369]Hai-Ming Zhang†
School of Astronomy and Space Science, Nanjing University, Nanjing 210023, People's Republic of China; hmzhang@nju.edu.cn


Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, People's Republic of China

0000-0003-3554-2996]Jin Zhang†
School of Physics, Beijing Institute of Technology, Beijing 100081, People's Republic of China; j.zhang@bit.edu.cn

§ ABSTRACT 

Pictor A is a γ-ray emitting radio galaxy and has a bright hotspot called WHS, located ∼4 arcmin away from the nucleus. In this letter, we present an analysis of its 16-year Fermi-LAT data and report the first Imaging X-ray Polarimetry Explorer (IXPE) observation for this source. Our analysis of the Fermi-LAT observations reveals evidence of two components in the average γ-ray spectrum of Pictor A, exhibiting a statistically significant hardening from Γ^1_γ=3.25±0.15 to Γ^2_γ=1.81±0.07 at a break energy of 2.46±0.09 GeV. The evident variability of γ-rays is observed in Pictor A. Interestingly, the variability is dominated by the component below the break energy, and the component above the break energy shows no variability. Furthermore, we find that a power-law function can adequately fit the spectrum during high-flux states, whereas a broken power-law is still required to explain the spectrum during low-flux state. We suggest that the low-energy component originates from the nucleus, while the high-energy component primarily stems from WHS. The broadband spectral energy distributions of both nucleus and WHS can be well represented by a simple leptonic model, with both γ-ray components attributed to the synchrotron-self-Compton (SSC) process. The analysis of IXPE data on the nucleus yields an upper limit to the polarization degree Π_ X<8.9% in the 2–8 keV band, agreeing with its X-ray emission originating from SSC. However, Π_ X=23.5%±5.6% is observed at a confidence level of >99% in the 5–7 keV band, and the possible physical origin of this narrow-energy-band polarization signal is discussed. 







§ INTRODUCTION


Radio galaxies (RGs), a subset of radio-loud Active Galactic Nuclei (AGNs), are distinguished by the presence of jet substructures that extend from the radio core to scales of kiloparsecs to megaparsecs (kpc–Mpc). Hotspots, characterized by high surface brightness at the edges of extended radio lobes in RGs, are considered to represent the jet termination, where magnetic fields are amplified and particles are accelerated to high energies <cit.>. Thanks to Chandra observations, numerous hotspots within RG jets have been detected in the X-ray band ( for a review), however, ongoing debate persists regarding their X-ray radiation mechanisms (e.g., ). In addition, both the inverse Compton scattering and second synchrotron radiation models of X-rays predict the observable γ-ray emission for some large-scale jet substructures <cit.>. 

The detection of γ-rays from large-scale radio lobes of RGs Cen A <cit.>, Fornax A <cit.>, and NGC 6251 <cit.> confirms that the large-scale jet substructures are acceleration sites of high-energy particles. To date, the Fermi Large Area Telescope (Fermi-LAT) has detected numerous RGs <cit.>; however, the limited spatial resolution of Fermi-LAT makes it difficult to judge the location of the γ-ray emission for the majority of RGs. It has been proposed that some large-scale jet substructures may have the detectable γ-ray emission through the search for a steady γ-ray emission component and modeling of the broadband spectral energy distributions (SEDs; ). 

Pictor A is a γ-ray emitting RG, located at a redshift of z=0.035 <cit.>. It has a bright primary hotspot, known as the western hotspot (WHS), which is situated about 4 arcmin (∼165 kpc) away from the radio core <cit.>. Its nucleus is a strong X-ray source and has been widely studied; a PL spectrum is observed <cit.>, maybe with a weak Fe Kα line <cit.>. Extensive research has also been conducted on this bright WHS <cit.>. The observed flat spectrum and flux variability in the X-ray band suggest that the X-rays from WHS are produced by synchrotron radiation <cit.>. X-ray polarization observations can serve as a valuable tool to verify its radiation mechanisms. 

The γ-rays detected by Fermi-LAT for Pictor A are generally attributed to the contribution of the nucleus; adding the X-ray emission from the nucleus, they are produced through the synchrotron-self-Compton (SSC) process <cit.>. However, it is also possible that the γ-rays stem from the WHS <cit.>. In this paper, we perform a comprehensive analysis of the ∼16-year Fermi-LAT observation data to investigate the origins of γ-ray emission from Pictor A, and we examine the X-ray emission mechanisms of the nucleus and WHS using the first Imaging X-ray Polarimetry Explorer <cit.> observation for Pictor A in conjunction with the broadband SED modeling. Observations and data analysis are given in Section <ref>, followed by the presentation of results in Section <ref>. The SEDs of the nucleus and WHS are constructed and modeled in Section <ref>. A discussion and conclusions are presented in Section <ref>. H_0=71 km s^-1 Mpc^-1, Ω_ m=0.27, and Ω_Λ=0.73 are adopted in this paper.



§ OBSERVATIONS AND DATA ANALYSIS




 §.§ Fermi-LAT


Pictor A is associated with γ-ray source 4FGL J0519.6-4544 in the 4FGL-DR4 <cit.>. We select the data within 15^∘ region of interest (ROI) centered on the radio position of Pictor A (R.A.=79^∘.957, Decl.=-45^∘.779) and download the PASS 8 data covering from 2008 August 4 to 2024 July 10 (MJD 54682–60501) within the energy range of 0.1–500 GeV from the Fermi Science Support Center[<https://fermi.gsfc.nasa.gov/cgi-bin/ssc/LAT/LATDataQuery.cgi>]. 

The publicly available software  (v.2.2.0)  and  (v.1.1)<cit.> are used in our analysis. We use event class “SOURCE” (evclass=128) and event type “FRONT+BACK” (evtype=3) for the binned likelihood analysis based on LAT data selection recommendations[<https://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone/Cicerone_Data_Exploration/Data_preparation.html>] and set the maximum zenith angle of 90^∘ in order to eliminate the contamination of γ-ray emission from the earth limb. A standard filter expression “(DATA_QUAL0)&&(LAT_CONFIG==1)” and the instrument response function of P8R3_SOURCE_V3 are used in our analysis. All of the γ-ray source models included in the 4FGL-DR4 <cit.> within the ROI and two background model including the isotropic emission ("iso_P8R3_SOURCE_V3_V1.txt") and the diffuse galactic interstellar emission ("gll_iem_v07.fits") are added to the model. Only the parameters of the sources within 6.5^∘ centered on Pictor A and the normalization of two background model are left free. 

We use the test statistic (TS) to evaluate the significance of a γ-ray source signal, TS=2(logℒ_ src-logℒ_ null), where ℒ_ src and ℒ_ null are the likelihood values of the background with and without the target source, respectively. The TS map of the Pictor A region is shown in the left panel of Figure <ref>, the cyan cross show the best-fit position of the γ-ray source in this work, i.e., R.A.=79.91±0.02 and Decl.=-45.77±0.02. This best-fit position is consistent with that of 4FGL J0519.6–4544 and spatially associated with Pictor A. 

Note that there is a bright flat-spectrum radio quasar named PKS J0515–4556 (4FGL J0515.6–4556) located 0.75 away from Pictor A. As shown in the right panel of Figure <ref>, the maximum TS value of the residual TS map is ∼4, indicating that the γ-ray emission from this source is well fitted by the model and no new γ-ray sources are found.

Pictor A is identified as a point-like source with a power-law (PL) spectrum in 4FGL-DR4 <cit.>. However, as shown in Figure <ref>(a), we observe that the spectrum of Pictor A is better described by a broken power-law (BPL) function than a PL function with a value of Δ TS=38.88 (corresponding to a significance level of 5.8σ). Δ TS is identified as Δ TS=2(logℒ_ BPL-logℒ_ PL), where ℒ_ BPL and ℒ_ PL are the likelihood values of BPL and PL functions, respectively. The PL function is defined as 

    dN(E)/dE=N_0×(E/E_0)^-Γ_γ,

and the BPL function is defined as

    dN(E)/dE=N_0×{[ (E/E_ b)^-Γ^1_γ                          E<E_ b; (E/E_ b)^-Γ^2_γ                         E⩾ E_ b;                 ],.

where E_ b represents the break energy, Γ^1_γ and Γ^2_γ are the index below and above the break energy, respectively. By analyzing the ∼16-year Fermi-LAT observation data for Pictor A, we obtain TS∼773.10 and an average flux of  (1.06±0.08)×10^-11  erg cm^-2 s^-1 with a BPL spectral form in the 0.1–500 GeV energy band. The derived spectral parameters are E_ b=2.46±0.09 GeV, Γ^1_γ=3.25±0.15, and Γ^2_γ=1.81±0.07. The fitting parameters are also listed in Table <ref>.

In order to investigate the γ-ray variability of Pictor A, we generate the ∼16-year light curve in the 0.1–500 GeV energy band with a time bin of three months. As shown in Figure <ref>(a), evident flux variations are presented in the γ-ray light curve. Based on a comparison of these fluxes with the average value, we tentatively identify two periods (F1 and F3) as high-flux states and one period (F2) as a low-flux state for Pictor A. We also calculate the light curves in the 0.1–2.46 GeV band (below the spectral break) and the 2.46–500 GeV band (above the spectral break) using an adaptive-binning method based on a criterion of TS≥9 for each time bin, where the minimum time bin is three months. The results are displayed in Figures <ref> (b) and (c). The light curves with different intervals are obtained by freeing the normalization of Pictor A and 4FGL J0515.6–4556 while keeping all other spectral parameters fixed at their best-fit values obtained from the ∼16-year data analysis.



 §.§ IXPE


Pictor A was observed by IXPE from 2024 June 15 to 2024 July 5, with a net exposure of ∼ 960 ks. The WHS is the primary target of that IXPE observation. However, the nucleus of Pictor A was also captured within the field of view (FOV) of IXPE, albeit in close proximity to the FOV edge of instrument. 

The nucleus and WHS are both identified as point-like sources under a ∼30^'' angular resolution of IXPE. We first apply coordinates correction to publicly available Level-2 event files to remove the detector pointing misalignment. We calculate the X-ray polarization parameters from the IXPE observation using two different methods: (1) the <cit.> method implemented in the  algorithm within the  task of the  (v.31.0.1) simulation and analysis software <cit.> and (2) a spectropolarimetric analysis using  <cit.>, which is described in <cit.>. The source region of the nucleus is defined as a circle with a radius of 60^'' centered on its radio position. Similarly, the source region for the WHS is also defined as a circle on the Chandra pointing position in <cit.>, but with a smaller radius of 30^'' due to its faintness compared to the nucleus. The background photons for both the nucleus and WHS are extracted from a nearby circle region with a radius of 90^'', taking into account the issue of the FOV edge.

The polarization parameters of the nucleus and WHS are first calculated using the  algorithm within  in the 2–8 keV energy range. No significant polarization is observed for either the nucleus or WHS. The analysis with  yields a minimum detectable polarization at 99% significance (MDP_99) of 7.2% for the nucleus based on the three combined IXPE detector units (DUs). However, due to a limited number of detected photons, this estimated MDP_99 value exceeds the theoretical maximum polarization degree of 70%, thereby constraining further investigation into the polarization properties of WHS.

The spectropolarimetric analysis is performed to the nucleus using  in the  (v.6.30.1). We generate Stokes parameter spectra (I, Q and U) for both the source and background using the ,  and  algorithms within . The I spectrum is rebinned with a minimum of 20 counts per spectral channel via the  task, while a constant energy binning of 0.2 keV is applied to both the Q and U spectra. We simultaneously fit the I, Q and U spectra of three DUs using an absorbed PL model with a constant polarization of the form  ×  ×  × . The PL function is

    dN(E)/dE = N_0×(E/E_0)^-Γ_ X  ,

where E_0=1 keV is the scale parameter of photon energy, N_0 is the PL normalization, and Γ_ X is the photon spectral index <cit.>. The  and  models account for the uncertainties in the absolute effective area of the three DUs and the Galactic photoelectric absorption, respectively. During the spectropolarimetric fit, the column density is fixed to N_ H = 3.6 × 10^20 cm^-2 <cit.>. The polarization model  assumes constant polarization parameters within the operating energy range and has only two free parameters: polarization degree (Π_ X) and polarization angle (ψ_ X). The best-fit parameters are presented in Table <ref>. 

We further perform an energy-resolved spectropolarimetric analysis on the nucleus following the methodology provided in <cit.> and <cit.>. The results of this analysis are described in section <ref>.



§ RESULTS




 §.§ Complex γ-ray Behaviors of Pictor A


As illustrated in Figure <ref>(a), the average spectrum of Pictor A over the full time shows a noticeable break at 2.46±0.09 GeV, with the spectral index hardening from Γ^1_γ=3.25±0.15 to Γ^2_γ=1.81±0.07. The BPL spectral model is preferred over the single PL spectral model for fitting this spectrum at a confidence level of 5.8σ. Moreover, both energy bands (0.1–2.46 GeV and 2.46–500 GeV) can be well described by a PL function, yielding photon spectral indices of 3.12±0.06 and 1.97±0.18, respectively, which are consistent with those obtained from fitting the entire energy band using a BPL model. These results are summarized in Table <ref>. These findings suggest the presence of statistically significant hardening in the observed γ-ray spectrum of Pictor A, similar to what has been observed in the other two RGs: Cen A <cit.> and M87 <cit.>. Such an “unusual” break could most naturally be explained by a superposition of different spectral components. In order to further investigate the properties of GeV emission from Pictor A, additional data analysis on Fermi-LAT observations is conducted and the main findings are summarized below.



 
  * The full energy light curve. The derived ∼16-year Fermi-LAT light curve of Pictor A in the 0.1–500 GeV band exhibits significant variability, particularly during periods F1 and F3, as illustrated in Figure <ref>(a). To quantify the variability of the γ-ray light curve, we employ the χ^2 test <cit.> to the light curve, which yields a probability of P(χ^2)=6.14×10^-10, corresponding to a confidence level of 6.1σ.

 
  * The light curves below and above the spectral break. To investigate the temporal characteristics of the two spectral components, we also derive the long-term γ-ray light curve of Pictor A above and below the spectral break, i.e., in the 0.1–2.46 GeV and 2.46–500 GeV bands, respectively, as displayed in Figures <ref>(b) and <ref>(c). Interestingly, the low-energy light curve exhibits evident flux variability with a probability of P(χ^2)=1.04×10^-32, corresponding to a confidence level of 11.9σ, while no significant flux variability is observed in the high-energy light curve. This indicates that the dominant statistical contribution to the full energy light curve comes from the low-energy band component. 

 
  * The spectra of low-flux and high-flux states. The spectrum shown in Figure <ref>(a) is a ∼16-year average spectrum, covering the different flux states of source. We thus extract the time-integrated spectra in the 0.1–500 GeV band for three different stages: MJD 55762–58732 (period F2) for the low-flux state, together with MJD 54952–55762 (period F1) and MJD 58822–59272 (period F3) for the two high-flux states, respectively. As illustrated in Figure <ref>(b), both spectra from the high-flux states can be well fitted using a PL function; however, they exhibit different spectral indices: Γ_γ=2.59±0.21 for F1 state and Γ_γ=3.30±0.17 for F3 state. Nonetheless, to account for the low-flux spectrum (F2 state), a BPL function is still necessary (Δ TS=11.51, corresponding to a significance of 3.4σ), characterized by Γ^1_γ=3.13±0.08, Γ^2_γ=1.92±0.16, and a break energy of 2.15±0.27 GeV. 

 

The aforementioned findings provide support for the idea that the spectrum depicted in Figure <ref>(a) results from the superposition of two distinct physical components. One component dominates the low-energy band, particularly during periods of high-flux emission, while another component primarily contributes to the high-energy band, which is more prominent during low-flux state of the source. The contrasting variability patterns above and below the spectral break could potentially serve as additional evidence for the existence of these two separate components in the γ-ray emission of Pictor A. 



 §.§ Possible γ-ray Emission from WHS


The analysis results of the Fermi-LAT data clearly indicate that the γ-rays detected from Pictor A have two distinct origins: one component dominates the observed flux and exhibits significant variability, primarily contributing to the emission at low energies (below the spectral break) with a steep spectrum; while another component contributes to the high-energy emission (above the spectral break) without evident variability, displaying a hard spectrum. Through broadband SED fitting of the nucleus and WHS in Pictor A, <cit.> proposed that there is detectable γ-ray emission in WHS. The combined spectrum from both the nucleus and WHS in the GeV band would manifest as a BPL feature, with a harder spectrum at higher energy ranges, as illustrated in Figure 3 of their study. Therefore, we suggest that the high-energy component observed above the spectral break in Pictor A is predominantly due to the emission from WHS, while radiation from the nucleus primarily accounts for low-energy emission below this break and exhibits evident variability in Pictor A.



 §.§ X-Ray Polarization of the nucleus and WHS


By analyzing the observational data from IXPE on Pictor A, only an upper limit of polarization degree is derived for the nucleus, i.e., Π_ X<8.9% at a 99% confidence level in the 2–8 keV band. To investigate whether detectable polarization exists within a narrow energy band, we divide the entire energy range of 2–8 keV into energy bins with intervals of 1 keV, 1.5 keV, or 2 keV. Interestingly, the polarization parameters can be estimated with a confidence level exceeding 99% in the energy bands of 5–6 keV, 5–6.5 keV, and 5–7 keV. Based on our spectropolarimetric fitting results at respective confidence levels of 3.4σ, 3.9σ, and 4.0σ, we find Π_ X=23.9%±7.1% with ψ_ X=131.1±8.5 in the 5–6 keV band, Π_ X=23.5%±6.0% with ψ_ X=128.4±7.3 in the 5–6.5 keV band, and Π_ X=23.5%±5.6% with ψ_ X=131.2±6.9 in the 5–7 keV band. The derived value of ψ_ X is slight larger than the jet position angle, which is about 100^∘ <cit.>.

The radio <cit.> and optical bands <cit.> have detected high polarization in WHS, reaching up to the theoretical maximum of approximately 70%. Furthermore, there is a remarkable similarity in structures between the radio and optical bands. However, due to a limited number of detection photons, no statistical X-ray polarization information can be obtained from this IXPE observation for WHS.



§ CONSTRUCTING AND MODELING THE SED


The observed BPL spectral pattern in Pictor A, exhibiting a harder spectrum at higher energy bands, implies the existence of two distinct sources contributing to its γ-ray emission. In order to further investigate the γ-ray emission characteristics of Pictor A, we construct broadband SEDs for both the nucleus and WHS regions, covering the energy band from radio to γ-rays, as illustrated in Figure <ref>. The data for both the nucleus and WHS, ranging from radio to X-rays, are obtained from <cit.> and references therein (for more details, please refer to the caption of Figure <ref>), while the γ-ray spectrum is derived through our own analysis in this study. As described in Section <ref>, the low-energy and high-energy components are proposed to primarily originate from the nucleus and WHS, respectively. The results appear to be consistent with their broadband SEDs, as shown in Figure <ref>; the combinations with their respective radio to X-ray data result in smooth features in their broadband SEDs. 

The leptonic model is commonly used to fit the broadband SEDs of the jet emission from AGNs. The radiation region is assumed to be a sphere with a radius R, magnetic field strength B, the bulk Lorentz factor Γ, and the Doppler boosting factor δ. The electron distribution in the emission region is taken as a PL or a BPL, characterized by an electron density parameter (N_0), a break energy (γ_ b), and indices (p_1 and p_2) ranging from γ_min to γ_max. In the case of a PL electron distribution, p_1=p_2 and γ_ b=γ_max. Additionally, the synchrotron-self absorption, Klein–Nishina effect, and extragalactic background light absorption <cit.> are considered during the SED modeling.



 §.§ Nucleus Region


The synchrotron and SSC processes of relativistic electrons with a BPL distribution are commonly used to explain the observed SED of the nucleus in γ-ray emitting RGs <cit.>, which is also used to represent the observed SED of the nucleus in Pictor A. The emission region radius is fixed at R=10^16 cm, as also mentioned in <cit.> and <cit.>. We adopt Γ=5 with a viewing angle of 30^∘ <cit.>, corresponding to δ=1.3. The values of p_1 and p_2 are determined using the observed spectral indices in X-rays and optical–UV bands, respectively. We set γ_min=1, while γ_max is approximately constrained by the observed GeV spectrum, showing a cutoff at the break frequency. By adjusting the parameters B, N_0, and γ_ b to match the broadband SED of the nucleus, we obtain B=7 G, γ_ b=1.6 × 10^3, and γ_ max=1.6 × 10^4. The modeling parameters are also listed in Table <ref>. 

The fitting result is presented in Figure <ref>; the one-zone synchrotron+SSC model well represents the broadband SED of the nucleus. It is evident that this model fails to account for the high-energy component observed in the Fermi-LAT spectrum, even without considering the low-energy component of the γ-ray spectrum. If the high-energy component also stems from the inner jet of Pictor A, a more complex radiation model is required to reproduced the observed broadband SED of nucleus, such as a spine-layer model <cit.> or hadronic processes (e.g., ).

It should be noted that there exists a degeneracy between B and δ during the SED modeling process (e.g., ). Taking a similar value of R∼10^16 cm, <cit.> reported δ=1.3 with B=10 G, whereas <cit.> found δ=1.6 with B=7 G, and <cit.> obtained δ=2.5 with B=4.2 G. Considering the dominance of the low-energy component in the high-flux γ-ray emission state of Pictor A, as illustrated in Figure <ref>(b), the X-ray spectrum obtained during the high-flux state <cit.> into account in our SED modeling. Consequently, the obtained value of B differs from theirs due to the different X-ray and γ-ray data.



 §.§ WHS


Considering the complex radio structure <cit.> and flat X-ray spectrum of WHS (e.g., ), we use a two-zone leptonic model to fit its broadband SED from radio to X-rays, same model as taken in <cit.> and <cit.>. This model includes synchrotron and SSC radiations originating from both a diffuse region and a compact substructure, where both regions are treated as stationary blobs characterized by Γ=δ=1. The energy density of the synchrotron radiation field in this scenario surpasses that of the cosmic microwave background (CMB) by more than one order of magnitude. Consequently, the contribution of the inverse Compton scattering of CMB photons by relativistic electrons can be considered negligible when compared to the SSC process. The emission of WHS from radio, optical to ultraviolet is primarily attributed to the synchrotron process occurring in the whole diffuse region, while the X-rays are produced through synchrotron radiation emitted by a compact substructure contained within the diffuse region. The γ-ray spectrum above the break in Figure <ref>(a) is taken into consideration in the SED modeling of the WHS.  

For the diffuse region of WHS, a radius of R=500 pc is adopted, which corresponds to half of the overall size (1.4 arcsec, ) of WHS in the optical band. The electron distribution follows a BPL, with p_1 and p_2 being well constrained by observed data across radio to UV bands. γ_min cannot be determined and is fixed at 1, while γ_max is roughly constrained by the last data point in the UV band. Considering the equipartition condition (U_B=U_ e), where U_B and U_ e are the energy densities of the magnetic fields and electrons in the radiation region, it is found that the prediction flux from SSC process fails to explain the γ-ray spectrum at GeV energies. The fitting of the highest-energy point in the GeV band necessitates a significantly small magnetic field that deviates substantially from the equipartition condition, while also resulting in the model's prediction line surpassing the two data points above the break. Therefore, we abandon this hypothesis.

For the compact substructure of WHS, a radius of R=30 pc is taken, which is half the average size of the five compact substructures reported by <cit.>. Considering the limited observational data, we adopt a PL electron distribution. p_1 is determined based on the spectral index between radio and soft X-rays. γ_min is also fixed at 1. When considering equipartition conditions, the predicted line from the SSC process is significantly lower than the γ-ray fluxes observed at the GeV band, similar to that of the diffuse region. We propose that the γ-rays primarily originate from the compact substructure; therefore, the equipartition condition in the compact substructure is not taken into account during SED modeling of WHS. 

We adjust the values of N_0 and γ_ b for the diffuse region under the equipartition condition, along with adjusting the parameters N_0, B, and γ_ max of the compact substructure to represent the observed SED of WHS from radio to γ-ray bands. The fitting result is presented in Figure <ref> and the parameters are given in Table <ref>. 
The parameter values for the diffuse region are found to be similar when compared with those in <cit.>. However, the B value for the compact substructure is one order of magnitude smaller, accompanied by a slightly higher value of γ_ max in order to match the γ-ray spectrum. Additionally, we observe that the predicted flux of SED modeling is over the sensitivity curve of Cherenkov Telescope Array (CTA) south array (CTA-S, 50 hr), but remains below the sensitivity of High Energy Stereoscopic System (H.E.S.S.), where the sensitivity curves are obtained from <cit.> and the CTA webpage[<https://www.ctao.org/for-scientists/performance/>], respectively. Considering the hard spectrum in the GeV band, it could potentially be considered as a candidate for CTA.



§ DISCUSSION AND CONCLUSIONS


By analyzing the Fermi-LAT observation data, we find that the ∼16-year average spectrum of Pictor A in the GeV band exhibits statistically significant hardening feature. The BPL spectral form is more preferred than a simple PL function for describing the γ-ray spectrum of Pictor A with a confidence level of 5.8σ. Similar behavior featuring significant hardening in the GeV γ-ray spectrum has also been observed in two other RGs: Cen A <cit.> and M 87 <cit.>. In both cases, it was proposed that the GeV emission consists of two distinct physical components. For M 87, the hardening γ-ray spectrum was observed at a significance level of ∼2.66σ, and both components were suggested to stem from the inner jet <cit.>. For Cen A, the significance level of the hardening γ-ray spectrum increased from 2.3σ  <cit.> based on the 4-year Fermi-LAT observation data to >5σ <cit.> when utilizing the 8-year Fermi-LAT observation data. They suggested that this additional high-energy component beyond the common SSC emission may be related with the interactions between relativistic protons and ambient gas in the large-scale jet <cit.>, or the dark matter around the black hole, or a population of millisecond pulsars in the core region <cit.>.    

The dominant low-energy component with a steep spectrum, which exhibits significant variability and accounts for the majority of the emission flux in Pictor A, is likely to originate from its nucleus. Contrarily, the high-energy component with a hard spectrum remains in a steady emission state, which is favored by the extended γ-ray production scenario. We propose that this high-energy component stems from the WHS and that the observed BPL spectrum in the GeV band of Pictor A results from the combined emission originating from both its nucleus and WHS.

We use a single-zone leptonic model to explain the broadband SED of the nucleus, while adopting a two-zone leptonic model to represent the broadband SED of WHS. The two components in the GeV spectrum of Pictor A smoothly connect with the X-ray spectra of the nucleus and WHS, respectively; both can be naturally represented by the SSC process occurring in two different regions. Moreover, our SED fitting predicts that CTA-S will be able to detect very high energy (VHE) emission from WHS. Notably, H.E.S.S. observations have unveiled VHE γ-ray emission along the large-scale jet of RG Cen A <cit.>, providing compelling evidence supporting large-scale jets as sites for accelerating ultrarelativistic electrons.

By analyzing the first IXPE observation data for Pictor A, we obtain an upper limit (<8.9%) for the polarization degree of the nucleus at a 99% confidence level. This finding is consistent with the low-polarization expectation that the X-ray emission of the nucleus is produced by the SSC process <cit.>, similar to the IXPE observation results of RG Cen A <cit.> and other low and intermediate spectral peak blazars <cit.>. We also note that the nucleus of Pictor A generally exhibits low polarization in the radio band, typically less than 5% ( and references therein). However, no optical polarimetry data is available in the archived records.

Although no significant polarization is detected in the 2–8 keV band for the nucleus, a polarization degree of 23.5%±5.6% is observed at a confidence level exceeding 99% in the 5–7 keV band. As described in Section <ref>, the X-ray spectrum of the nucleus can be adequately explained by a simple PL function. This is also consistent with recent studies employing other X-ray detectors (e.g., ). It is implausible that this narrow-energy-band polarization arises from superimposed components originating from other regions or radiation processes. The presence of a weak Fe Kα line with an energy of 6.36±0.02 keV has been reported in the X-ray spectrum of the nucleus by <cit.>. Hence, we speculate that this polarization signal may be associated with this emission line; however, <cit.> suggested that the absence of significant polarization detection for Circinus galaxy in the 6–8 keV band is likely due to the prominent iron lines present in its spectrum. The confidence level of this narrow-energy-band polarization signal in Pictor A is no greater than 5σ,  therefore, we suggest that it represents a tentative detection.

Additionally, numerous studies have suggested that the X-rays emitted by WHS originate from synchrotron radiation within a compact substructure <cit.>. Considering the significant polarization observed in both radio and optical bands, with some measurements reaching up to the theoretical maximum of approximately 70% <cit.>, it is expected that high polarization detection of WHS in the X-ray band, which would provide a valuable tool for investigating radiation mechanisms of X-rays in WHS. Unfortunately, due to the limited number of detected photons, this IXPE observation is unable to provide any statistical polarization information for WHS.



This work is supported by the National Key R&D Program of China (grant 2023YFE0117200) and the National Natural Science Foundation of China (grants 12203022, 12022305, and 11973050).




aasjournal







ccc
    
    3
    Results of the Spectropolarimetric Analysis in the 2–8 keV Band for the Nucleus
    Model     Parameter     Value
         C_ DU1     1.0 (fixed) 

        C_ DU2     0.995±0.008 

        C_ DU3     0.944±0.008 

         N_ H (10^20 cm^-2)     3.6 (fixed) 

         Π_ X (%)     <8.9% 

        ψ_ X ()     

         Γ_ X     1.86±0.01 

        Norm (10^-3 photon keV^-1 cm^-2 s^-1)     5.09±0.07 

    3cχ^2/dof = 580/546 = 1.06
    















 Paper Title: Flare Accelerated Electrons in Kappa-Distribution from X-Ray Spectra with Warm-Target Model.





Yingjie Luo
Yingjie.Luo@glasgow.ac.uk

0000-0002-5431-545X]Yingjie Luo (骆英杰) School
of Physics & Astronomy, University of Glasgow, G12 8QQ, Glasgow, UK


0000-0002-8078-0902]Eduard P. Kontar
School of Physics & Astronomy, University of Glasgow, G12 8QQ,
Glasgow, UK

0000-0002-2651-5120]Debesh Bhattacharjee
School of Physics & Astronomy, University of Glasgow, G12 8QQ,
Glasgow, UK

§ ABSTRACT 
X-ray observations provide essential and valuable insights into the acceleration
and propagation of non-thermal electrons during solar flares. Improved X-ray
spectral analysis requires a deeper understanding of the dynamics of energetic
electrons. Previous studies have demonstrated that the dynamics of accelerated
electrons of a few thermal speeds are more complex. To better describe
the energetic electrons after injection, a model considering energy diffusion
and thermalization effects in flare conditions (warm-target model) has recently
been developed for Hard X-ray spectral analysis. This model has demonstrated how
the low-energy cut-off, which can hardly be constrained in cold-target
modeling, can be determined. However, the power-law form may not be the most
suitable representation of injected electrons. The kappa distribution, which is
proposed as a physical consequence of electron acceleration, has shown
successful application in RHESSI spectral analysis. In this study, we employ the
kappa-form injected electrons in the warm-target model to analyze two M-class
flares, observed by RHESSI and STIX, respectively. The best-fit results show
that the kappa-form energetic electron spectrum generates lower non-thermal
energy when producing a similar photon spectrum in the fit range compared to the
power-law form. We also demonstrated that the fit parameters associated with
kappa-form electron spectrum can be well determined with small fit
uncertainty. Further, the kappa distribution, which covers the entire electron energy range, enables the determination of key electron
properties such as total electron number density and average energy in the flare
site, providing valuable information on electron acceleration processes.







§ INTRODUCTION
 

Solar flares are explosive energy-release events of which the emissions span
over a wide electromagnetic wavelengths ranging from radio to γ-ray. In
order to understand the acceleration mechanism and the transport of energetic
electrons, it is crucial to know where and how the energy is released
<cit.>. Energetic electrons from solar flares can be
indirectly observed through X-ray and radio emissions. Due to the unique
advantages, Hard X-ray (HXR) diagnostics is essential for determining the
electron properties (see <cit.> for
reviews). HXR emission observed from solar flares is dominantly due to
electron-ion bremsstrahlung. The intensity of HXR emission  is linearly
proportional to the non-thermal electron density, so the observed HXR photon
spectrum provides a straightforward relationship with the energetic electrons.
Furthermore, HXR emission is less affected by propagation effects than radio
emissions, resulting in minimal modification of the original electron properties
during observations.

Over the past two decades, advanced X-ray instruments, such as the Reuven Ramaty
High-Energy Solar Spectroscopic Imager <cit.>
and the Spectrometer/ Telescope for Imaging X-rays
<cit.>, have facilitated the collection of
high-quality spectra and spectroscopic imaging. The frequently observed X-ray
sources in the corona and chromosphere footpoints
<cit.>
strongly supports the 'thick-target' interpretation
<cit.>, where flare-accelerated
electrons propagate down to the dense chromosphere and deposit all their energy
within the chromosphere. According to the thick target model, the observed X-ray
photon spectrum is determined by the injected electron spectrum, bremsstrahlung
cross-section, and the energy loss-rate within the target
<cit.>. When the energy of electrons is high compared to
the target, i.e. E ≫ k_BT (k_B is the Boltzmann constant,
T is the temperature), the model is referred to as a cold target. In
cold-target conditions, the kinetic energy loss is predominantly due to Coulomb
collisions <cit.>, and while the other
effects, such as return current Ohmic losses
<cit.>,
could affect the results. While such a cold target model is widely utilized for
the basis of HXR spectral analysis, observations necessitate that the coronal
plasma during flares can reach high temperatures over 10^7 K, highlighting
the need to improve the model. Studies have demonstrated that the behavior of
electrons in the warm-target condition (E∼ k_BT) is more complex and
significantly different from what is predicted by the cold-target model.
<cit.> highlighted a rapid drop in the energy loss rate in
warm-target conditions, indicating a substantial overestimation of electron flux
by the cold-target model in generating observed photon spectra. Additionally,
collisional energy diffusion and thermalization <cit.> have
a significant impact on the dynamics of injected electrons. Furthermore, a
physically self-consistent model requires consideration of both energy and
spatial diffusions during electron transportation <cit.>.
Taking into account these advances, <cit.> developed a warm-target model that provides a physically
sound treatment of electron dynamics in the warm-target condition (referred to
as the warm-target model in the following). This model includes collisional
energy diffusion, suggesting that injected electrons thermalize within the
target rather than losing all their energy and disappearing. The HXR spectrum
generated by the warm-target model includes both the contributions from the the
electrons in the cold-target chromosphere as well as the the component from the
corona.

The warm-target model has been utilized to analyze RHESSI flares to more
precisely fit the characteristics of non-thermal electrons
<cit.>. One
significant challenge in inferring non-thermal electron properties is that the
non-thermal emission is typically masked by the thermal emissions below
20-30 keV <cit.>. This makes it difficult to determine the
low-energy cut-off, E_c, which is essential in non-thermal electron energetics
<cit.>.  
This parameter, along with the closely-related non-thermal power are the key
parameters to characterize the acceleration mechanism. The cold-target condition
E ≫ k_BT is typically applicable in the range over 20 keV, providing
solely an upper limit for the low-energy cut-off E_c. On the other hand, the
warm-target model considers the thermalization of injected electrons, which
enables effectively constraining electron properties, as the spectra of
thermalized electrons are proportional to the injected electrons.
<cit.> have demonstrated that using the warm-target model
with a power-law distribution can significantly enhance the accuracy of
determining the low-energy cut-off E_c compared to the cold-target model.

However, it is important to recognize that the power-law form of the injected
electron spectrum is not unique and not always the most suitable representation
in many cases <cit.>. The power-law distribution requires a
low energy cut-off (E_c) to prevent infinite electron number. The presence of
the low energy cut-off in the injected electron spectrum, 
while suggested to have physical significance in terms
of the critical velocity of run-away and free acceleration
<cit.>, 
results in a non-physical discontinuous electron distribution of
Maxwellian plus the power law. 
Around the low-energy cut-off E_c, where positive distribution slope (df/dv>0) exists, Langmuir waves are expected to generate and grow <cit.>. The interaction between the waves will result in a flattened distribution around
the low-energy cut-off <cit.>, suggesting the injected (accelerated) 
electron distribution should be nonincreasing function 
for all energies. On the other hand, the kappa distribution, consisting of a Maxwellian core and smoothly merged power-law tail, has been studied to represent accelerated electrons. 
The kappa distribution is often used
to model the entire energetic electron spectrum from in-situ observations
<cit.>
and has emerged as a promising alternative for describing the accelerated
electrons in HXR spectral fitting
<cit.>.
Previous studies have shown that while spatially integrated spectra cannot be
well-fitted only with the electrons obeying a kappa distribution, kappa-form
accelerated electrons with a prominent thermal component yields promising
results. 

In addition to not requiring an arbitrary energy cut-off E_c, the adoption of
a kappa-form injection offers several advantages over the power-law
distribution. First, the kappa-form electron spectrum is believed to result from
acceleration out of thermal equilibrium, and its parameters can provide crucial
information on electron acceleration. Various studies have delved into the
origin of the kappa distribution from different perspectives, including
collisional relaxation along with wave-particle interaction
<cit.>, beam-plasma interactions
<cit.>. Moreover,
originating from the Fokker-Planck equation,  has
pointed out that the kappa electron distribution results from stochastic
acceleration in the presence of Coulomb collisions and velocity diffusion,
specifying insight into electron kinetics during solar flares. Similarly,
simulations of electron acceleration during magnetic reconnection lead to a
distribution resembling a kappa-distribution <cit.>.
Another advantage of the electron spectrum in kappa distribution over the
power-law distribution is its entire electron kinetic energy coverage. Kappa
distribution can provide electron information at low energy (∼keV), which
is not sensitive to X-ray instruments but can be observed through other
diagnostics such as EUV observations. This allows for a more comprehensive study
of electron distribution through multi-wavelength observations, providing
additional constraints <cit.>.
Furthermore,  due to its whole electron energy coverage and convergent nature,
the kappa distribution allows for the determination of parameters such as
electron number density and average electron energy, which is not achievable
with the power-law distribution. The kappa distribution is closely related to
the thermal properties of the ambient plasma. Parameters such as electron number
density and average electron energy can be directly compared with the
corresponding thermal properties of the pre-accelerated plasma, serving as an
important constraint and cross-check.


The paper investigates the spectrum of accelerated (injected) electrons in solar
flares. Section <ref> discusses relationship between kappa-form
injected electron spectrum and the mean electron flux based on the warm-target
model. We utilize the warm-target model with kappa form injected electrons to
analyze solar flares observed by RHESSI on February 24, 2011, and by STIX on
March 28, 2022. The best-fit results and a comparison between the kappa and
power-law form injection is provided in Section <ref>. Section
<ref> discusses the uncertainty of fitting parameters related to
the kappa distribution with different approaches. The resulting characteristics
such as electron number density and average electron energy are also shown. The
findings and the physical significance of the derived kappa parameters are
summarized and discussed in Section <ref>.



§ SOURCE-INTEGRATED SPECTRUM FROM KAPPA-FORM INJECTED ELECTRONS



<cit.> highlighted the concept of the source-integrated
density-weighted mean electron flux ⟨ nVF ⟩(E)  [electrons 
cm^-2s^-1keV^-1] and expressed the observed HXR spectrum
I(ε) [photons  cm^-2s^-1keV^-1] as:

    I(ε)=1/4π R^2∫_ε^∞ Q(ε,E)⟨ nVF ⟩(E)  dE ,


where R is the distance between the observer and the Sun, and
Q(ε,E) is angle-averaged bremsstrahlung cross-section (see
discussion in <cit.> for RHESSI). Determination of the mean
electron flux from the observed photon spectrum requires only the well-studied
bremsstrahlung cross-section <cit.>. To obtain the injected
electron rate spectrum Ṅ(E)  [electrons  s^-1keV^-1] from the
obtained mean electron flux, a model for the dynamics of injected electrons is
necessary. The commonly used cold-target model establishes the relationship
between mean electron flux and injected electron rate as 

    ⟨ nVF ⟩(E) = E/K∫_E^∞Ṅ(E_0) dE_0 ,

were K = 2 π e^4 ln(Λ) is collision parameter, e  [esu] is
the elementary charge and ln(Λ) is the Coulomb logarithm
<cit.>. To account for the electron dynamics in the
warm-target corona and cold chromosphere, <cit.> included
the energy diffusion, transport and thermalization of the accelerated electrons,
rewriting Equation (<ref>) 

    ⟨ nVF ⟩(E) = 1/2K E  e^-E/k_BT ∫_E_min^Ee^E'/k_BT/E'G(√(E'/k_BT)) dE' ×∫_E'^∞Ṅ(E_0) dE_0 .


<cit.> further simplify equation (<ref>) and
re-write as:

    ⟨ nVF ⟩(E) ≈Δ EM √(8/π m_e)E/(k_B T)^3/2 e^-E/k_BT + E/K∫_E^∞Ṅ(E_0) dE_0 ,

where Δ EM, referred as 'thermalized emission measure', is due to the
thermalization of accelerated electrons, can be explicitly written as 

    Δ EM ≈π/K√(m_e/8)(k_B T)^2Ṅ_0/E_min^1/2 ,
            E_min≈ 3 k_BT(5λ/L)^4 ,

and Ṅ_0  [electrons  s^-1]= ∫_0^∞Ṅ(E)  dE
is the total electron injection (acceleration) rate,
λ=(k_BT)^2/2Kn is the collisional mean free path. The half loop
length L, number density n, and temperature T are the parameters
associated with  the coronal part of the loop. Equation <ref> consists
of two parts, refer as the thermalized and non-thermal components, corresponding
to the warm- (coronal) and cold-target conditions, respectively. It is worth
noting that the thermalized component is proportional to the total injected
electron rate Ṅ_0, which is why this model is effective in constraining
the nonthermal component when fitting the observed X-ray spectrum. This
warm-target model 'f_thick_warm', which utilizes Equations <ref> and <ref>, is implemented in Solar Software (SSW) and
 package <cit.>. Function
'f_thick_warm' utilizes the commonly employed broken power-law form
of injected electrons, and it is worth noting that the warm-target model
(Equations <ref> and <ref>) can accept any electron
injection rate form. As mentioned earlier, the use of a kappa distribution for
injection is a promising alternative and has been applied to fit the RHESSI
spectra.


 derives the kappa distribution as a result of
stochastic acceleration in collisional plasma:

    f_k(v)=n_k/π^3/2v_te^3κ^3/2Γ(κ)/Γ(κ-3/2)(1+v^2/κ v_te^2)^-κ ,

where v_te is the thermal speed (1/2 m_e v_te^2=k_B
T_κ), and κ is the kappa index, which importantly represents the
ratio between the acceleration τ_acc(v) and collisional deceleration
τ_c(v) time scales κ=τ_acc(v)/2τ_c(v).
While different formats of the kappa distribution exist <cit.>, they are identical in essence
and only differ in their format and use of parameters. Furthermore, the physical
interpretation of κ vary when additional factors are are taken into
account. In this study, we have chosen to adopt Equation <ref>
for the kappa distribution.


The relationship between injected electron rate spectrum and electron
distribution for velocity is Ṅ(E)dE=A v f_k(v) d^3v, A is the
injection area. For isotropic electron distribution (d^3v=4π v^2dv), the
injected electron rate spectrum becomes (see red curves in Figure <ref>):

    Ṅ(E)=An_kΓ(κ)/Γ(κ-3/2)κ^3/2√(8/π m_ek_BT_κ)E/k_BT_κ/(1+E/κ k_BT_κ)^κ ,

where n_k [cm^-3]= ∫f_k(v) d^3v is the total electron number
density, Γ(x) is the gamma function. The total injection rate becomes:

    Ṅ_0= ∫_0^∞Ṅ(E)  dE = 2A n_k √(2k_BT_κ/m_e)κ^1/2/(κ-2)B(κ-3/2,1/2),

where B(x,y) is the beta function.  
Equation <ref> can be expressed as:

    Ṅ(E)=Ṅ_0/k_BT_κ(κ-1)(κ-2)/κ^2E/k_BT_κ/(1+E/κ k_BT_κ)^κ.


According to Equation <ref>, the mean electron flux from a kappa-form
injected electrons is given by <cit.>:


    ⟨ nVF ⟩(E) ≈Δ EM √(8/π m_e)E/(k_B T)^3/2 e^-E/k_BT + Ṅ_0E/K1+(1-1/κ)E/k_BT_κ/(1+E/κ k_B Tκ)^κ-1.


For high energy electrons, the kappa  distribution can be well approximated by a
power-law distribution with a spectral index κ-1. Figure <ref>
shows that the two distributions have an identical injection rate near 100 keV.
However, in the energetically important deka-keV range, kappa distribution is
distinct from a power-law distribution, and thus, a different photon spectrum
emerges. Figure <ref> illustrates the electron injection rate (top
panels, red and black for kappa- and power-law injection ,respectively) and the
resulting photon spectra (bottom panels) generated by the warm-target model in a
typical flare coronal condition (L_loop = 30  Mm, n_loop
=  5× 10^10 cm^-3, and k_B T_loop=1.5
 keV). 



§ APPLICATION OF THE WARM-TARGET MODEL TO OBSERVATIONS
 

In this section, we apply the warm-target model to analyze two well-observed
flare event. One on 2011 February 24, observed by the RHESSI and one on 2022
March 28 observed by the STIX, both events are situated in the limb and
providing spectroscopic images with two footpoint and looptop sources. The
warm-target model with kappa-form electron injection (function
f_thick_warm_kappa, based on Equation <ref>)
encompasses parameters of kappa distribution (total electron injection rate
Ṅ_0, kappa temperature T_κ, and kappa index κ) and thermal
properties of the target coronal loop (the half loop length L, number density
n_loop, and temperature T_loop). 

In this study, the thermal properties of the target plasma are determined by the
HXR observations of the thermal component. Following the approach outlined in
<cit.>, the thermal component of the X-ray spectra prior to
the HXR burst can provide the plasma temperature (T) and emission measure
(EM). Moreover, the HXR spectroscopic imaging can estimate the coronal loop
length (L) and coronal X-ray loop source size for determining the electron
number density (n_th). While this method is straightforward, we recognize
that other methods, such as differential emission measure analysis <cit.> from EUV
flare context or combining X-ray data with EUV data
<cit.>, can yield thermal properties
<cit.>. 



 §.§ 2011 February 24 RHESSI flare
 









The first flare of study is a GOES-class M3.5 flare located on the eastern limb
from the Earth's perspective. The flare occurred on February 24, 2011 with GOES
Soft X-ray flux peaked at ∼07:35 UT <cit.>. Our
attention is drawn to the first HXR peak at approximately 07:30 UT (red shadowed
time range in Figure <ref> left panel). The HXR spectroscopic
imaging of the burst displays a 'classical' configuration of the loop top source
and two HXR footpoints (Figure <ref> right panel). The X-ray
observations from the RHESSI of the selected HXR burst benefits from the
low/negligible pile-up effect. Furthermore, RHESSI was in optimal working
condition during the flare, providing more functional detectors. 


For spectral analysis, we first gather the thermal properties of the target
coronal loops according to the images of the burst (07:30:00 to 07:30:44 UT,
Figure <ref> left panel red shadowed time range) and spectra before
the interval analysed (short time interval closely prior to the burst analysed,
refer as 'pre-burst' in the following, in this case 07:29:52 to 07:30:00 UT,
Figure <ref> left panel black shadowed time range). In the analysis
of the pre-burst X-ray spectra, we utilize functions f_vth
(iso-thermal) and f_thick2 (cold-thick-target). In this stage, we
focus solely on the thermal plasma parameters and can overlook the constraints
associated with using the cold-target model. f_vth from the best-fit
result of the pre-burst spectra provides us with the plasma temperature
(k_BT_0=1.30  keV) and emission measure (EM_0=0.127×10^49 cm^-3) before the injection/acceleration of nonthermal electrons.
Additionally, we use X-ray spectroscopic imaging (Figure <ref>
right panel) to estimate the size of the X-ray source and target coronal loop.
We determine the source size in various dimensions (see pink and red solid lines
in Figure <ref> right panel, r=l/2=3.7 Mm and d=15.3  Mm, respectively) by considering the 50% contour of X-ray loop top
sources. The cross-area can be calculated as A=π r^2=4.21×10^10 cm^2, and the source volume can be obtained through the cylindrical
formula: V=A× d=π r^2 d=6.46×10^26 cm^3. Thus the
thermal electron number density can be estimated using
n_loop=√(EM_0/V)=4.4×10^10 cm^-3. To obtain the
coronal loop length, we construct a semi-circular loop (see cyan dashed line in
Figure <ref> right panel) that connects two footpoints and passes
through the loop top source. Consequently, we determine the thermal properties:
loop temperature (k_B T_loop=k_BT_0=1.30  keV),
loop number density (n_loop=√(EM_0/V)=4.4×10^10 cm^-3), and the the half-loop length (L=15.8  Mm).


Once the thermal properties are determined, we proceed to fit the spectra of the
burst time interval (07:30:00 to 07:30:44 UT) in 6–100 keV using functions
f_vth and f_thick_warm_kappa. In function
f_vth, the temperature and emission measure are fixed according to
the pre-burst spectra (T=T_0, EM=EM_0). The warm-target algorithm propose
that the emission measure during the burst interval is a combination of the
emission measure from the pre-burst and the emission measure of thermalized
injected electrons (EM=EM_0+Δ EM, photon spectrum associated with EM_0
and Δ EM are plotted as the red solid curve and dashed curves in Figure
<ref> right panel). Since the plasma is slowly changing during
the 40 sec interval (electron diffusion time along the loop ∼56 seconds,
use <cit.> Equation 11), the parameters in
f_thick_warm_kappa related to thermal properties of the loop are
fixed in the fit (the half loop length L, number density n_loop,
temperature T_loop, and the relative elemental abundances, default to
be 1 <cit.>). The parameters of kappa-form injection
spectrum (total electron injection rate Ṅ_0, kappa temperature
T_κ, and kappa index κ, except the high energy cut_off
E_high fix to 10^4  keV) are determined from the fit. Moreover,
fit with a single power-law electron injection in the warm-target model
'f_thick_warm' was performed for comparison: the total injection rate
Ṅ_0, power-law index δ, and low-energy cut_off E_c are free,
the break-energy E_break and high energy cut_off E_high is
fixed to 10^4  keV. The best-fit parameters (see Table <ref>) and generated photon spectra are plotted in Figure <ref> (blue and green curves for warm-target model with kappa and
power-law form injection, respectively, same color representation for the
normalized residuals in the bottom panel).
 

The fit using the warm-target model, with kappa and power-law forms of electron
injection, both produces favorable fit results with reasonable parameters. With
the systematic uncertainty of 0.02, which is default for RHESSI in
, the reduced χ^2 is 1.93 and 2.11 for the fit with kappa and
power-law forms of electron injection, respectively. From Figure <ref> left bottom panel, the significant normalized residuals remain
below 12 keV, where thermal bremsstrahlung dominates. This may arise from the
use of the iso-thermal assumption f_vth for the thermal emission
<cit.>.
From the photon spectrum made by individual component (Figure <ref> right panel), we found that for this HXR burst, the emission
measure contributed by the thermalized injected electron (Δ EM, dashed
curves) is of comparable magnitude to the iso-thermal component (EM, red solid
curve). Here, we note that although the ratio of Δ EM/EM is affected by
the pre-burst timing chosen, all reasonable selections of pre-burst timing can
yield acceptable fit results. The different ratios only arise from the
identification of pre-accelerated plasma. 

In Figure <ref> left panel, we plot the injected electron rate
spectrum (blue and green curves represent the best-fit kappa and power-law form
injection, respectively). We noticed that the power-law and kappa forms electron
injection exhibit similarly between 40-100 keV but behave differently
approximately below 30 keV. At the power-law low energy cut-off E_c, the
electron injection rate Ṅ(E_c) of the kappa distribution is almost one
order of magnitude smaller (1.11 vs. 7.88 ×10^35 electrons sec^-1 keV^-1 for kappa and power-law form injection,
respectively). Additionally, the fitted injected electron spectrum in the kappa
distribution decreases more rapidly (κ>δ+1) at higher energy.
Although the best-fit electron spectrum in kappa and power-law form have a
similar total injection rate (Ṅ_0 = 22.5 vs. 20.3 ×10^35 electrons sec^-1 for kappa and power-law form injection,
respectively), the power-law electron spectrum produces a more substantial
nonthermal power than the kappa form electron spectrum (see Table <ref>, 3.62 vs 2.23 ×10^28 erg/s,
P_κ/P_power-law≈ 0.61). Moreover, for kappa form
electron spectrum, the nonthermal power produced by the electrons above
power-law low energy cut_off E_c is 1.41 ×10^28 erg/s. The
equations used for calculating the power are listed below, for power-law
distribution:



    P=∫_E_c^∞ E'N(E')  dE'=δ-1/δ-2E_CṄ_0.


For kappa distribution:


    P     = ∫_E_1^E_2 E'N(E')  dE' = Ṅ_0/(k_BT_κκ)^2(κ-3)(k_BT_κκ+E”)×
        (E”/k_BT_κκ+1)^-κ[2(k_BT_κκ)^2+2k_BT_κ(κ-1)κ E”+(κ^2-3κ+2)E”^2]|_E_1^E_2.


Thus


    P=∫_0^∞ E'N(E')  dE'= 2Ṅ_0k_BT_κκ/κ-3;



    P     = ∫_E_c^∞ E'N(E')  dE'= Ṅ_0/(k_BT_κκ)^2(κ-3)(k_BT_κκ+E_c)×
        (E_c/k_BT_κκ+1)^-κ[2(k_BT_κκ)^2+2k_BT_κ(κ-1) κ E_c+(κ^2-3κ+2)E_c^2].


We also include a distribution plot (f_k(E), f_k(E)dE=⟨
f_k(v) ⟩ d^3v) of the accelerated kappa-form electrons (blue curve in
Figure <ref> right panel, n_k=1.67×10^9 cm^-3, calculated from Equation <ref>, here A=4.21×10^10 cm^2). The
Maxwellian thermal distribution, which is take from the iso-thermal component
f_vth (total electron number density n_k=n_th=4.4×10^10 cm^-3, k_BT=1.30 keV), is also shown in red curve for
comparison. We will provide a more detailed discussion of the parameters
obtained in the following section.






 §.§ 2022 March 28 STIX flare
 









We conducted the same spectral analysis to the other well-observed GOES class
M4.0 flare on March 28, 2022. GOES Soft X-ray flux peaks at ∼ 11:29 UT; the
flare-associated HXR bursts were detected by STIX onboard the Solar Orbiter
<cit.>. The STIX temporal profile is shown in
Figure <ref> left panel, with the time difference of 335.7 seconds
between the Sol-O and the Earth has been applied to the STIX data. The flare
site is situated in the eastern limb from Sol-O's perspective. Extreme
Ultraviolet Imager <cit.>/Full Sun Imager (FSI)
provides EUV context from the same viewpoint (Figure <ref> right
panel). This flare comprises several HXR bursts, and our focus is on the burst
occurring at approximately 11:21 UT (red shadowed time range in Figure <ref> left panel). 


The selected burst and pre-burst time ranges for this flare are
11:21:24–11:21:44 UT and 11:21:12–11:21:17 UT (Figure <ref> left
panel red and black shadowed regions), respectively. For this particular event,
bottom pixels have a noticeable larger counts than the top pixels. Thus,
spectrum for fitting and spectroscopic imaging relies solely on the data from
the bottom pixels. The fit range is set to be 6 – 50 keV. The pre-burst spectra
yield emission measure and temperature values of k_BT_0=1.67  keV
and EM_0=0.0532×10^49 cm^-3. Figure <ref> right
panel displays X-ray spectroscopic images using the CLEAN algorithm
<cit.> and are overlaid on EUI/FUI 174 Å flare context.
Two looptop sources (red contours, 8-12 keV), of which the southern one existed
before the flare <cit.>, and two footpoint sources (blue contours, 20-50
keV) are observed. According to the 50% contour of the two loop top sources, we
obtain the source size in different dimensions (r_1=l_1/2=3.4  Mm,
r_2=l_2/2=3.2  Mm, pink lines, and d_1=12.1  Mm, d_1=9.6
 Mm, red lines). The cross-area (using northern source, A=π
r_1^2=3.64×10^10 cm^2) and source volume (V=π r_1^2 d_1+π
r_2^2 d_2=7.59×10^26 cm^3) can be calculated accordingly. The
constructed semi-elliptical coronal loop (cyan dashed line in Figure <ref> right panel), which rooted at two HXR footpoint sources and passing
through the loop top sources, features a half-loop length of L=18.4  Mm.
As a result, we can determine the target loop's thermal properties (k_B
T_loop=k_BT_0=1.67  keV and
n_loop=√(EM_0/V)=2.4×10^10 cm^-3).

The best-fit results using f_vth and
f_thick_warm_kappa/f_thick_warm are shown in Figure
<ref> and Table <ref>. To account for STIX
calibration uncertainty, which is still unknown, 6% systematic error was added
in  following  
<cit.>. From the Figure <ref> left panel,
distinct difference between the kappa and power-law form injection electron
spectrum was noted below 30 keV, consistent with the findings from the 2011
February 24 flare. For this event, low energy cut_off E_c obtained from
f_thick_warm is 12.46 keV. For comparison, the injection rate
Ṅ(E=E_c) is 0.641 vs. 4.21 ×10^35 electrons sec^-1 keV^-1 for kappa and power-law form injection,
respectively. The kappa form electron distribution, same as 2011 February 24
flare, will decrease more rapidly at higher energy (κ>δ+1). The
nonthermal power generated by the injected electron spectrum (3.66 vs 2.95
 ×10^28 erg/s, for electrons in power-law and kappa form
distribution, respectively) also exhibits significant dissimilarities. The kappa
form electrons generate less nonthermal power P_κ/P_power-law≈ 0.81 despite larger total injection rates: 20.5 vs. 13.6  [10^35 electrons sec^-1]. Figure <ref> also
illustrates a comparison between the distribution of accelerated kappa-form
electrons (blue curve, total electron number density n_k=1.1×10^9 cm^-3) and ambient thermal Maxwellian distribution (red curve, total
electron number density n_th=2.4×10^10 cm^-3,
k_BT=1.67 keV) in the right panel.






§ ANALYSIS TO THE FIT PARAMETERS AND FIT RESULTS
  In
this section, we will further investigate how well the three free kappa
parameters (Ṅ_0, κ, and T_κ) can be constrained by the fit
with the warm-target model. We will provide the uncertainty of the fit
parameters using different methods. Additionally, we will compare the total
electron density n_k and average electron energy E_avg derived from
the fit kappa-form spectrum with the corresponding thermal properties.





 §.§ 2011 February 24 RHESSI flare
 







In the previous section, we show that fitting using the functions
f_vth and f_thick_warm_kappa involves three free
parameters: Ṅ_0, κ, and T_κ, all associated with the kappa
distribution. To estimate the uncertainty of each parameter at different
confidence levels, we apply the method used in <cit.>. We
began by assessing the uncertainty of total injection rate Ṅ_0. We carry
out spectral fits with Ṅ_0 set at a fixed value while setting two other
kappa parameters (κ, and T_κ) for free. By doing this, we can
obtain the best-fit full χ^2 as well as the corresponding kappa index
κ and kappa temperature T_κ at this fixed Ṅ_0.
Subsequently, we repeat these fits multiple times with different fixed
Ṅ_0 values and record the corresponding full χ^2 values. Further,
we define Δχ^2=χ^2-min(χ^2), where min(χ^2) is the minimum
of all the obtained full χ^2 values. Δχ^2 reaches 0 at the
Ṅ_0 value of 22.5 ×10^35 electrons sec^-1
(Figure <ref> panel a), consistent with the best-fit results
shown in Table <ref>. The Δχ^2–Ṅ_0 curve
(Figure <ref> panel a) displays a prominent minimum, and the
Δχ^2 value changes rapidly in the vicinity of this minimum. The
Ṅ_0 within the 1-σ uncertainty level (corresponding to a 68%
confidence level, red horizontal dashed line in Figure <ref>
panel a) ranges 22.5^+0.8_-0.6 ×10^35 electrons sec^-1 (red vertical dashed lines in Figure <ref>
panel a), and the Ṅ_0 within the 3-σ uncertainty level
(corresponding to a 99% confidence level, green horizontal dashed line in
Figure <ref> panel a) ranges 22.5^+2.4_-1.8
×10^35 electrons sec^-1 (green vertical dashed lines in
Figure <ref> panel a). The Δχ^2–Ṅ_0 curve
and the estimated uncertainty suggest that the Ṅ_0 can be effectively
constrained by the fit with the warm-target model.


The same approach is used to analyze two other kappa parameters, κ, and
T_κ, for uncertainty. Both the Δχ^2–T_κ (Figure
<ref> panel c) and Δχ^2–κ (Figure
<ref> panel d) curves reveal a distinct minimum, similar to
the Δχ^2–Ṅ_0 curve. The range of k_BT_κ for
1-σ and 3-σ uncertainty levels (red and green horizontal dashed
lines in Figure <ref> panel c) is 1.29^+0.04_-0.04 keV
and 1.29^+0.13_-0.12 keV (red and green vertical dashed lines in Figure
<ref> panel c), respectively. The range of κ for
1-σ and 3-σ uncertainty levels (red and green horizontal dashed
line in Figure <ref> panel d) is 5.14^+0.04_-0.04 and
5.14^+0.12_-0.12 (red and green vertical dashed lines in Figure <ref> panel d), respectively.  The small fit uncertainties for all
three kappa parameters indicate that all parameters can be accurately determined
using the warm-target model.

In this study, we utilize the 3-σ uncertainty level to define the
acceptable range of the fit parameters and investigate the physical properties
within this acceptable range. One valuable property we examine is the average
energy of the electrons E_avg. Here, we first adopt the acceptable
range of Ṅ_0, which falls between 20.3 and 24.9 ×10^35 electrons sec^-1. Each given Ṅ_0 within the acceptable
range corresponds to a fit kappa-form electron spectrum. The average electron
energy for the kappa-form electron spectrum E_avg is given by:

    E_avg  =∫_0^∞ Ef_k(E)  dE/∫_0^∞ f_k(E)  dE 
    =3/2κ/κ-2.5k_B T_κ.

For comparison, the average electron energy of the Maxwellian electron spectrum
is E_avg=3/2k_B T. The obtained
E_avg–Ṅ_0 curve is depicted by the black curve in Figure
<ref> panel b. It is worth noting that, within the
acceptable range (green vertical dashed lines in Figure <ref> panel b), the average electron energies derived from the
kappa-form electron spectrum E_avg-κ are consistently higher than the
average electron energy of the thermal plasma in the target loop
(E_avg-loop=3/2k_B T_loop=1.95 keV, black horizontal
dashed lines in Figure <ref> panel b). According to Equation
<ref>, the total electron number density n_k can be
determined once the injection area A is known. We use the cross-area of the
loop top X-ray coronal source (A=π r^2=4.21×10^10 cm^2,
red outlier contour in Figure <ref> right panel) as the injection
area A to calculate the total electron number density n_k. The
n_k–Ṅ_0 curve is shown in blue curve in Figure <ref> panel b. Within the acceptable range of Ṅ_0, the total
electron number density n_k is found to be lower than the thermal electron
density of the target coronal loop n_loop (blue horizontal dashed lines
in Figure <ref> panel b, 4.4×10^10 cm^-3).
While the electron injection area cannot be precisely determined, and it may be
smaller than the observed loop cross area. n_k is unlikely to be higher than
the loop density n_loop, which requires a extremely small injection
area A<1.60×10^9 cm^2. The results indicate that the values
of E_avg-κ and n_k are physically reasonable when compared with the
ambient thermal plasma. The fitted kappa-form electron spectrum and associated
properties (E_avg-κ and n_k) within the acceptable range for
T_κ and κ closely resembles those obtained within the acceptable
range for Ṅ_0, and consistent with the previous findings. Consequently,
we have opted not to display them in the plot.


In order to further evaluate how well the fit parameters can be determined and
cross-check their uncertainty, we utilized the Monte Carlo analysis provided by
. Detailed information on this method is available in the 'Fit
Parameter Uncertainty Analysis' section from <cit.>. In
addition, <cit.> discussed commonly used methods, including
the Monte Carlo method, for estimating the constraint of fit parameters. The
Monte Carlo analysis requires a predetermined best-fit result, which is
available in Table <ref>. The Monte Carlo sampling explores
the multidimensional parameter space (Ṅ_0, κ, and T_κ)
surrounding the best-fit results by assuming a Gaussian probability function for
each parameter. Figure <ref> shows the posterior probability
density function for the three free parameters obtained from the Monte Carlo
analysis. The final fit results of the Monte Carlo analysis, as well as the
predetermined best-fit results from the forward fit, are also depicted in red
and blue dots, respectively. The coroner plot diagonal panels show the
one-dimensional projection of the probability density function. For the 2011
February 24 M3.5 flare, each parameter exhibits a prominent peak and a
relatively narrow width, indicating that they are well-constrained around the
best-fit results. The 1-σ uncertainty level range for each parameter is
shown in Figure <ref>: Ṅ_0=22.5^+0.6_-0.6
×10^35 electrons sec^-1,
κ=5.14^+0.03_-0.03, and
k_BT_κ=1.29^+0.04_-0.03 keV, comparable with the uncertainty
obtained from the previous method. The non-diagonal panels display the
probability density function projected for different parameter pairs, which
illustrate the correlations between different parameters. We observed that
κ and T_κ are positively correlated, while Ṅ_0 is
anti-correlated with κ and T_κ. The photon spectrum and normalized
residuals generated by Monte Carlo sampling, Monte Carlo final fit result, and
best forward fit result are shown in gray, red, and blue curves in Figure
<ref>.





 §.§ 2022 March 28 STIX flare
 







We conducted the same analysis on the 2022 March 28 M4.0 flare observed by STIX.
Similar to the 2011 February 24 M3.5 flare, the Δχ^2–Ṅ_0,
Δχ^2–T_κ, and Δχ^2–κ curves (black curves in
Figure <ref> panel a, c, and d, respectively) all exhibit a
prominent minimum. The ranges for each parameter at 1-σ and 3-σ
uncertainty levels are listed below: for Ṅ_0, 20.5^+1.3_-1.2
×10^35 electrons sec^-1, and 20.5^+3.9_-3.8
×10^35 electrons sec^-1; for k_BT_κ,
2.97^+0.17_-0.16 keV, and 2.97^+0.55_-0.45 keV; for κ,
8.85^+0.41_-0.38, and 8.85^+1.45_-1.01. The spectral fit for the
2022 March 28 M4.0 flare, due to a narrow fit range (6–50 keV) and a 6%
systematic error from STIX (2% for RHESSI), results in relatively larger
uncertainties, particularly for T_κ and κ. However, these
uncertainties still demonstrate that the warm target fit effectively constrains
the fit parameters. We adopt the 3-σ uncertainty level as the acceptable
range, consistent with our approach for the 2011 February 24 M3.5 flare. The
derived average electron energy E_avg and total electron density n_k
both satisfy E_avg-κ>E_avg-loop and n_k<n_loop within the
acceptable range (Figure <ref> panel b). Additionally, the
Monte Carlo analysis results are displayed in Figures <ref> and
<ref>. The 1-σ uncertainty level range for each
parameter is: Ṅ_0=20.5^+0.2_-0.2 ×10^35 electrons sec^-1, κ=8.85^+0.17_-0.14, and
k_BT_κ=2.97^+0.06_-0.05 keV. The correlations between these
parameters are consistent with the findings from the 2011 February 24 M3.5 flare
(Figure <ref> diagonal panels).






§ SUMMARY
  

In this study, we utilized the warm-target model to analyze two GOES M-class limb
flares. One M3.5 class flare occurred on February 24, 2011, and is observed by
RHESSI, while STIX observed the other M4.0 class flare on March 28, 2022. We
used the warm-target model due to its ability to accurately determine nonthermal
electron properties, such as nonthermal power, which can hardly be constrained
by the commonly used cold-target model. Unlike previous investigations, 
the kappa distribution is used to
characterize the accelerated/injected electrons. 
Unlike the power-law distribution, the kappa distribution is finite
as the electron speed approaches zero, 
thereby effectively covering the entire electron energy range without the
requirement of an arbitrary low energy cutoff. 
Consequently, the fitted
kappa-form electron spectrum can characterize electrons of which the kinetic
energy falls below the X-ray instrument's sensitive range. 
The results of the fitting and subsequent analysis are summarized below:




  * The best-fit results using the warm-target model in kappa-form injected
electrons feature reasonable electron spectrum. For comparison, we also employed
the warm-target model in the power-law electron distribution to fit the HXR
spectrum. We found that the obtained power-law and kappa-form electron spectrum
generate similar photon spectrum within the fitting range. However, the
kappa-form electron spectrum produces less nonthermal power despite having a
comparable or higher total injection rate Ṅ_0. This energy difference
is due to the behavior of the electron spectrum. The obtained
kappa-form and power-law-form electron spectra exhibited similar behavior at the
deka-keV level, while the kappa-form spectrum displayed a significantly lower
flux below ∼30 keV. Moreover, the kappa-form electron spectrum decreased
more rapidly than the power-law spectrum (κ>δ+1) at higher energies
(typically above 100 keV, out of the fit range).


  * The fit with the warm-target model in kappa-form electron injection
involves three free parameters (Ṅ_0, κ, and T_κ). 
Δχ^2-Ṅ_0 curve reveals a prominent minimum, 
suggesting an accurate determination of Ṅ_0 within a narrow range. 
For the M3.5 flare on February 24, 2011, Ṅ_0 within 1-σ 
uncertainty level ranges 
22.5^+0.8_-0.6 ×10^35 electrons sec^-1, and the
3-σ uncertainty level range is 22.5^+2.4_-1.8 ×10^35 electrons sec^-1. For the M4.0 flare on March 28, 2022,
Ṅ_0 within 1-σ uncertainty level ranges 20.5^+1.3_-1.2
×10^35 electrons sec^-1, and the 3-σ
uncertainty level is 20.5^+3.9_-3.8 ×10^35 electrons sec^-1. We apply the same analyses to the other two fit parameters,
κ and T_κ, for their uncertainty. 
For the M3.5 flare on February 24, 2011, κ
within 1-σ and 3-σ uncertainty level range is
5.14^+0.04_-0.04 and 5.14^+0.12_-0.12, respectively.
k_BT_κ within 1-σ and 3-σ uncertainty level range is
1.29^+0.04_-0.04 keV and 1.29^+0.13_-0.12 keV, respectively. For the
M4.0 flare on March 28, 2022, κ within 1-σ and 3-σ
uncertainty level range is 8.85^+0.41_-0.38 and 8.85^+1.45_-1.01,
respectively. k_BT_κ within 1-σ and 3-σ uncertainty
level range is 2.97^+0.17_-0.16 keV and 2.97^+0.55_-0.45 keV,
respectively. All the three parameters, Ṅ_0, T_κ and κ are
tightly constrained within a narrow range. Additionally, we carry out a Monte
Carlo analysis to further confirm the effectiveness in determining the fit
parameters with the warm-target model. The resulting posterior probability
density function for all three kappa parameters demonstrates distinct peaks and
relatively narrow widths, indicating well-constrained fit parameters. The Monte
Carlo analysis also provide a 1-σ level uncertainty for each parameter:
for 2011 February 24 M3.5 flare, Ṅ_0=22.5^+0.6_-0.6 ×10^35 electrons sec^-1, κ=5.14^+0.03_-0.03, and
k_BT_κ=1.29^+0.04_-0.03 keV; for 2022 March 28 M4.0 flare,
Ṅ_0=20.5^+0.2_-0.2 ×10^35 electrons sec^-1, κ=8.85^+0.17_-0.14, and
k_BT_κ=2.97^+0.06_-0.05 keV.



  * Unlike the power-law distribution, which requires a low energy cut-off
E_c, the kappa-form electron spectrum works for all energies, anchored at zero
speed, and extend to the speed of light. Therefore, the kappa distribution
facilitates the determination of the accelerated electron number density n_k
(with the information of the injection area A) and the average electron energy
E_avg. For both flares in this study, we find that, the derived total electron energy density is lower
than the thermal number density of the target loop (n_k<n_loop,
1.7×10^9 cm^-3<4.4×10^10 cm^-3 and
1.1×10^9 cm^-3<2.4×10^10 cm^-3 for 2011
February 24 and 2022 March 28 flare, respectively), while the average electron
energy surpasses that of the ambient Maxwellian thermal plasma (
E_avg-κ>E_avg-loop, 3.77 keV > 1.95 keV and 6.21 keV > 2.51 keV
for 2011 February 24 and 2022 March 28 flare, respectively).



In this study, we have demonstrated that the fit with the warm-target model
provides a convincing kappa-form accelerated electron spectrum. The associated
kappa distribution parameters are valuable for understanding electron
acceleration and transport. Here, we follow the model proposed by
<cit.>, which uses the Fokker-Planck equation to analyze
the evolution of flare-associated electrons. 
proposed that the kappa distribution arises from stochastic acceleration in the
presence of Coulomb collisions and velocity diffusion. The kappa index κ
is given by κ=τ_acc(v)/2τ_c(v). The collisional
deceleration time can be estimated as: τ_c(v)≈v^3/Γ, where
the collisional parameter Γ=2Kn/m_e^2. Here, we take the parameters
from the 2011 February 24 M3.5 flare event for study. For the upper limit of
electron energy E_l enabling the complete stop within the target, we have
E_l=√(2KnL)≈19.0 keV. At E_l, we have the electron speed
v_l=8.0×10^9 cm/s and the collisional deceleration time
τ_c(v_l)≈0.71 seconds. Thus the acceleration time can be
determined by τ_acc(v_l)=2κτ_c(v_l)≈7.33
seconds. The collisional diffusion time
τ_d(v_l)=2v_l^5/Γ v_te^2≈19.77 seconds, where
thermal speed v_te=√(2k_BT_κ)/m_e≈2.1×10^9
cm/s.

The ratio between the density of accelerated nonthermal electrons
(n_nth) and ambient thermal electrons (n_th) can provide
valuable insights into electron acceleration during solar flares. Previous
studies have shown that the ratio of nonthermal electrons to protons
(n_p=n_nth+n_th in the hydrogen plasma) is on the order
of approximately 1% <cit.>. In this study, we determined
the density of accelerated nonthermal electrons in corona by integrating the
kappa electron spectrum over 10 keV(n_nth[>10 ]∼1.2×10^8 cm^-3, and 2.0×10^8 cm^-3 for 2011 February 24 M3.5 and 2022 March 28 M4.0 flare,
respectively). For the thermal electron density in the flare site, We take the
loop electron density obtained from the X-ray thermal component
(n_th=n_loop). The ratio between the density of coronal
accelerated nonthermal electrons and ambient protons is found to be less than
1% (n_nth/n_p∼0.003, and 0.008, for 2011 February 24
M3.5 and 2022 March 28 flare, respectively), consistent with recent studies
<cit.>.

Here, we note the electron number density is calculated using Equation <ref> and is dependent on the injection area A. We estimated the
injection area A using 50% of the loop top X-ray source contour. The
resulting total injected electron density is approximately an order of magnitude
lower than the estimated coronal loop thermal electron density (1.7×10^9 cm^-3<4.4×10^10 cm^-3 and 1.1×10^9 cm^-3<2.4×10^10 cm^-3 for 2011 February 24 and 2022
March 28 flare, respectively). 
It is important to note that the electron injection
in the coronal region may be non-uniform in height.
According to <cit.>, 
the actual injection might be smaller.
In this study, we are utilizing the spatially integrated spectrum for analysis. At present, the signal-to-noise ratio does not sufficiently support precise spatially resolved spectral diagnostics at the arcsecond scale. Once we obtain the spatially resolved spectral information from the improved X-ray observations, the electron density derived from the kappa distribution will be more persuasive.

Since both the RHESSI and Sol-O/STIX are not sensitive to photons below 
∼3-6 keV, the thermal properties obtained from
RHESSI X-ray spectral diagnostics could differ from those obtained from SDO/AIA EUV
diagnostics <cit.>. Furthermore, thermal properties 
inferred from the observed X-ray spectrum are likely to be height 
dependent <cit.>. The obtained kappa-form energetic electrons 
allows for a more comprehensive combined study of the X-ray and EUV diagnostics. 
Potential future applications of the derived kappa distribution also include studying 
acceleration mechanisms based on the average electron energy and nonthermal power. 


The work was supported via the STFC/UKRI grant ST/T000422/1.



aasjournal




 Paper Title: Weak rates in strongly coupled cold quark matter.

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             Weak rates in strongly coupled cold quark matter
                Carlos Hoyos^1[hoyoscarlos@uniovi.es], Andrea Olzi^2[andrea.olzi@unifi.it],
            David Rodriguez-Fernandez^3[david.rfernandez@upm.es]
                August 5, 2024
            ====================================================================================================================================
            
            
            
            0.75[][s]^1Department of Physics and Instituto de Ciencias and Tecnologías Espaciales de Asturias (ICTEA)[][l] Universidad de Oviedo, c/ Leopoldo Calvo Sotelo 18, ES-33007 Oviedo, Spain[][s]^2INFN, Sezione di Firenze and Dipartimento di Fisica e Astronomia, Università di Firenze,[][l] Via G. Sansone 1,I-50019 Sesto Fiorentino (Firenze), Italy.[][s]^3Departamento de matemática aplicada a las TIC, Universidad Politécnica de Madrid[][l] Nikola Tesla, s/n, ES-28031 Madrid, Spain
            
            § ABSTRACT 
            The rates of flavor-changing weak processes are crucial in determining the conditions of beta equilibrium in neutron stars and mergers, influencing the damping of oscillations, the stability of rotating pulsars, and the emission of gravitational waves. We derive a formula for these rates at nonzero temperature, to leading order in the Fermi coupling and exact in the QCD coupling. Utilizing a simple phenomenological holographic model dual to QCD, we study massless unpaired quark matter at high densities. We numerically compute the rate for small deviations from beta equilibrium and derive an analytic approximation for small temperatures. Our findings reveal that, compared to the perturbative result, the rate is suppressed by logarithmic factors of the temperature.
            
            
            
            
            
            
            
            
            § INTRODUCTION
            
            
            
            Weak interactions play a crucial role in determining the properties of neutron stars and their binary mergers. The equation of state, which dictates the mass and radius of the star, is constrained by the condition of beta equilibrium. This condition establishes relationships among the relative amounts of different quark (or baryon) and lepton species. In mergers, the beta equilibrium condition may be modified due to the temperature dependence of the rates <cit.>. When a perturbation in the density drives the matter of the star out of beta equilibrium, weak processes work to restore it and dampen the perturbation. This results in an effective bulk viscosity that is significantly enhanced when the characteristic time of the perturbation resonates with the timescale determined by the rate of weak processes and the density in the star <cit.> (see <cit.> for a review of transport in neutron stars).
            
            Oscillations of a rotating neutron star produce gravitational waves that cause the star to lose angular momentum and spin down, a process that is more pronounced for rapidly rotating stars. Despite this, fast-rotating pulsars with periods on the order of milliseconds are observed within a limited range of temperatures. The effective bulk viscosity induced by beta equilibration is likely necessary to explain the existence and location of this stability window <cit.>. This viscosity also affects the continuous emission of gravitational waves by a quiescent star <cit.> and could impact the gravitational wave signal produced during the inspiral <cit.> and post-merger evolution of a neutron star collision <cit.>.
            
            Although most matter in a neutron star and the remnant of a merger is expected to be primarily composed of hadrons, there are indirect hints of quark matter in the core of heavy neutron stars <cit.>. Simulations of binary mergers suggest that pockets of quark matter may be produced just after the collision <cit.>. Therefore, it is interesting to evaluate the bulk viscosity of quark matter induced by weak processes. At the densities involved, quantum chromodynamics (QCD) cannot be treated perturbatively, and lattice QCD suffers from the sign problem, necessitating other non-perturbative approaches. Holographic models can circumvent these issues and have been applied to the study of dense matter in neutron stars (see <cit.> for comprehensive reviews). Specifically, the bulk viscosity induced by weak processes was computed using holographic models in <cit.>[The holographic estimate of the QCD contribution to the bulk viscosity was computed previously and found to be much smaller than the expected weak contribution <cit.>.]. However, the bulk viscosity formula involves weak reaction rates that have been computed only perturbatively, so it is not a completely self-contained approach. Furthermore, the existing formulas for weak rates are obtained through tree-level scattering matrix elements of quarks and leptons <cit.>. Since the observables obtained from gauge/gravity duality are correlators of gauge-invariant operators, there is no straightforward way to obtain the rates by extrapolating the tree-level formulas. Unfortunately, a main feature of the bulk viscosity, namely its temperature dependence, is highly sensitive to the behavior of the rate. Thus, the present estimation of the bulk viscosity is hindered by significant uncertainty in this respect.
            
            As a step towards improving the bulk viscosity estimates, we will provide a non-perturbative formula for the rates based on correlators of gauge-invariant operators. We will then use it to obtain the temperature dependence of the rates in a simple holographic model of unpaired quark matter. The model, a phenomenological holographic dual description of QCD <cit.>, has been used extensively, with recent related works including the description of nuclear matter in neutron stars and mergers <cit.> and the calculation of neutrino emissivities from flavor current correlators <cit.>. Our goal is not to obtain completely realistic values for the rates, for which more refined models would be needed, but to ascertain possible qualitative changes in the temperature dependence at strong coupling.
            
            Holographic models, despite their utility, come with inherent limitations. They are typically constructed as approximations that capture certain features of QCD, but they do not encompass all the complexities of the theory. One significant limitation is the lack of a first-principles derivation from QCD, leading to uncertainties in their applicability to specific phenomena. Furthermore, these models often rely on large-N_c (number of colors) approximations, which may not accurately reflect the behavior of QCD at the finite N_c relevant to real-world scenarios. The models also face challenges in accurately representing the dynamics at low temperatures and densities where confinement and chiral symmetry breaking are essential. Therefore, while holographic models provide valuable insights and a non-perturbative framework, their predictions should be interpreted with caution and validated against other approaches and experimental data.
            
            Keeping the limitations above in mind, for direct application to the description of neutron stars and their binary mergers, the rate should be computed in the model used to describe matter in the star. The most employed holographic models so far have been those inspired by top-down brane intersections, namely the D3-D7 <cit.> and Witten-Sakai-Sugimoto models <cit.>, and a bottom-up model designed to fit lattice QCD data and other known observables, the V-QCD model <cit.>. The D3-D7 model is the simplest but is limited to describing quark matter in the star <cit.> and must be combined with other nuclear matter models to construct hybrid equations of state for the matter inside the neutron star. The V-QCD model can describe confined matter but is used primarily at large densities, requiring a hybrid approach to describe matter at lower densities outside the core of the star <cit.>. Hybrid equations of state have also been constructed with the Sakai-Sugimoto model <cit.>, but there are proposals to describe nuclear matter in neutron stars at all densities purely holographically <cit.>. The Sakai-Sugimoto model has also been used to describe more exotic quark stars <cit.>. Other proposals to describe dense matter in neutron stars include Einstein-Maxwell-Dilaton theories and other holographic phenomenological models <cit.>. Computing the weak rate and bulk viscosity to estimate the location of the stability window of rotating pulsars would serve as an additional constraint on these models, as well as on any other phenomenological model for which such a calculation is possible.
            
            The outline of the paper is as follows. In Section <ref>, we introduce the weak reaction rates for beta equilibrium of quark matter and derive a non-perturbative formula in terms of correlators of flavor currents. Next, in Section <ref>, we use a simple holographic model to compute flavor current correlators in a high-density state and study their dependence on frequency and spatial momentum. Finally, in Section <ref>, we combine the results of the previous sections to produce analytical and numerical estimates of the weak rates at low temperatures. We compare these to the perturbative results and discuss potential improvements in Section <ref>. Technical details regarding the calculation of correlators in the holographic model are provided in the appendices.
            
            
            
            
            
            
            § WEAK REACTION RATES
            
            
            
            In this section, we derive non-perturbative formulas in the coupling of the strong interactions for the weak reaction rates of quark matter. The formulas correspond to the leading contribution of the electroweak interactions, whose corrections could in principle be computed perturbatively. We also assume that the times and length scales involved are much larger than the scale set by the inverse of the W boson mass.
            
            For the densities expected in the interior of neutron stars and mergers, one should be able to restrict to three quark flavors (q=u,d,s) and two lepton flavors (ℓ=e,μ). The relevant weak processes, or reactions, are produced by a W-boson exchange or emission:
            
                u+d⇌ u+s,   u+ℓ^- →  d,s+ν_ℓ,    d,s → u+ℓ^- +ν̅_ℓ,  ℓ_1^- →ℓ_2^-+ν_ℓ_1+ν̅_ℓ_2 .
            
            Assuming neutrinos are not trapped, the beta-equilibrium conditions in quark matter implied by these reactions are
            
                μ_u+μ_ℓ=μ_d=μ_s,    μ_ℓ=μ_e=μ_μ .
            
            Even if quarks are confined to hadrons the same reactions will change their type among protons, neutrons, and hyperons. In this situation, the beta equilibrium conditions should be given in terms of baryon chemical potentials. Note that in order for any of the species of fermions to be present it is necessary that the chemical potential is larger than the mass of the corresponding particle.
            
            
            If a perturbation takes the chemical potentials out of their equilibrium values, the weak reactions will not be in balance, and the densities will change according to the weak rates Γ:[We have omitted the label on the neutrinos when there is just one lepton involved and put the same numerical sub-index as the corresponding lepton when there are two.]
                d n_u/dt   =   ∑_ℓ=e,μ( Γ_d→ uℓν̅-Γ_uℓ→ dν+ Γ_s→ u ℓν̅-Γ_uℓ→ sν) ,
                d n_d/dt   =   Γ_su→ ud-Γ_ud→ su+∑_ℓ=e,μ( Γ_uℓ→ dν-Γ_d→ uℓν̅) ,
                d n_s/dt   =   Γ_ud→ su-Γ_su→ ud+ ∑_ℓ=e,μ(Γ_uℓ→ sν-Γ_s→ u ℓν̅) ,
                d n_ℓ_1/dt   =   Γ_d→ uℓ_1ν̅-Γ_uℓ_1→ dν+Γ_s→ u ℓ_1ν̅-Γ_uℓ_1→ sν+Γ_ℓ_2→ℓ_1ν_2ν̅_1-Γ_ℓ_1→ℓ_2ν_1ν̅_2 .
            
            It is easy to confirm that the total baryon number and electric charge are conserved by these processes
            
                n_B=1/3(n_u+n_d+n_s),   n_Q=2/3n_u-1/3n_d-1/3n_s-n_e-n_μ,    dn_B/dt=dn_Q/dt=0 .
            
            
            Assuming the deviation of the chemical potentials from their beta-equilibrium values δμ_f^a is small compared to the baryon chemical potential μ_B=μ_u+μ_d+μ_s, we can expand the rates to linear order in the differences
            
                Γ_ud→ su-Γ_su→ ud   ≈   λ_ds (δμ_s-δμ_d),
                Γ_uℓ→ dν-Γ_d→ uℓν̅   ≈   λ_ud^ℓ (δμ_d-δμ_u-δμ_ℓ),
                Γ_uℓ→ sν-Γ_s→ uℓν̅   ≈   λ_us^ℓ (δμ_s-δμ_u-δμ_ℓ),
                Γ_e→μν_2ν̅_1-Γ_μ→ eν_1ν̅_2   ≈   λ_eμ (δμ_μ-δμ_e).
            
            
            In some cases, for instance when quark matter is close to being symmetric, the lepton chemical potential could be below the muon mass μ_ℓ<m_μ. When this happens, muons quickly decay and the only leptons present are electrons. The equilibration equations simplify somewhat
            
                d n_u/dt   =   Γ_d→ u eν̅-Γ_ue→ dν+ Γ_s→ u eν̅-Γ_u e→ sν ,
                d n_d/dt   =   Γ_su→ ud-Γ_ud→ su+Γ_u e→ dν-Γ_d→ u eν̅ ,
                d n_s/dt   =   Γ_ud→ su-Γ_su→ ud+ Γ_u e→ sν-Γ_s→ u eν̅ ,
                d n_e/dt   =   Γ_d→ ueν̅-Γ_ue→ dν+Γ_s→ u eν̅-Γ_ue→ sν .
            
            For small deviations from beta-equilibrium, the rates are expanded as (<ref>), but taking into account only the electron contribution. Moreover, the coefficient λ_eμ does not involve strong coupling physics, so we will ignore it in the following, as it can be computed using the standard methods. Our strategy to derive the other coefficients is to arrive at an expression like (<ref>) from Ward identities capturing the non-conservation of flavor currents due to the coupling to the weak sector. 
            
            
            
            
             §.§ Flavor Ward identities
            
            
            The theory of strong interactions, QCD, respects flavor symmetries so that there are no flavor-changing processes induced by them. Weak interactions on the other hand are not flavor symmetric, and processes such as those in (<ref>) are possible. At energy scales much below the W mass, the flavor-changing processes are captured by adding Fermi's interaction term to the effective action
            
                L_ Fermi=-2√(2) G_F (J_ch^μ)^† J_ch μ,
            
            where the J_ch^μ is the left-charged current
            
                J_ch^μ=ν_e Lγ^μ e_L+ν_μ Lγ^μμ_L+cosθ_C u_Lγ^μ d_L+sinθ_C u_Lγ^μ s_L,
            
            with G_F Fermi's constant and θ_C the Cabibbo angle giving the change of basis between quark mass and flavor eigenstates. In addition to Fermi's term, flavor symmetry can also be broken by flavor-preserving mass terms and electromagnetic and weak neutral interactions.
            
            In the absence of any flavor-breaking terms, there would be a SU(3)_L× SU(3)_R global symmetry for flavor rotations among quarks and a SU(4)_L× SU(4)_R flavor global symmetry for leptons. To each of these symmetries, we can associate a conserved current J^μ_f χ, where f=l,q for leptons or quarks and χ=L,R for left or right chirality. Since the symmetry is non-Abelian the current components transform under flavor rotations. Denoting by T^A_f χ the generators of the flavor symmetry, the transformation of the current is
            
                δ_θ_f χ J^μ_f χ=iθ_f χ^A[J^μ_f χ,T^A_f χ] .
            
            We will take the usual normalization for the generators tr(T_f χ^A T_f χ^B)=1/2δ^AB. 
            
            A vector flavor transformation corresponds to a transformation with equal angles for both chiralities θ_f L^A=θ_f R^A=θ_f^A. In the following vector currents and generators will be denoted as J^μ_f and T^A_f. If there are flavor-symmetry breaking terms in the action L_ br, the Ward identity associated with a vector transformation becomes
            
                2θ_f^A tr(∂_μ J^μ_f T^A_f)=-δ_θ^A_f L L_ br-δ_θ^A_f R L_ br|_θ_f L^A=θ_f R^A=θ_f^A.
            
            In the following, it will be convenient to introduce the notation for the currents
            
                (J_f χ^μ)^a_ b=f̅_χ bγ^μ f_χ^a ,     f^a=q^a,l^a;   q^a=u,d,s;  l^a=ν_e,e,ν_μ,μ.
            
            
            We are interested in the flavor-changing contributions introduced by Fermi's interaction. We can simplify the analysis by noticing that since Fermi's interaction is symmetric under a rotation between the two lepton generations (ν_e,e)↔ (ν_μ,μ), we can actually restrict lepton transformations to SU(2)× SU(2)⊂ SU(4), with each SU(2) rotating a charged lepton and its corresponding neutrino within each generation. We leave details of the calculation, which is straightforward but tedious, to Appendix <ref>. The Ward identities for the diagonal components of the vector flavor currents are
            
                ∂_μ(J_q^μ)^u_ u   =     i√(2)G_F η_μν∑_ℓ=e,μ[cosθ_C(J_q L^μ)^u_ d(J_l L^ν)^d_ u-(J_q L^μ)^d_ u(J_l L^ν)^ν_ℓ_ ℓ.
                
                        . +sinθ_C(J_q L^μ)^u_ s(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^s_ u(J_l L^ν)^ν_ℓ_ ℓ],
                
                ∂_μ(J_q^μ)^d_ d    =    i√(2) G_Fη_μν[cosθ_C sinθ_Cη_μν(J_q  L^μ)^u_ s (J_q L^ν)^d_ u-(J_q L^μ)^s_ u(J_q L^ν)^u_ d.
                
                        .-cosθ_C∑_ℓ=e,μ(J_q L^μ)^u_ d(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^d_ u(J_l L^ν)^ν_ℓ_ ℓ],
                
                ∂_μ(J_q^μ)^s_ s   =    i√(2) G_Fη_μν[-cosθ_C sinθ_Cη_μν(J_q  L^μ)^u_ s (J_q L^ν)^d_ u-(J_q L^μ)^s_ u(J_q L^ν)^u_ d.
                
                        .-sinθ_C∑_ℓ=e,μ(J_q L^μ)^u_ s(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^s_ u(J_l L^ν)^ν_ℓ_ ℓ],
                
                ∂_μ(J_l^μ)^ν_ℓ_ ν_ℓ   =    i√(2)G_F η_μν[-cosθ_C(J_q L^μ)^u_ d(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^d_ u(J_l L^ν)^ℓ_ ν_ℓ.
                
                        . -sinθ_C(J_q L^μ)^u_ s(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^s_ u(J_l L^ν)^ν_ℓ_ ℓ.
                
                        .-(J_l L^μ)^ν_ℓ'_ ℓ'(J_l L^ν)^ℓ_ ν_ℓ-(J_l L^μ)^ℓ'_ ν_ℓ'(J_l L^ν)^ℓ_ ν_ℓ],
                
                ∂_μ(J_l^μ)^ℓ_ ℓ   =    i√(2)G_F η_μν[cosθ_C(J_q L^μ)^u_ d(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^d_ u(J_l L^ν)^ν_ℓ_ ℓ.
                
                        . +sinθ_C(J_q L^μ)^u_ s(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^s_ u(J_l L^ν)^ν_ℓ_ ℓ.
                
                        .+(J_l L^μ)^ν_ℓ'_ ℓ'(J_l L^ν)^ℓ_ ν_ℓ-(J_l L^μ)^ℓ'_ ν_ℓ'(J_l L^ν)^ν_ℓ_ ℓ].
            
            Where (ℓ,ℓ')=(e,μ) or (μ,e). Note that other terms that break the non-Abelian flavor symmetry, but are diagonal in flavor, do not contribute to the Ward identities of the diagonal components. One can easily confirm that the expressions above are consistent with the conservation of baryon and lepton numbers as well as the electric charge:
            
                ∂_μ(J_q^μ)^u_ u+(J_q^μ)^d_ d+(J_q^μ)^s_ s=0,  ∂_μ(J_l^μ)^ν_ℓ_ ν_ℓ+(J_l^μ)^ℓ_ ℓ=0, 
                   ∂_μ2/3(J_q^μ)^u_ u-1/3(J_q^μ)^d_ d-1/3(J_q^μ)^s_ s-∑_ℓ=e,μ (J_l^μ)^ℓ_ ℓ=0.
            
            The time components of the diagonal components of the flavor vector currents equal the corresponding quark or lepton densities
            
                n_u=(J_q^0)^u_ u, n_d=(J_q^0)^d_ d, n_s=(J_q^0)^s_ s, n_ℓ=(J_l^0)^ℓ_ ℓ.
            
            Compared with (<ref>) for homogeneous configurations, the rates are determined by the expectation value of a non-diagonal operator quadratic in the flavor currents appearing on the right-hand side of the Ward identities. We can think of such a quadratic operator as obtained from a two-point correlator in the (properly regularized) limit of coincident points. In the absence of flavor-breaking terms, the most general form of the flavor current correlators would be
            
                (J_f  L^μ)^a_ b (J_f L^ν)^c_ d= G_1 δ^a_bδ^c_d+G_2 δ^a_d δ^c_b .
            
            If there are mass terms in the action of the form L_m=f̅_a M^a_f b f^b, or chemical potentials introduced through a background flavor gauge field A^a_ f b, there could also be terms where we replace some or all of the deltas in (<ref>) by M_f or A_f, or contractions like M^a_f c M^c_f b,  M^a_f c A^c_f b, etc. This potentially can produce many new terms, however, for diagonal mass matrices and chemical potentials, the non-diagonal current correlators appearing on the right-hand side of the Ward identities would still vanish, and similarly for contributions originating from interactions that are diagonal in flavor. Therefore, the non-diagonal current correlators contributing to the Ward identities must be proportional to the flavor-changing coupling G_F. We will derive in the following a closed expression for the O(G_F^2) contribution to the rates in thermal field theory.
            
            
            
            
            
             §.§ Leading contribution in the Fermi coupling
            
            
            We will assume the theory is at finite temperature and use the real-time formalism to compute the current correlators. In general, thermal correlation functions are computed by insertions in a path integral defined along a contour in complex time C. The two-point function of two operators O_1 and O_2 is
            
                T_C[ O_1(x_1) O_2(x_2)].
            
            The complex time path, or Schwinger-Keldysh path, extends along (t_i,t_f)∪ (t_f,t_f-iσ)∪ (t_f-iσ,t_i -iσ)∪ (t_i-iσ,t_i-iβ), with σ>0 and  t_i→-∞, t_f→+∞. We take the point x_1 to be at the first leg of the complex time path on the real axis. Then we have two possible contributions, if x_2 is on the same leg, then the thermal correlator coincides with the usual time-ordered Feynman correlator
            
                T_C[ O_1(x_1) O_2(x_2)]=T[ O_1(x_1) O_2(x_2)]=G^F_ O_1 O_2 (t_1-t_2,x_1-x_2).
            
            If, on the other hand, x_2 is on the third leg (t'=-σ), the correlator is the Wightman correlator
            
                T_C[ O_1(x_1) O_2(x_2)]= O_2(x_2) O_1(x_1)=G^<_ O_1 O_2 (t_1-t_2+iσ,x_1-x_2).
            
            With this in mind, let us find the effect of Fermi's interaction (<ref>) when we treat it perturbatively. The thermal expectation value of an operator O will be, to leading order in the Fermi coupling,
            
                O(x)_G_F=T_C[ O(x) e^i∫_C   L_ Fermi]_0≈ O(x)_0+i∫_C d^4 x'T_C[ O(x)   L_ Fermi(x')]_0.
            
            In the expression above X_0 indicates that the expectation value of X is computed in the absence of Fermi's interaction. Since x is in the first leg of the complex time contour and  x' runs over its full extent, we have contributions from both the first and third leg. 
            
                ∫_C d^4 x'T_C[ O(x)   L_ Fermi(x')]_0=∫ d^4x' [G_ O L_ Fermi^F(t-t',x-x')+G_ O L_ Fermi^<(t-t'+iσ,x-x')].
            
            In the evaluation of the right-hand side of the Ward identities (<ref>)O is quadratic in the flavor currents, as well as L_ Fermi. Therefore, the leading O(G_F^2) contribution is proportional to correlators of the form
            
                [(J_f^μ)^a_ b(x) (J_f^ν)^c_ d(x)][(J_f'^α)^a'_ b'(x') (J_f'^β)^c'_ d'(x')]_0≈
                   δ_ff'[(J_f^μ)^a_ b(x)(J_f^α)^a'_ b'(x')_0(J_f^ν)^c_ d(x) (J_f^β)^c'_ d'(x')_0+(J_f^μ)^a_ b(x) (J_f^β)^c'_ d'(x')_0(J_f^ν)^c_ d(x)(J_f^α)^a'_ b'(x')_0].
            
            The factorization above is an approximation. For holographic models dual to a large-N_c theory it will happen naturally as part of the large-N_c expansion. More generally, the non-factorizable contributions would originate from effective eight-fermion interactions, so it is perhaps not unreasonable to assume that they would introduce relatively small corrections even in the finite-N_c strongly coupled theory. 
            
            As we discussed before, in the absence of the Fermi interaction, non-vanishing current correlators should be diagonal in flavor, thus
            
                G^μα_f;ab,a'b'(x-x')≡(J_f^μ)^a_ b(x)(J_f^α)^a'_ b'(x')_0≠ 0,  ⇒  (a=b   a'=b')   (a=b'   a'=b).
            
            This limits the number of non-zero contributions to the four-current thermal correlators. We will now write the leading contributions in the Fermi coupling to the Ward identities (<ref>) in terms of the Fourier transforms of the Feynman and Wightmann correlators of the flavor currents
            
                G^K μα_f;ab,a'b'(x-x')=∫d^4 q/(2π)^4e^iq· (x-x') G^K μα_f;ab,a'b'(q),     K=F,<  .
            
            It will be convenient to define
            
                γ__f_1^a f_2^b→ f_1^cf_2^d=γ^F__f_1^af_2^b→ f_1^c f_2^d+γ^<__f_1^af_2^b→ f_1^c f_2^ d,
            
            with (K=F,<)
            
                γ_f_1^a f_2^b→ f_1^c f_2^d^K=η_μνη_αβ∫d^4 q/(2π)^4[G^K μα_f_1;ac,ca(q)G^K νβ_f_2;bd,db(-q)-G^K μα_f_1;ca,ac(q)G^K νβ_f_2;db,bd(-q)] .
            
            In terms of these quantities, the leading order contributions in the Fermi coupling to the flavor Ward identities are
            
                ∂_μ(J_q^μ)^u_ u   ≈    -4 G_F^2∑_ℓ=e,μ(cos^2θ_C γ_uℓ→ dν+sin^2θ_C γ_uℓ→ sν),
                ∂_μ(J_q^μ)^d_ d   ≈    4 G_F^2 (-sinθ_C^2cos^2θ_C γ_ud→ su+cos^2θ_C ∑_ℓ=e,μγ_uℓ→ dν),
                ∂_μ(J_q^μ)^s_ s   ≈    4 G_F^2 (sinθ_C^2cos^2θ_C γ_ud→ su+sin^2θ_C ∑_ℓ=e,μγ_uℓ→ sν),
                ∂_μ(J_l^μ)^ν_ℓ_ ν_ℓ   ≈    4 G_F^2 (cos^2θ_C γ_uℓ→ dν+sin^2θ_C γ_uℓ→ sν+γ_ℓν'→νℓ'),
                ∂_μ(J_l^μ)^ℓ_ ℓ   ≈    -4 G_F^2 (cos^2θ_C γ_uℓ→ dν+sin^2θ_C γ_uℓ→ sν+γ_ℓν'→νℓ').
            
            Where in the last two rows (ℓ,ℓ')=(e,μ) or (μ,ℓ) and ν,ν' are the neutrinos corresponding to ℓ,ℓ' respectively.
            
            
            
            
             §.§ Formulas for the rates
            
            
            A direct comparison of the equilibration equations  (<ref>) and the Ward identities (<ref>) results in the identification of the leading order contribution in the Fermi coupling to the rates as
            
                Γ_uℓ→ dν-Γ_d→ u ℓν̅   ≈    4 G_F^2cos^2θ_C γ_uℓ→ dν  ,
                Γ_u ℓ→ sν-Γ_s→ u ℓν̅   ≈     4 G_F^2sin^2θ_C γ_uℓ→ sν  ,
                Γ_ud → su-Γ_su → ud   ≈     4 G_F^2sinθ_C^2cos^2θ_C γ_ud→ su  ,
                Γ_ℓ_1→ℓ_2ν_1ν̅_2-Γ_ℓ_2→ℓ_1ν_2ν̅_1   ≈    4 G_F^2γ_ℓ_1ν_2→ν_1ℓ_2 .
            
            With the factors γ defined in (<ref>) and (<ref>) in terms of thermal correlation functions. We can further simplify the expressions by first noting that, if the system has parity and time reversal invariance, the Feynman correlator satisfies
            
                G_f;ab,ba^F μν(-q_0,-q)=G_f;ba,ab^F νμ(q_0,q).
            
            A straightforward calculation then shows that γ^F__f_1^af_2^b→ f_1^c f_2^d=0 in this case. The Wightmann correlator on the other hand is proportional to the spectral function, defined as the imaginary part of the retarded correlator:
            
                G^< μν_f;ab,ba(q_0,q)=n_f;ab(q_0)ρ_f;ab^μν(q_0,q),  ρ_f;ab^μν=-2 (G^R μν_f;ab,ba),
             
            where the factor multiplying the spectral function is the thermal distribution
            
                n_f;ab(q_0)=1/e^β(q_0+μ_f^a-μ_f^b)-1 ,
            
            with β=1/T the inverse of the temperature and μ_f^a the chemical potential for the fermion species f^a. We now introduce (<ref>) in (<ref>), and use that
            
                n_f;ab(-q_0)=-(1+n_f;ba(q_0)),     ρ_f;ab^μα(-q_0,-q)=-ρ_f;ba^αμ(q_0,q) ,
            
            where the last identity follows from time-reversal invariance and parity. After this, one arrives at
            
                γ__f_1^a f_2^b→ f_1^cf_2^d=γ^<__f_1^af_2^b→ f_1^c f_2^ d=η_μνη_αβ∫d^4 q/(2π)^4[(n_f_1;ac(q_0)-n_f_2;db(q_0))ρ^μα_f_1;ac(q_0,q)ρ^βν_f_2;db(q_0,q)] .
            
            This quantity will vanish when the chemical potentials satisfy the relations
            
                μ_f_1^a-μ_f_1^c=μ_f_2^d-μ_f_2^b.
            
            Then, demanding that each of the rates in (<ref>) vanish gives the conditions
            
                μ_u-μ_d    =    μ_ν_ℓ-μ_ℓ,
                μ_u-μ_s    =   μ_ν_ℓ-μ_ℓ,
                μ_u-μ_s    =   μ_u-μ_d,
                μ_ν_ℓ_1-μ_ℓ_1   =   μ_ν_ℓ_2-μ_ℓ_2.
            
            Setting μ_ν_ℓ=0, it is easy to check that this coincides with the beta equilibrium conditions in (<ref>). We want to expand (<ref>) to the lowest order around the equilibrium values, μ_f^a=μ_f^a^eq+δμ_f^a. The thermal distributions have a pole at small frequencies, that we need to deal with. Let us introduce an arbitrary frequency ω_0 such that δμ_f_1^a-δμ_f_1^c∼δμ_f_2^d-δμ_f_2^b≪ω_0 ≪ T. Then, there is a contribution that we can expand to linear order in the chemical potentials plus a correction from the poles
            
                γ__f_1^a f_2^b→ f_1^cf_2^d≈(δμ_f_2^d-δμ_f_2^b-(δμ_f_1^a-δμ_f_1^c))Λ_f_1^a f_2^b→ f_1^cf_2^d^0+δγ__f_1^a f_2^b→ f_1^cf_2^d ,
            
            where, after defining Δμ^eq=μ_f_1^a^eq- μ_f_1^c^eq=μ_f_2^d^eq- μ_f_2^v^eq, the coefficient Λ is
            
                Λ__f_1^a f_2^b→ f_1^cf_2^d^0=   η_μνη_αβ∫d^4 q/(2π)^4ρ^μα_f_1;ac(q_0,q)ρ^βν_f_2;db(q_0,q) 
                   ×[1/4Tsinh^2(q_0+Δμ^eq/2T)-T/(q_0+Δμ^eq)^2Θ(ω_0^2-(q^0+Δμ^eq)^2)].
            
            We remind the reader that in this formula the fermion species are f_1,f_2=q (quarks) or l (leptons) and the possible flavors are q^a=u,d,s, l^a=ν_e,e,ν_μ,μ. For q_0+Δμ^eq≳ T the function decays exponentially ∼ e^-(q_0+Δμ^eq)/T, thus most of the contributions in the frequency integral will come from an interval of width ∼ T around q^0+Δμ^eq. 
            
            The correction to the rate originating from the poles is approximately
            
                δγ__f_1^a f_2^b→ f_1^cf_2^d=   η_μνη_αβ∫_|q^0+Δμ^eq|<ω_0d^4 q/(2π)^4ρ^eq μα_f_1;ac(q_0,q)ρ^eq βν_f_2;db(q_0,q) 
                   ×[T/q^0+Δμ^eq+δμ_f_1^a-δμ_f_1^c -T/q^0+Δμ^eq+δμ_f_2^d-δμ_f_2^b] .
            
            We will use the principal value integral
            
                ∫_-ω_0-Δμ^eq^ω_0-Δμ^eqd q^0/2πT/q^0+Δμ^eq+δμ S(q^0) ≈T/2πS(-Δμ^eq)logω_0+δμ/ω_0-δμ≈T/πω_0S(-Δμ^eq) δμ.
            
            Thus, we can write
            
                γ__f_1^a f_2^b→ f_1^cf_2^d≈(δμ_f_2^d-δμ_f_2^b-(δμ_f_1^a-δμ_f_1^c))Λ_f_1^a f_2^b→ f_1^cf_2^d ,
            
            with
            
                Λ__f_1^a f_2^b→ f_1^cf_2^d≈Λ__f_1^a f_2^b→ f_1^cf_2^d^0+η_μνη_αβT/πω_0∫d^3 q/(2π)^3ρ^eq μα_f_1;ac(-Δμ^eq,q)ρ^eq βν_f_2;db(-Δμ^eq,q).
            
            If the spectral function vanishes at q^0=-Δμ^eq the last term in (<ref>) is zero and one can take the ω_0→ 0 limit, in which case the expression for the rate simplifies
            
                Λ__f_1^a f_2^b→ f_1^cf_2^d≈η_μνη_αβ∫d^4 q/(2π)^4ρ^eq μα_f_1;ac(-Δμ^eq,q)ρ^eq μα_f_2;db(-Δμ^eq,q)/4Tsinh^2(q_0+Δμ^eq/2T).
            
            
            Combining the expression (<ref>) with (<ref>) and (<ref>), we arrive at the following formulas for the coefficients of the rates
            
                λ_ds   ≈    4 G_F^2 sin^2θ_Ccos^2θ_C Λ_ud→ su,
                λ_ud^ℓ   ≈    4 G_F^2 cos^2θ_C Λ_uℓ→ dν,
                λ_us^ℓ   ≈    4 G_F^2 sin^2θ_C Λ_uℓ→ sν,
                λ_eμ   ≈    4 G_F^2 Λ_μν_e→ν_μ e.
            
            When leptonic contributions can be neglected, as it is usually the case in the evaluation of the bulk viscosity in neutron stars, the only relevant coefficient is λ_1=λ_ds, following the conventions in <cit.>. In the remaining of the paper, we will give an estimate of λ_1 at strong coupling using a holographic model.
            
            
            
            
            
            
            § HOLOGRAPHIC MODEL
            
            
            
            We will use a holographic model dual to a SU(N_c) gauge theory with N_f flavors to compute the rate. We use a bottom-up approach, where we construct a gravitational action with the necessary ingredients to capture the physics of interest, but that does not correspond to a string theory construction where both sides of the duality are known.
            
            The glue sector is given by an action consisting of just the Einstein-Hilbert action plus a negative cosmological constant
             
            S_c = 1/16π G_5∫ d^5 x  √(-g)[ R - 2Λ],  Λ=-6/L^2 . 
             
            In the absence of matter, the maximally symmetric solution is an AdS_5 space with radius L. Including the flavor sector will modify the geometry, but it will still asymptote to the same AdS_5, so we can apply the usual holographic dictionary between the fields in the gravity theory and sources and operators in the gauge theory dual.
            
            The flavor sector consists of U(N_f)_L× U(N_f)_R gauge fields, L_M and R_M, dual to the flavor currents, and a complex scalar field X in the bifundamental representation dual to a quark bilinear. The action for the flavor fields is the one commonly used for holographic QCD models, originating with <cit.>
            S_f = 1/16π G_5N_f/N_c∫ d^5 x √(-g)  Tr[- | D X|^2 + 3 | X|^2 -L^2/4( F_(R)^2 + F_(L)^2) ] .
             
            The field strength is defined as F_(A)MN = ∂_M A_N - ∂_N A_M - i[ A_M,A_N]. The covariant derivative of the scalar field is defined as follows 
             
            D_N X = ∂_N X - i L_N X + i X R_N ,      D_N X^† = ∂_N X^† + i X^† L_N - i  R_N X^† .
             
            The mass of the scalar is fixed in such a way that the field is dual to a scalar operator of scaling dimensions Δ=3, as corresponds to a quark bilinear.
            
            The action is fixed up to an overall normalization of the scalar and gauge fields. In <cit.> (see also <cit.>) the normalization was fixed by matching the large Euclidean momentum form of current correlators derived from the holographic model to the OPE of QCD. The gauge fields and scalars are rescaled to put the flavor sector of the action in the form  
            
            S_f =∫ d^5 x √(-g)  Tr[g_X^2(- | D X|^2 + 3 | X|^2) -1/4 g_5^2( F_(R)^2 + F_(L)^2) ] .
             
            Then, the matching to QCD in the UV fixes 
            
                1/g_5^2=N_c/12π^2,   g_X^2=N_c/4π^2 .
            
            
            The equations of motion for the metric are Einstein's equations with a cosmological constant and the energy-momentum tensor sourced by the gauge fields and the scalars
            
                R_MN-1/2g_MNR+ g_MNΛ=N_f/N_cT_MN.
            
            The equations of motion for the gauge and scalar fields read
            1/√(-g) D_M [√(-g) F^MN_(R)]    =     j^N_(R) ,
            1/√(-g) D_M [√(-g) F^MN_(L)]    =    j^N_(L) ,
            1/√(-g) D_M [√(-g)  D^M X  ]    =      -3 X .
            
            
            where the bulk currents appearing in the equations of the gauge fields are
            
            j^M_(R)    =    i [( D^M X)^† X - X^† D^M X] , 
            
            j^M_(L)    =    i [ ( D^M X) X^† - X (D^M X)^†] .
            
            We will look for solutions dual to a deconfined state of the gauge theory at nonzero temperature and baryon density, which are charged black brane geometries. In order to simplify the analysis we will turn off the scalar field, which amounts to having zero quark mass and condensate for all the flavors. In this case, imposing the condition of beta equilibrium fixes all quark chemical potentials to be equal, since deviations from symmetric quark matter due to nonzero quark masses are absent. Since we are ultimately interested in comparing with QCD results, we will fix the rank of the flavor fields N_f=3.
            
            
            
            
             §.§ Background solution
            
            
            The metric of the charged black brane takes the general form, in Poincaré coordinates,
            
                ds^2 = L^2z^2(-f(z)dt^2+dz^2f(z)+dx ^2),      f(z)=1-M z^4/z_h^4+Q^2 z^6/z_h^6 .
            
            The AdS_5 asymptotic boundary is at z=0, and the horizon is at the value where the black body factor vanishes f(z_h)=0. The values of M and Q depend on the background scalar and gauge fields. The leading term in the asymptotic expansion of the background scalar at the boundary determines the quark masses 
            
                X ∼([ m_u        ;     m_d    ;         m_s ])z+O(z^3) .
            
            We will set all quark masses to zero m_u=m_d=m_s=0, which allows to have a trivial solution of the scalar X^ bkg=0. The background gauge fields dual to the quark densities are
            
                (L_0^ bkg)^a_ b=(R_0^ bkg)^a_ b=μ_q/2(1-z^2/z_h^2)δ^a_ b .
            
            Where μ_q=μ_u=μ_d=μ_s is the quark chemical potential, which is the same for all flavors at beta equilibrium. The backreaction over the metric is limited to the gauge fields, so the geometry is the AdS Reissner-Nordstrom charged black brane (<ref>), with 
            
                M=1+Q^2 ,      Q^2=z_h^2μ_q^2/2 .
            
            The temperature in the dual field theory is equal to the Hawking temperature, given by
            
                T = 1π z_h(1-Q^22) .
            
            One can use these formulas to obtain the position of the horizon as a function of the temperature and quark chemical potential
            
                z_h=2/μ_q(√(1+(π T/μ)^2)-π T/μ) .
            
            Later we will often use
            
                T̅= z_h T ≈2 T/μ_q .
            
            
            
            
             §.§ Two-point functions of flavor currents
            
            
            
            The spectral function entering the formulas for the rates can be obtained from the imaginary part of the retarded correlator of the left flavor currents. Following the usual holographic dictionary, we can extract the correlator from the solutions to the linearized equations of fluctuations. We will impose the boundary conditions 
            
                lim_z→ 0 L_μ(x,z)=l_μ(x),  lim_z→ 0R_μ(x,z)=0,  lim_z→ 01/zX(x,z)=0.
            
            This turns off the sources for the right flavor currents and the quark bilinears and sets l_μ(x) as the source for the left flavor currents. The boundary asymptotic expansion of the Fourier transform of the left gauge fields is
            
                (L_μ)^a_ b(ω,k,z)∼( δ^ ν_μ+(c_μ^ ν)_ab z^2 logz/L+z^2 (g_μ^ ν)_ab+⋯)(l_ν)^a_ b(ω,k).
            
            Where, keeping the dependence on the quark masses,
            
                (c_μ^ ν)_ab = 12((k_α k^α+m_a^2+m_b^2)δ_μ^ν - k_μ k^ν).
            
            The two-point function of the left flavor current can be obtained from the one-point function identifying the coefficient of the source l_μ. The one-point function is extracted from the boundary value of the regularized canonical momentum
            
                J_q L^μ=-lim_z→ 0L^2/16π G_5N_f/N_c[√(-g) F_L^zμ+(logz/L+log(LΛ))(L D_α(√(-γ) F_L^αμ)-1/L√(-γ) j_(L)^α)],
            
            where γ_μν=g_μν is the induced metric on a radial slice and Λ is an energy scale of the theory. The additional terms are obtained from the variation of the counter-term action
            
                S_c.t=L/16π G_5N_f/N_c(logz/L+log(LΛ))∫ d^4 x√(-γ)[ L^2/4 F_L^μνF_L μν+(D_μ X)^† D^μ X] .
            
            
            Using the expansion (<ref>), the one-point function to linear order in the fluctuation is
            
                (J_q L^μ)^a_ b= L^3/8π G_5N_f/N_c G^μν_ab(l_ν)^a_ b,     G^μν_ab=-η^μα((g_α^ ν)_ab-(c_α^ ν)_ablogΛ L/e^1/2)   .
            
            Taking the variation with respect to the source gives the two-point function
            
                (J_q L^μ)^a_ b (J_q L^ν)^a'_ b'= L^3/8π G_5N_f/N_c G_ab^μνδ^a_ b'δ^a'_b  .
            
            In cases where the holographic dual is a known gauge theory L^3/8π G_5∼ N_c^2, so the overall factor appearing on the correlator is expected to be proportional to N_c N_f. In the following, we will denote the normalization constant simply as
            
                N=L^3/8π G_5N_f/N_c .
            
            The fit of the model to QCD at large Euclidean momentum (<ref>) and (<ref>) fixes
            
                N= N_c/6π^2 .
            
            The spectral function is obtained from the imaginary part of the retarded correlator, which can only receive contributions from the coefficient of the O(z^2) term in the boundary expansion (<ref>)
                ρ^μν_ab,ba=2 Nη^μα Im (g_α^ ν)_ab .
            
            In order to compute the correlators of flavor currents involved in the formulas for the weak reaction rates, we must consider linearized fluctuations of the gauge fields around the background solution. We can restrict to fluctuations that are off-diagonal in flavor, and take into account that the background is diagonal. When the quark masses are zero and the scalar profile is trivial, the energy-momentum tensor of the gauge fields is at least quadratic in the fluctuations. Therefore, metric fluctuations can be turned off. Furthermore, the fluctuations of the left gauge fields decouple from the right gauge field and the scalar, so it is enough to restrict the analysis to the left gauge fields.
            
            
            
            
             §.§ Linearized fluctuations
            
            
            We will work in the radial gauge L_z=R_z=0 and expand the fluctuations in Fourier modes
            
                Y(x,z)=∫dω/2π∫d^3 k/(2π)^3 e^-iω x^0+ik·x Y(ω,k,z) ,  
                Y=L_μ,R_μ,X,X^† .
            
            Our fluctuations correspond to the off-diagonal flavor components Y≡ Y^a_ b, but we will omit flavor indices unless there is some ambiguity.
            
            It will be convenient to rescale coordinates by the horizon position 
            
                z=z_h Z,    x^μ=z_h X^μ ,
            
            and work with the dimensionless frequency and  momentum  
            
                ω̅=z_h ω,  k̅=z_h k ,  (k̅=|k̅|) .
            
            
            We are interested in temperatures much smaller than the chemical potential T/μ_q ≪ 1. Since the integral over frequencies that determines the rates (<ref>) is concentrated on frequencies smaller or of the order of the temperature, we can use a small frequency approximation ω/μ≪ 1. We will use the approach of <cit.> to compute low-frequency correlators. When the temperature is zero, the background solution is an extremal black brane that develops an AdS_2 throat at the horizon. One can then divide the geometry into an inner and outer region relative to this throat. When the temperature is non-zero, but very small compared to the chemical potential, there is still a throat but the inner geometry is a non-extremal AdS_2 with a horizon. From (<ref>), we see that the near-extremal solutions are for Q^2≲ 2. Let us parametrize
            
                Q^2=2/Z_*^6,     Z_*-1=|ω̅| Z_*^2/12 ζ_0,     Z_*-Z=|ω̅| Z_*^2/12 ζ .
            
            The temperature in units of the horizon position is πT̅=1-Z_*^-6. This gives the approximate relation
            
                ζ_0≈|ω̅|/2πT̅=|ω|/2π T .
            
            Expanding the metric (<ref>) for |ω̅|→ 0 and Z_*≈ 1 (since T̅≪ 1), one gets
            
                ds^2≈L^2/12 ζ^2(-F(ζ)dτ̅^2+dζ^2/F(ζ))+L^2dX ^2 ,      F(ζ)=1-ζ^2/ζ_0^2,  τ̅=|ω̅| X^0=|ω| x^0  .
            
            This is AdS_2×ℝ^3 Schwarzschild geometry with an AdS_2 radius L^2__AdS_2=L^2/12 and a horizon at ζ=ζ_0. In the zero temperature limit Z_*→ 1, ζ_0→∞ the throat extends to values ζ→∞.
            
            The equations for linearized fluctuations can be solved in the inner region imposing an ingoing boundary condition at the horizon ζ→ζ_0. In the region towards the end of the AdS_2 throat, the solutions have an expansion
            
                L_μ∼ A_μ^-ζ^Δ_-+A_μ^+ ζ^Δ_+ ,    Δ_+>Δ_-.
             
            This matches with the behaviour of solutions in the outer region close to the horizon Z→ 1
                L_μ = C^-_μη_-(Z)+C^+_μη_+(Z),  η_±(Z)∼ (1-Z)^-Δ_± ,
            
            which allows the identification
            
                C^±_μ=A_μ^±(|ω̅|/12)^Δ_± .
            
            The ingoing condition fixes a relation between A_μ^+ and A_μ^- that translates into a relation between C_μ^+ and C_μ^-:
            
                C_μ^+=𝒢(ω,k) C_ν^- ,
            
            with 𝒢(ω,k) usually interpreted as an infrared (IR) Green's function associated with a 0+1 CFT dual to the AdS_2 gravitational theory. This IR function determines the Green's function of the full theory. In order to see this, we need to translate the above relation between coefficients to the outer normalizable and non-normalizable solutions
            
                L_μ = l_μϕ_NN(Z)+g_μϕ_N(Z),
            
            with expansions close to the boundary Z→ 0
                ϕ_NN(Z)∼ 1+k̅^2/2Z^2log Z+O(Z^4),  ϕ_N(Z)=Z^2+O(Z^4) .
            
            The two sets of outer solutions are related by the connection coefficients
            
                η_±(Z)=α_±ϕ_NN(Z)+β_±ϕ_N(Z) .
            
            Thus, the full solution is
            
                L_μ =C_μ^-( η_-(Z)+𝒢η_+(Z))=C_μ^-(α_- +α_+𝒢)ϕ_NN(Z)+C_μ^-(β_- +β_+𝒢)ϕ_N(Z) .
            
            Fixing the coefficient of the non-normalizable solution to be the source l_μ, we get that the coefficient of the normalizable term is
            
                g_μ=β_- +β_+𝒢/α_- +α_+𝒢 l_μ .
             
            More precisely, the connection coefficients and IR Green's function depend on whether the fluctuation is transverse or longitudinal. The calculation of the solutions and connection coefficients up to O(ω^2/μ_q^2) corrections is detailed in Appendix <ref>. It turns out that at leading order in ω/T the longitudinal components do not contribute.
            
            
            
            
             §.§ Spectral function
            
            
            Since only the transverse components of the left gauge fields are relevant at low frequencies, the spectral function determined by (<ref>) is (ρ^00≈ 0, ρ^0i≈ 0),
            
                ρ^ij_ab,ba≈ Nμ_q^2/2(δ^ij-k^i k^j/k^2)ρ_L(ω/2π T,k̅,T̅) ,  ρ_L(ω/2π T,k̅,T̅)=β_+α_- - β_-α_+(α_-+ α_+Re 𝒢_L)^2 + (α_+)^2(Im 𝒢_L)^2Im 𝒢_L .
            
            The IR Green's function of the transverse modes of the left gauge fields is
            
                𝒢_L(ω̅,k̅,T̅)≈Γ(-ν_k) Γ(1/2-i  ω/2π T+ν_k)/Γ(ν_k) Γ(1/2-i  ω/2π T-ν_k)(πT̅/12)^2ν_k.
            
            Up to corrections suppressed by factors of T/μ_q. At large spatial momentum the IR Greens' function approaches
            
                𝒢_L(ω̅,k̅,T̅)∼ -cos(πk̅/√(12)+iω/2T)/sin(πk̅/√(12))(πT̅/12)^k̅/√(3) .
            
            At low temperatures, there is a strong suppression due to the T̅-dependent factor. When T/μ_q→ 0 (T̅→ 0), there is a strong suppression of the correlator already at small momenta
            
                𝒢_L∼(πT̅/12)e^-k̅^2/12log12/πT̅ .
            
            The support of the Green's function is concentrated on values k̅≲ |log(T/μ_q)|^-1≪ 1. In this case, we can approximate the Green's function by
            
                𝒢_L(ω̅,k̅,T̅)≈(πT̅/12)e^-(2ν_k-1)log12/πT̅(iω/π T+ (2ν_k-1))+O((2ν_k-1)^2,iω̅(2ν_k-1)) .
            
            The connection coefficients are real and only depend on the momentum. We have computed them numerically and determined their asymptotic behaviour at large spatial momentum (c.f. Appendix <ref>)
            
                α_±∼∓ 6^∓k̅/√(12)k̅^-1/2 ,
                   β_±∼∓ 6^∓k̅/√(12) k̅^3/2 logk̅.
            
            Considering the large momentum asymptotic behaviour of the IR Green's function (<ref>) and connection coefficients (<ref>), we see that the spectral function vanishes exponentially fast at large spatial momentum. A special point is k̅=3 (ν_k=1), where 𝒢_L diverges and the spectral function has its first zero. For a numerical estimate, one can thus restrict the momentum integral to smaller values.
            
            When T/μ_q→ 0 we can approximate the connection coefficients by their k̅→ 0 values, that we can compute analytically
            
                α_- ≈ 1+O(ν_k-1/2),   β_-≈ -3/2(2ν_k-1)+O[(ν_k-1/2)^2],
                   α_+≈4/3(2ν_k-1)+O(1),  β_+≈ 4+O(ν_k-1/2).
            
            Thus β_+α_–β_-α_+≈ 6 and
            
                ρ_L(ω/2π T,k̅,T̅)≈6(2ν_k-1)^2(2ν_k-1)^2 + (4 A_k(T̅) ω/3π T)^2  A_k(T̅)ω/π T≈ 6A_k(T̅)ω/π T .
            
            Where we have defined
            
                A_k(T̅)=(πT̅/12)e^-(2ν_k-1)log12/πT̅ .
            
            
            Numerically, at low frequencies the IR Green's function is well approximated by
            
                𝒢_L(ω̅,k̅,T̅)≈   Γ(-ν_k) Γ(1/2+ν_k)/Γ(ν_k) Γ(1/2-ν_k)(πT̅/12)^2ν_k(1-i/2tan(πν_k)ω/T +
                   + π ^2 tan ^2(πν_k )-ψ ^(1)(1/2-ν_k )+ψ ^(1)(1/2+ν_k)8π^2ω^2/T^2)+O(ω^3/T^3),
            
            where ψ^(1) is a polygamma function. In order to see how good is the approximation we plot the real and the imaginary part of the full IR Green function and its approximated expression. We define a reduced IR Green function
            
                g_L(ω̅,k̅) = Γ(ν_k)/Γ(-ν_k) (12/πT̅)^2ν_k𝒢_L(ω̅,k̅,T̅),
            
            so that the plots will be independent of T̅. In Figure <ref> we plot the full imaginary part of g_L (dashed lines) with the approximated formula (solid lines) for some values of ν_k, as a function of ω/T ∈ (0,2). Then, we plot the full real part of g_L (dashed lines) with the approximated formula (solid lines), as a function of ω/T ∈ (0,2), keeping just the first term in the expansion, Figure <ref>, and then adding also the quadratic correction, Figure <ref>. The values of ν_k are 0.6,0.7,0.75,0.8,0.9 which correspond to colors from blue to light purple. We observe that as k becomes smaller, the real part becomes better approximated by the leading term.
            
            
            
            
            
            § ESTIMATE OF WEAK REACTION RATES AT STRONG COUPLING
            
            
            
            We will proceed to estimate the weak rate. Since all the quark chemical potentials are equal at beta equilibrium and the spectral function (<ref>) vanishes at zero frequency, we can use the simplified formula (<ref>) to compute the rate. Taking into account that only the transverse components contribute to leading order, and rescaling the variables in the momentum integral q^0=2 π T s, k=μ_q k̅/2, 
            
                Λ_ud→ us≈ N^2μ_q^7/2^7π^2∫_-∞^∞ ds ∫_0^∞ dk̅ k̅^2 ρ_L^2(s,k̅,T̅)/sinh^2(π s) .
            
            Introducing (<ref>), and changing variables to k̅^2=3(4ν^2-1),
            
                Λ_ud→ us≈ N^26^2μ_q^7/2^7π^22(πT̅/12)^2
                  ∫_0^∞ ds (2s)^2/sinh^2(π s)∫_1/2^∞ dν 12 ν√(3(4ν^2-1))  e^-2(2ν-1)log12/πT̅ .
            
                Λ_ud→ us≈ N^23√(3)μ_q^5 T^2/4∫_0^∞ ds s^2/sinh^2(π s)∫_1/2^∞ dν ν√((4ν^2-1))  e^-2(2ν-1)log12/πT̅ .
            
            We have
            
                ∫_0^∞ ds s^2/sinh^2(π s)=1/6π
            
            Changing variables ν=1/2+u,
            
                Λ_ud→ us≈ N^2√(3)μ_q^5 T^2/8π∫_0^∞ du  (1+2u) √(u(u+1))  e^-4ulog12/πT̅ .
            
            We use that the main contributions to the integral come from u≪ 1:
            
                ∫_0^∞ du  (1+2u) √(u(u+1))  e^-4ulog12/πT̅≈∫_0^∞ du  u^1/2  e^-4ulog12/πT̅=Γ(3/2)/(4log12/πT̅)^3/2=√(π)/2(4log12/πT̅)^3/2 .
            
            Then,
            
                Λ_ud→ us≈ N^2/128√(3/π)μ_q^5 T^2(log6μ_q/π T)^-3/2 .
            
            We have computed numerically the rate for values of the temperature T/μ_q=10^-1,10^-2,10^-3,10^-4, considering the full dependence of the connection coefficients, and several approximations for the IR Green's function, namely the leading order (<ref>), the second order approximation (<ref>), and the full result (<ref>). In the numerical integration over momentum, we cut the integral at ν_k=1, where the spectral function has its first zero. The exponential suppression already present at this value of the spatial momentum makes higher momentum contributions to be exceedingly small, so we believe this is a good approximation. We show our numerical results together with the analytic approximation at low temperatures (<ref>) in Figure <ref>. We see that indeed the analytic formula approximates well the numerical result at low temperatures, even up to T/μ_q ≲ 10^-2, but for T/μ_q∼ 10^-1 the deviations are already sizeable, although the analytic formula still serves as an estimate of the order of magnitude.
            
            
            
            
            
            
            § DISCUSSION AND OUTLOOK
            
            
            
            In this work, we have derived formulas (<ref>) and (<ref>) for the weak equilibration rate of quark flavor densities n_a (a=u,d,s) to leading order in the Fermi coupling and exact in the QCD coupling. Although we have focused on quark matter, the formulas are derived in an explicitly gauge-invariant way and rely solely on symmetry arguments. Thus, they should be valid for any state of QCD at nonzero temperature, including hadronic (confined) phases. In a confined phase, the natural degrees of freedom are baryons rather than quarks. Baryons are composed of three quarks and can be grouped according to their flavor composition. Therefore, the chemical potential for a baryon B^abc∼ q^a q^b q^c would be μ_B^abc = μ_a + μ_b + μ_c, with μ_a,b,c being the quark chemical potentials corresponding to each flavor. The associated baryon densities should satisfy the conditions
            
                n_a = 3 n_B^aaa + 2 ∑_b ≠ a n_B^aab + ∑_b,c ≠ a n_B^abc,
            
            where the order of the flavor labels is immaterial, and n_B^abc would, in principle, include all contributions from baryon resonances with the same flavor composition and with a mass below the value of the baryon chemical potential, assuming the decay channels to lower resonances become closed. With these identifications, one can combine the expressions in (<ref>) to obtain the rates for the evolution of the density of different baryon species, or rather for the evolution of some combinations of them.
            
            Returning to our analysis, we have obtained the leading order low-temperature behavior of the weak reaction rate in a large-density state of strongly coupled quark matter at beta equilibrium. For simplicity, we have only considered massless quarks, from which we were able to extract an analytic expression (<ref>) for the rate coefficient as defined in (<ref>) and (<ref>), resulting in
            
                λ_ds≈ G_F^2 sin^2 θ_C cos^2 θ_C  N^2/32√(3/π)μ_q^5 T^2 ( log6 μ_q/π T)^-3/2.
            
            A fit to the large momentum behavior of QCD correlators fixes the normalization constant to N = N_c / (6 π^2), Eq. (<ref>). We can compare this with the perturbative result <cit.> including the leading QCD correction <cit.>
                λ_ds≈ G_F^2 sin^2 θ_C cos^2 θ_C 64/4 π^2μ_q^5 T^2 (1 + 4 α_s/9 πlogκ α_s μ_q/T)^4,
            
            where κ≈  0.28(2/3π)^1/2≈ 0.13. The logarithmic enhancement of the rate at low temperatures originates from a non-Fermi-liquid behavior of quark matter due to long-range interactions (see also <cit.>). Although the logarithm should not be too large in order for the perturbative expansion to be valid, the fourth power with which it appears can produce a significant effect on the rate. Considering the strong coupling result, the holographic dual geometry has also been conjectured to describe a non-Fermi (“marginal”) liquid <cit.>. However, the logarithmic factor operates in the opposite way, suppressing the rate. Furthermore, compared to the perturbative result, the logarithm will be much larger, as there are no coupling-dependent factors in the argument suppressing the ratio μ_q/T. As shown in Figure <ref>, the analytic formula (<ref>) provides a good approximation for temperatures T/μ_q below 10^-2, but at higher temperatures, corrections to this formula start to become significant. However, their effect is to further suppress the rate.
            
            Several improvements could be applied to the model, with perhaps the most significant being the inclusion of non-zero quark masses. In the perturbative calculation of the rate, the mass enters the dispersion relation of the fermions in the scattering amplitude and removes the low-energy contributions from the massive quarks. In the holographic model, the scalar field dual to the quark bilinears would be non-zero, and its backreaction can change the near-horizon geometry, thus affecting the rate behavior. A generic expectation is that the AdS_2 throat in the near-horizon geometry would be replaced by a hyperscaling Lifshitz geometry (see <cit.> and references therein), which would probably result in the rate becoming proportional to a different power of the temperature depending on the dynamical exponent and the hyperscaling parameters.
            
            Another extension of the model would be to study phases of paired quark matter, as quark pairing can affect transport properties <cit.>. A paired state would have a quark condensate with a non-trivial flavor structure, which could be captured by turning on a profile of the scalar field reproducing the same structure.
            
            In conclusion, our study provides a framework for understanding weak interaction rates in strongly coupled quark matter and highlights the importance of non-perturbative methods in studying dense QCD phases. Future work should focus on refining these models, incorporating more realistic features such as non-zero quark masses and quark pairing, to enhance our understanding of neutron star interiors and their observational signatures.
            
            
            
            
            
            § ACKNOWLEDGMENTS
            
            
            We thank Niko Jokela for useful discussions. This work is partially supported by the AEI and the MCIU through the
            Spanish grant PID2021-123021NB-I00. A.O. thanks the University of Oviedo for the kind hospitality during the preparation of this work. 
            
            
            
            
            
            
            
            
            § NON-CONSERVATION EQUATIONS OF FLAVOR CURRENTS
            
            
            
            With three massless flavors of quarks q^i=(u,d,s) and one flavor of massless leptons l^a=(ν_e,e) (with only left-handed component for the neutrino), the flavor symmetry group is SU(3)_q_L× SU(3)_q_R× SU(2)_l_L× U(1)_e_R. Unequal mass terms for the quarks and the electron break explicitly the flavor group to U(1)^5, where the quarks and electron U(1)s are vector-like.
            
            The EW sector, in particular, W-boson exchange introduces a term in the effective action of the form (Fermi model)
            
                Δ L_EW=-2√(2) G_F (J_ch^μ)^† J_ch μ,
            
            where the J_ch^μ is the left-charged current
            
                J_ch^μ=ν_e Lγ^μ e_L+cosθ_C u_Lγ^μ d_L+sinθ_C u_Lγ^μ s_L,
            
            with θ_C the Cabbibo angle, which gives the change of basis between mass and flavor eigenstates. Let us introduce the notation
            
                (J_q^μ)^i_ j=q̅_j γ^μ q^i,  (J_l^μ)^a_ b=l̅_b γ^μ l^a .
            
            Then, in terms of current components,
            
                J_ch^μ=(J_l L^μ)^2_ 1+cosθ_C (J_q L^μ)^2_ 1+sinθ_C  (J_q L^μ)^3_ 1,
            
            Where J_l L^μ is the current for the SU(2)_L lepton symmetry and J_q L^μ is the current for the SU(3)_L flavor symmetry. The EW term becomes
            
                Δ L_EW=   -2√(2) G_F [ cos^2θ_C (J_q L^μ)^ 2_ 1 (J_q L μ)^1_ 2+sin^2 θ_C  (J_q L^μ)^ 1_ 3 (J_q L μ)^3_ 1.
                   +cosθ_C sinθ_C( (J_q L^μ)^ 2_ 1 (J_q L^μ)^ 1_ 3+(J_q L^μ)^3_ 1(J_q L^μ)^ 1_ 2)
                   +cosθ_C ((J_l L^μ)^2_ 1(J_q L μ)^1_ 2+(J_l L^μ)^1_ 2(J_q L μ)^2_ 1)
                   .+sinθ_C ((J_l L^μ)^2_ 1(J_q L μ)^ 1_ 3+(J_l L^μ)^ 1_ 2(J_q L μ)^3_ 1)] .
            
            Let us neglect the terms coupling quarks and leptons. We can rewrite the remaining terms in the following way
            
                Δ L_EW=   -2√(2) G_F [ cos^2θ_C  tr(P_11 J_q L^μ P_22 J_q L μ) +sin^2 θ_C  tr(P_11 J_q L^μ P_33 J_q L μ).
                   .+cosθ_C sinθ_C( tr(P_11 J_q L^μ P_32 J_q L μ)  + tr(P_11 J_q L^μ P_23 J_q L μ) )] .
            
            Where we have defined the projectors (P_kl)^i_ j=δ^i_kδ_j^l.
            
            By doing SU(3)_L× SU(3)_R rotations of equal angle we can derive Ward identities for the vector SU(3) flavor currents. The currents transform as
            
                δ_θ J_q^μ=iθ_A [T^A,J^μ_q]
            
            
            Defining J_q^μ=J_q L^μ+J_q R^μ and J_l^μ=J_l L^μ+J_l R^μ. The covariant derivative under a background flavor gauge potential A_μ^f is
            
                D_μ J_q^μ=∂_μ J_q^μ-i[A_μ^f,J_q^μ] .
            
            We will take the normalization for the gauge group generators to be tr(T^A T^B)=1/2δ^AB. Then, the Ward identities are
            
                2tr(D_μ J_q^μ T^A)=
                   2i√(2) G_F[ cos^2θ_C  (tr(P_11 [T^A, J_q L^μ] P_22 J_q L μ)+tr(P_11 J_q L^μ P_22 [T^A,J_q L μ]  )).
                    +sin^2 θ_C  ( tr(P_11 [T^A,J_q L^μ] P_33 J_q L μ)+tr(P_11J_q L^μ P_33 [T^A,J_q L μ]  ))
                   +cosθ_C sinθ_C( tr(P_11 [T^A,J_q L^μ] P_23 J_q L μ)+tr(P_11 J_q L^μ P_23 [T^A,J_q L μ]  ). 
                   .. + tr(P_11 [T^A,J_q L^μ] P_32 J_q L μ)+ tr(P_11 J_q L^μ P_32[T^A, J_q L μ]  ) )] .
            
            Combining the traces and using the cyclic property we have that
            
                D_μ J_q^μ =
                   i√(2) G_F( cos^2θ_C  [ J_q L μ , P_11 J_q L^μ P_22 +P_22 J_q L^μ P_11 ].
                    +sin^2 θ_C  [ J_q L μ , P_11 J_q L^μ P_33 +P_33 J_q L^μ P_11 ]
                   .+cosθ_C sinθ_C(  [ J_q L μ , P_11 J_q L^μ P_23 +P_23 J_q L^μ P_11 ]+  [ J_q L μ , P_11 J_q L^μ P_32 +P_32 J_q L^μ P_11 ])) .
            
            In general, there will also be contributions on the right-hand side depending on the quark masses, but they should vanish for diagonal components.
            
            We can do a similar derivation taking into account the leptons if we introduce projectors
            
                (P̃_ai)^b_ j=δ_a^bδ^i_j,  (P̃^T_ia)^j_ b=δ^a_bδ^j_i,
            
            Then
            
                P̃J^μ_qP̃^T∼ J_l^μ,  P̃^T J^μ_lP̃∼ J_q^μ.
            
            And introduce the SU(2)_L× SU(2)_R rotations of the leptons, such that the diagonal rotations transform the vector currents as
            
                δ_ϕ J_l^μ=iϕ_α[τ^α,J^μ_l].
            
            With similar normalization for the generators tr(τ^ατ^β)=1/2δ^αβ.
            
            
            
            
            
            
            § LINEARIZED EQUATIONS OF MOTION OF FLUCTUATIONS
            
            
            
            The transverse spatial components of the gauge fields k^i L_i=k^i R_i=0 decouple from the other fluctuations. Their equations are
             
            0    =   L_i”+ (f'/f-1/Z)L_i'+
            ( ω̅^2/f^2-k̅^2/f)L_i  ,
            
            
             0    =   R_i”+ (f'/f-1/Z)R_i'+
            ( ω̅^2/f^2-k̅^2/f)R_i  , 
            
            When the quark masses are nonzero, the time L_0, R_0 and longitudinal L_∥ =k^i/k L_i, R_∥=k^i/k R_i components of the gauge fields are coupled to the scalar fluctuations, but they decouple otherwise. The left gauge field equations are
            
                0   =   L_∥”+(f'/f-1/Z)L_∥'+ω̅^2/f^2L_∥+k̅ω̅/f^2L_0, 
                
                     0   =   L_0”-1/ZL_0'-k̅^2/fL_0-k̅ω̅/fL_∥ ,
                
                     0   =   ω̅L_0'+k̅ f L_∥' .
            
            The right gauge field equations are
            
                0   =   R_∥”+(f'/f-1/Z)R_∥'+ω̅^2/f^2R_∥+k̅ω̅/f^2R_0,
                
                     0   =   R_0”-1/ZR_0'-k̅^2/fR_0-k̅ω̅/fR_∥,
                
                     0   =   ω̅R_0'+k̅ f R_∥' .
            
            The scalar equations are     
            
                0    =   X”+(f'/f-3/Z)X'+(ω̅^2/f^2-k̅^2/f+3/Z^2 f)X  .
            
            We can simplify the equations in the longitudinal sector by introducing the electric fields
            
                E_L=k̅ L_0+ω̅L_∥,     E_R=k̅ R_0+ω̅R_∥ .
            
            The constraints (<ref>) and (<ref>) fix the time and longitudinal components of the gauge fields in terms of the electric fields and the scalars
            
                L_∥'=ω̅E_L'/ω̅^2-k̅^2 f ,     L_0'=- k̅ fE_L'/ω̅^2-k̅^2 f ,
                   R_∥'=ω̅E_R'/ω̅^2-k̅^2 f ,     R_0'=- k̅ fE_R'/ω̅^2-k̅^2 f ,
            
            The combinations ω̅f/Z(<ref>)+k̅/Z(<ref>), and  ω̅f/Z(<ref>)+k̅/Z(<ref>) produce the equations for the left and right electric fields respectively
            
                0   =   ∂_Z (f/ZE_L'/ω̅^2-k̅^2 f)+1/f ZE_L, 
                
                0   =   ∂_Z (f/ZE_R'/ω̅^2-k̅^2 f)+1/f ZE_R .
            
            
            
            
            § SOLUTIONS IN ADS REISSNER-NORSDTROM
            
            
            
            We want to solve (<ref>) and (<ref>) without the terms depending on the masses, in the charged black brane geometry (<ref>) with parameters (<ref>). We can solve the equations of motion in the near-extremal AdS_2 throat (<ref>) and in the asymptotically AdS_5 metric (<ref>). When |ω̅|→ 0 the two sets of solutions have an overlapping region ζ→ 0 in the throat and Z→ 1 in the outer region. To leading order in the small |ω̅| expansion, the equations of motion of the left gauge fields (<ref>) and (<ref>) in the inner region are
            
                0    =   ∂_ζ^2 L_i +F'F∂_ζ L_i +1/ζ^2 F^2(ζ^2 -k̅^2/12F)L_i ,
                
                 0    =   ∂_ζ^2 E_L + ( F'/F-2/ζk̅^2/121/ζ^2-k̅^2/12F)∂_ζ E_L +1/ζ^2 F^2(ζ^2-k̅^2/12F)E_L .
            
            Outside the AdS_2 throat, we can approximate the geometry by the extremal (zero temperature) Reissner-Norstrom solution, with f(Z)=1-3Z^4+2Z^6. The equations in the outer region to leading order in the small frequency expansion are
            
                0    =   ∂_Z(f(Z)Z∂_Z L_i ) -k̅^2ZL_i+O(ω̅^2) ,
                
                 0    =   ∂_Z(∂_Z E_LZ) -k̅^2Z f(Z)E_L +O(ω̅^2) .
            
            
            
             §.§ Inner solutions of the transverse components
            
            
            Let us first find the solutions to the equations in the inner region (<ref>). At T=0 (F=1), there is a simple solution to the equation of the transverse components
            
                L_i,I(ζ) =√(ζ) ( C_i^(1)Θ(ω) H_ν_k^(1)(ζ)+C_i^(2)Θ(-ω)  H_ν_k^(2)(ζ)) ,
            
            where the coefficients have been chosen so that the solution satisfies ingoing boundary conditions and we have defined
            
                ν_k=√(1/4+k̅^2/12) .
            
            These solutions should match with the solutions of the outer region for ζ→ 0. The leading terms in the expansion are
            
                L_i,I(ζ) ∼  A_i ζ^1/2-ν_k+B_i ζ^1/2+ν_k ,
            
            where
            
                A_i    =      -i2^ν_kΓ(ν_k)/π(C_i^(1)Θ(ω)-C_i^(2)Θ(-ω)) ,
                
                B_i    =   1/2^ν_kΓ(ν_k+1)(C_i^(1)Θ(ω)+C_i^(2)Θ(-ω)+i(πν_k)(C_i^(1)Θ(ω)-C_i^(2)Θ(-ω))  ) .
            
            In terms of the original coordinate Z, related to ζ through (<ref>), the expansion (<ref>) is
            
                L_i,I(ζ) ∼  A_i(|ω̅|/12)^1/2-ν_k[(1-Z)^-1/2+ν_k+𝒢_L(ω̅,k̅) (1-Z)^-1/2-ν_k) ,
            
            where the IR Green's function is 
            
                𝒢_L(ω̅,k̅)=π2^2ν_kΓ(ν_k)Γ(1+ν_k)(-(πν_k)+i sign(ω))(|ω̅|/12)^2ν_k.
            
            At non-zero temperatures, it is still possible to find analytic solutions
            
                L_i,I(ζ) = A_i φ_ν_k^-(ζ)+B_iφ_ν_k^+(ζ) ,
            
            where
            
                φ_ν_k^±(ζ)=(1-ζ^2/ζ_0^2)^-i/2sign(ω)ζ_0(ζ/ζ_0)^1/2±ν_k_2 F_1(1/4-i/2sign(ω)ζ_0±ν_k/2 , 3/4-i/2sign(ω)ζ_0±ν_k/2 ,  1±ν_k ; ζ^2/ζ_0^2) .
            
            The condition that the solution is ingoing fixes
            
                B_i=Γ(-ν_k) Γ(1/2-i  sign(ω)ζ_0+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-i  sign(ω) ζ_0-ν_k)  A_i .
            
            Using that Z_*≈ 1, we arrive at 
            
                L_i,I(ζ) ∼  A_i(|ω̅|/12 ζ_0)^1/2-ν_k[(1-Z)^-1/2+ν_k+𝒢_L(ω̅,k̅,ζ_0) (1-Z)^-1/2-ν_k) ,
            
            where now
            
                𝒢_L(ω̅,k̅,ζ_0)=Γ(-ν_k) Γ(1/2-i  sign(ω)ζ_0+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-i  sign(ω) ζ_0-ν_k)(|ω̅|/12 ζ_0)^2ν_k.
            
            In terms of temperature the IR Green's function is approximately
            
                𝒢_L(ω̅,k̅,T̅)≈Γ(-ν_k) Γ(1/2-i  ω/2π T+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-i  ω/2π T-ν_k)(πT̅/6)^2ν_k.
            
            For frequencies smaller than the temperature, we can expand the IR Green's function as
            
                𝒢_L(ω̅,k̅,T̅)≈Γ(-ν_k) Γ(1/2+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-ν_k)(πT̅/6)^2ν_k(1-i/2tan(πν_k)ω/T)+O(ω^2/T^2).
            
            The imaginary term produces the leading contribution to the spectral function. We can obtain these first terms in the ω/T by first extracting the ingoing factor from the solutions
            
                L_i,I(ζ)=(1-ζ^2/ζ_0^2)^-i/2sign(ω)ζ_0ψ_i(ζ),
            
            and then expanding the equations in powers of ζ_0 and solving order by order. Changing variables to u=ζ^2/ζ_0^2, the equations for the first two orders ψ_i=ψ_i^0+ζ_0ψ_i^1+⋯ are
            
                0   =   ∂_u(√(u)(1-u)∂_u ψ_i^0)+1-4ν_k^2/16 u^3/2ψ_i^0 ,
                
                 0   =   ∂_u(√(u)(1-u)∂_u ψ_i^1)+1-4ν_k^2/16 u^3/2ψ_i^1+i sign(ω)√(u)(∂_uψ_i^0+1/4 uψ_i^0 ) .
            
            The solutions to zeroth order are
            
                ψ_i^0(u)= A_i^0 φ_ν_k^0,-(u)+B_iφ_ν_k^0,+(u),
            
            with
            
                φ_ν_k^0,±(u)= u^1/4(1± 2ν_k)_2 F_1(1/4±ν_k/2 , 3/4±ν_k/2 ,  1±ν_k ; u ) .
            
            Imposing regularity of this solution when u→ 1 (ζ→ζ_0), i.e. removing logarithmic terms, fixes the relation between the coefficients
            
                B_i^0=Γ(-ν_k) Γ(1/2+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-ν_k)  A_i^0 .
            
            The next-order solution can be obtained using a Green's function. First we identify the subleading solution when u→ 0 (ζ→ 0), φ^<, and the ingoing solution when u→ 1, φ^>,
            
                φ^<(u)=φ_ν_k^0,+(u) ,  φ^>(ζ)=φ_ν_k^0,-(u)+Γ(-ν_k) Γ(1/2+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-ν_k)φ_ν_k^0,+(u) .
            
            With these and the Wronskian of the two subleading and ingoing functions we construct a Green's function that maintains the boundary conditions of the leading order solution 
            
                G(u,u')=-1/ν_k{[ φ^<(u)φ^>(u')             ,          u<u'; φ^>(u)φ^<(u')             ,          u>u' ].
            
            Then, taking into account that ψ_i^0=A_i^0φ^>(u), the subleading solution is
            
                ψ_i^1(u)=-i sign(ω)A_i^0∫_0^1 du' G(u,u')√(u')(∂_u'φ^>(u')+1/4 u'φ^>(u') ) .
            
            We can now expand the leading and subleading solutions for u→ 0. One can check that all contributions from ψ_i^1 are subleading compared to the leading term ∼ u^1/4(1-2 ν_k). Also, there is just one contribution to the term ∼ u^1/4(1+2 ν_k), from a total derivative 
            
                ∫_0^1 du' G(u,u')√(u')(∂_u'φ^>(u')+1/4 u'φ^>(u') )∼φ^<(u)lim_u→ 0∫_u^1 du' ∂_u'(√(u')(φ^>(u'))^2/2)=φ^<(u)  (φ^>(1))^2/2 .
            
            Taking this into account, the relevant terms in the expansion are
            
                ψ_i^0    ∼   ( u^1/4(1- 2ν_k)+Γ(-ν_k) Γ(1/2+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-ν_k)  u^1/4(1+2 ν_k))A_i^0  ,
                ψ_i^1    ∼    -i sign(ω)πtan(πν_k) Γ(-ν_k) Γ(1/2+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-ν_k)  u^1/4(1+2 ν_k)  A_i^0 .
            
            It is straightforward to check that this reproduces the first two terms in the ω/T expansion of the  IR Greens' function (<ref>).
            
            
            
            
            
             §.§ Inner solutions of the longitudinal components
            
            
            The equation for the electric field in (<ref>) does not have a simple analytic solution, but since we are interested in frequencies ω/T≲ 1, we can apply the same expansion as for the transverse components to find the leading contribution to the spectral function.
            
            We first extract the ingoing factor from the solutions
            
                E_L,I(ζ)=(1-ζ^2/ζ_0^2)^-i/2sign(ω)ζ_0ψ_i(ζ),
            
            and then expand the equations in powers of ζ_0 and solve them order by order. Changing variables to u=ζ^2/ζ_0^2, the equations for the first two orders ψ_i=ψ_i^0+ζ_0ψ_i^1+⋯ are
            
                0   =   ∂_u(u^3/2∂_u ψ_i^0)+1-4ν_k^2/16 u^1/2(1-u)ψ_i^0 ,
                
                 0   =   ∂_u(u^3/2∂_u ψ_i^1)+1-4ν_k^2/16 u^1/2(1-u)ψ_i^1+i sign(ω)u^3/2/1-u(∂_uψ_i^0+3-u/4 u(1-u)ψ_i^0 ) .
            
            The solutions to zeroth order are
            
                ψ_i^0(u)= A_i^0 φ_ν_k^0,-(u)+B_iφ_ν_k^0,+(u),
            
            with
            
                φ_ν_k^0,±(u)= u^-1/4(1∓ 2 ν_k)_2 F_1(-1/4±ν_k/2 , 1/4±ν_k/2 ,  1±ν_k ; u ) .
            
            Imposing regularity of this solution when u→ 1 (ζ→ζ_0), i.e. removing logarithmic terms, fixes the relation between the coefficients
            
                B_i^0=Γ(-ν_k) Γ(3/2+ν_k)/2^2ν_kΓ(ν_k) Γ(3/2-ν_k)  A_i^0 .
            
            It turns out that this makes the solution vanish at the horizon, something that will be relevant below.
            
            The next-order solution can be obtained in terms of a Green's function. First, we identify the subleading solution when u→ 0 (ζ→ 0), φ^<, and the ingoing solution when u→ 1, φ^>,
            
                φ^<(u)=φ_ν_k^0,+(u) ,  φ^>(ζ)=φ_ν_k^0,-(u)+Γ(-ν_k) Γ(3/2+ν_k)/2^2ν_kΓ(ν_k) Γ(3/2-ν_k)φ_ν_k^0,+(u) .
            
            With these and the Wronskian of the two subleading and ingoing functions we construct a Green's function that maintains the boundary conditions of the leading order solution 
            
                G(u,u')=-1/ν_k{[ φ^<(u)φ^>(u')             ,          u<u'; φ^>(u)φ^<(u')             ,          u>u' ].
            
            Then, taking into account that ψ_i^0=A_i^0φ^>(u), the subleading solution is
            
                ψ_i^1(u)=-i sign(ω)A_i^0∫_0^1 du' G(u,u')u'^3/2/1-u'(∂_u'φ^>(u')+3-u'/4 u'(1-u')φ^>(u') ) .
            
            We can now expand the leading and subleading solutions for u→ 0. One can check that all contributions from ψ_i^1 are subleading compared to the leading term ∼ u^-1/4(1+2 ν_k). Also, there is just one potential contribution to the term ∼ u^-1/4(1-2 ν_k), from a total derivative 
            
                ∫_0^1 du' G(u,u')u'^3/2/1-u'(∂_u'φ^>(u')+3-u'/4 u'(1-u')φ^>(u') )
                       ∼φ^<(u)lim_u→ 0∫_u^1 du' ∂_u'(u'^3/2/1-u'(φ^>(u'))^2/2)=φ^<(u)  lim_u'→ 1(φ^>(u'))^2/2(1-u')=0 .
            
            Taking this into account, the relevant terms in the expansion are
            
                ψ_i^0    ∼   ( u^-1/4(1+2 ν_k)+Γ(-ν_k) Γ(3/2+ν_k)/2^2ν_kΓ(ν_k) Γ(3/2-ν_k)  u^-1/4(1-2 ν_k))A_i^0  ,
                ψ_i^1    ∼    0 ×  A_i^0  u^-1/4(1-2 ν_k) .
            
            Thus, the leading ω/T correction actually vanishes for the longitudinal components. This implies that for the calculation of the spectral function, it is enough to compute the transverse contribution.
            
            
            
            
            
             §.§ Outer solutions of the longitudinal components
            
            
            Although we will not need these solutions to compute the leading contributions to the rates, it will be useful to study them as they can be found analytically.
            
            We are looking for solutions of the electric field in (<ref>). First we will change coordinates u=Z^2, so the equations takes the form of a Schroedinger equation
            
                E_L,O”(u)-k̅^24u(2u+1)(u-1)^2E_L,O=0 .
            
            We can transform the above equation into a hypergeometric equation by the following change of coordinate and field redefinition
            
                E_L,O(u) = (1+2u)^c_±(1-u)^b_±t_±(u),      c_± = 12±ν_k,     b_± = 1-c_± = 12∓ν_k,
                   x=3u1+2u,           u =x3-2x.
            
             Then, we get the following hypergeometric equation
                
                x(1-x)t_±”(x)+(± 2ν_k -1)x t_±'(x) -k^212 t_±(x)=0.
            
                We can compare the above equation to the standard form x(1-x)t_±”(x) + (c_±-(a_±+b_±+1)x)t_±'(x) -a_± b_± t(x) =0, to get
            
                c_± = 0,         a_± = {1/2∓ν_k,-1/2∓ν_k},     b_± = {-1/2∓ν_k,1/2∓ν_k}.
            
            A general solution can be written as a linear combination of a normalizable and a non-normalizable mode at the boundary. In terms of the original radial variable, they have the form
            
                E_O^(0) = α ϕ_NN(Z) + β ϕ_N(Z),
                   ϕ_NN(Z) = (1-Z^2)^1/2+ν_k(1+2Z^2)^1/2-ν_k[((12-2ν_k^2)H_-1/2+ν_k-ν_k^2(-1+log 3)+
                             + 112(1+3log 3))3Z^21 + 2Z^2 _2F_1(12+ν_k,32+ν_k,2,3Z^21 + 2Z^2)+
                             +1Γ(-1/2+ν_k)Γ(1/2+ν_k)G_2,2^2,2(
                [ 3Z^21+2Z^2 ]|[ 1/2-ν_k,32-ν_k;            0,1;                ])],
                   ϕ_N(Z) = (1-Z^2)^1/2+ν_k(1+2Z^2)^1/2-ν_k3Z^21+2Z^2 _2F_1(12+ν_k,32+ν_k,2,3Z^21 + 2Z^2),
            
            where H_-1/2+ν_k is the (-1/2+ν_k)^th harmonic number and G is the Meijer function.
            
            The non-normalizable and normalizable solutions have been chosen so that their near-boundary expansion is
            
                ϕ_NN(Z) = 1 + 32(-1+4ν_k^2)Z^2log Z + O(Z^4log Z),               ϕ_N(Z) = Z^2 + O(Z^4) .
            
            We want to determine the coefficients α and β for solutions with a selected behaviour close to the horizon Z→ 1. The expansion close to the horizon is 
            
                E_L,O(Z) ∼ A(1-Z)^1/2+ν_k + B(1-Z)^1/2-ν_k +...         Z→ 1,
            
            These match with the expansions of the φ_ν_k^0,± inner solutions (<ref>) in the ζ→ 0 limit.
            By requiring that A = 1 and B = 0 we then determine the value of α and β for the minus mode while, if we impose A = 0 and B = 1 we find the coefficients for the plus mode. They turn out to be
            
                α_∓ =6^±ν_k -1/2Γ (±ν_k +1)√(π) Γ(±ν_k +3/2),
                   β_- = -2^ν_k -5/2 3^ν_k -1/2ν_k  Γ (ν_k ) ((6-24 ν_k ^2) H_ν -1/2-12 ν_k ^2 (1+log (3))+1+log (27))√(π) Γ(ν_k +3/2),
                   β_+ = -πν_k  ( (2 πν_k ) ((6-24 ν_k ^2) H_ν_k -1/2-12 ν_k ^2 (log (3)-1)+1+log (27))+3 π(4 ν_k ^2-1)  ^2(πν_k ))2^-ν_k +3/2 3^ν_k +1/2Γ(1/2-ν_k ) Γ(3/2-ν_k ) Γ (2 ν_k +1) .
            
            
            Since in order to compute the rates we have to perform an integral over all momentum, it is interesting to ascertain the behaviour of the spectral function at large spatial momentum, which will be in part determined by the behaviour of the connection coefficients. From the analytic formulas above one can see that, for ν_k→∞, their asymptotic form is
            
                α_- ∼ 6^ν_k-1/2(πν_k)^-1/2,  α_+ ∼ -6^-ν_k-1/2(πν_k)^-1/2(πν_k) ,
                   β_-∼ 6^ν_k+1/2ν_k^3/2log(3^1/2e^-1/2e^γ_Eν_k),
                   β_+ ∼  6^-ν_k+1/2ν_k^3/2√(π)(1-(πν_k)/πlog(3^1/2e^-1/2e^γ_Eν_k)) .
            
            We can recover this behaviour of the connection coefficients by using a WKB approximation at large spatial momentum. The equation of motion (<ref>) is already in Schroedinger form with a potential
            
                U_E(u)=k̅^24u(2u+1)(u-1)^2 .
            
            We expand the solutions as
            
                E_L,O(u) = Cexp(k̅ S_0(u) + S_1(u) + O(1/k̅)).
            
            The equation for the leading term in the 1/k̅ expansion is
            
                S_0'(u)^2-18 u^4-12 u^3+4 u = 0,
            
            which has an analytical solution
            
                S_0(u) = ±1√(3)tanh ^-1( √(3u2u+1)) + c_0.
            
            Then, the subleading term is given by
            
                S_1(u) = log(√(2) √(1- u) √(u (1+2u))) + c_1.
            
            Then the solution up to this order reads
            
                E_L,O(u)≈ψ(u)= C_± √(1- u) √(u (1+2u))exp(±k̅√(3)tanh ^-1( √(3u1+2u))).
            
            The WKB solution near the boundary has the following expansion
            
                ψ(u)∼ C_± u^1/4e^± k√(u)          u→ 0.
            
            Note that the two terms in the solution directly correspond to the "plus" and "minus" modes at the horizon
            
                ψ(u) ∼C_±(11-u)^±k̅/√(12)         u→ 1.
            
            We have to fix the overall coefficient to properly normalize the WKB solutions in the original coordinate z which are
            
                E(z) = C_± √(1- z^2) √(z) (1+2z^2)^1/4exp(±k̅√(3)tanh ^-1( √(3z^21+2z^2))).
            
            Then, the overall coefficients are found to be
            
                C_± = 6^∓k̅/√(12).
            
            Now we have to match these results with the solutions close to the boundary and read the boundary coefficients. Close to the boundary, the potential is to leading order
            
                U_E(u) ∼k̅^24u.
             
            A solution for this approximated potential can be given in terms of modified Bessel functions
            
                ψ(u)≈ c_1 √(u) K_1(k̅ √(u))+ c_2√(u) I_1(k̅ √(u)).
            
            The expansion for large k̅ is
            
                √(u) K_1(k̅ √(u))∼√(π2k̅)u^1/4e^-k̅√(u),                √(u) I_1(k̅ √(u))∼u^1/4√(2πk̅)e^k̅√(u),
            
            which is consistent with the WKB solution. We fix the coefficient c_1,2 with the formulas for C_± so that the WKB solution for z→ 0 and the solution close to the boundary for large k̅ coincide. Finally, we can read off from the z→ 0 expansion the coefficients α_± and β_± which are respectively the coefficients of the non-normalizable and normalizable mode. In a compact form, they are
            
                α_±∼∓ 6^∓k̅/√(12)k̅^-1/2 ,
                   β_±∼∓ 6^∓k̅/√(12) k̅^3/2 logk̅.
            
            Comparing to (<ref>), and taking into account ν_k∼k̅/√(12) in this limit, the WKB approximation captures the leading k̅ dependence up to the oscillatory factors.
            
            
            
            
             §.§ Outer solutions of the transverse components
            
            
            We are looking for solutions of the transverse components in (<ref>). We do not have a close analytic solution in this case, so we must resort to numerics. The general solution is a linear combination of two modes η_L,∓(Z) which when Z→ 1 match with the inner solutions (<ref>) for ζ→ 0,
            
                η_L,±(Z) ∼(1-Z)^-1/2∓ν_k + ...          Z→ 1.
            
            There is another basis of solutions we can use to write a general solution, consisting of the normalizable and non-normalizable modes, with a boundary expansion at Z→ 0
                ϕ_NN(Z) = 1 + 32(-1+4ν_k^2)Z^2log Z + O(Z^4log Z),               ϕ_N(Z) = Z^2 + O(Z^4)
            
            The change of basis is determined by the connection coefficients α_±,β_±:
            
                η_L,±(Z) = α_±ϕ_NN(Z) +β_±ϕ_N(Z).
            
            The connection coefficients can be determined from the solutions by matching the η_L ± solutions and their first derivative with a linear combination of the normalizable and non-normalizable solutions at any intermediate point 0<Ẑ<1:
            
                [ α_±; β_± ]
                =
                1W(Z)[ -∂_Zϕ_N(Z)     ϕ_N(Z); ∂_Zϕ_NN(Z)   -ϕ_NN(Z) ][    η_L,±(Z); ∂_Zη_L,±(Z) ]|_Z = Ẑ ,
            
            where W(Z) = ϕ_N(Z)∂_Zϕ_NN(Z)-∂_Zϕ_N(Z)ϕ_NN(Z).  We will obtain the connection coefficients using this formula, by finding numerically the η_L ± solutions using shooting from the horizon, and the ϕ_N,ϕ_NN solutions using shooting from the boundary. We use Ẑ=0.2 as the matching point, but we have checked that changing this value through the interval of numerical integration does not affect the result within our numerical precision.
            
            For the shooting from the boundary, we introduce a cutoff at Z=10^-5 and use a series expansion to give initial values for the solution and its first derivative at the cutoff. The expansion is 
            
                ϕ_N(Z) = ∑_n = 1^N_a a_n Z^2n,
                   ϕ_NN(Z) = b_0 + ∑_n = 2^N_b b_n Z^2n +log(Z)∑_n = 1^N_c c_n Z^2n.
            
            Where we take N_a=8, N_b=6 and N_c=7. The first coefficients of the expansion given that a_1 = b_0 = 1 are
            
                a_2 = k̅^28,    a_3 = 1 + k̅^4192,    a_4 = -12 + 5k̅^224 + k̅^69216, ...
                   b_2 = -3k̅^464,    b_3 = k̅^26 - 7k̅^62304  ...      c_1 =k̅^22,    c_2 = k̅^416, …
            
            
            For the shooting from the horizon we introduce a cutoff at Z=1-10^-5. We extract the leading power in the horizon expansion from the solutions
            
                η_L,∓(Z) = (1-Z)^-1/2±ν_kp_∓(Z) .
            
            The equation of motion we solve is
            
                ∂_Z^2 p_∓(Z) +(1+2Z(-1 ±ν_k)Z(-1+Z) + f'(Z)f(Z))∂_Z p_∓(Z)+
                   +(±ν_k-1)[(2-5Z ± 2Zν_k)f(Z) + 2(1-Z)Z(6(-1+Z)(1± 2ν_k) -f'(Z))]4 (Z-1)^2 Z f(Z) p_∓(Z).
            
            To give initial values at the cutoff we use the series expansion 
            
                p_∓ = ∑_n=0^N_j j_∓,n(1-Z)^n,
            
            and shooting it from the horizon towards the boundary. We expand up to N_j=8 terms, with the first coefficients in the expansion, given that j_∓,0=1, are
            
                j_∓,1 = 8√(3)√(3+k̅^2)∓ 24 ± 7k̅^212(√(3)√(3+k̅^2)± 3),    j_∓,2 = k̅^2 (49 √(3)√(k̅^2+3)±534)-156 (√(3)√(k̅^2+3)∓3)864 (√(3)√(k̅^2+3)±6), …
            
            
            Although we do not have an analytic formula for the connection coefficients, we can obtain analytic expressions for their asymptotic behaviour at large spatial momentum, by applying the same WKB approximation we used for the coefficients of the longitudinal components. First, we should put the equation of motion for L_i in (<ref>) in Schroedinger form. This is achieved by the change of variables u=Z^2 followed by the redefinition
            
                L_i(u)= (1-u) (1+2 u)^-1/2ψ_L_i(u) .
            
            The equation of motion is now
            
                ψ_L_i”(u)-U_L_i(u)ψ_L_i(u) = 0 ,      U_L_i(u) = k^2/4 (1-u)^2 u (2 u+1)-3 (u+1)/(1-u) (2 u+1)^2 .
            
            At large k̅ the potential is to leading order the same as for the longitudinal modes (<ref>)
                U_L_i(u)∼k^2/4 (1-u)^2 u (2 u+1)=U_E(u) .
            
            Therefore, the large k̅ WKB analysis proceeds in the same way as for the longitudinal components, and the connection coefficients have the save asymptotic behaviour given in (<ref>).
            
            
            We have plotted the connection coefficients in Figures <ref>- <ref>. In Figures <ref> and <ref> we plot α_+ and α_-. The coefficient of the non-normalizable mode, α_+, is multiplied by the factor 2(1-ν_k)(2ν_k-1) in order to remove the divergences in k̅ =0, 3. We will examine the origin of this divergence in more detail below. 
            In Figures <ref> and <ref> are shown the plots of 2(1-ν_k)β_+ and (2ν_k-1)^-1β_-. 
            
            
            
            
              §.§.§ Low momentum behavior of connection coefficients
            
            
            When k̅=0 (ν_k=1/2) the solutions in the outer region are
            
                η_L,-^0(Z)=1,    η_L,+^0(Z)=2/1-Z^2+4/3log[1+2Z^2/1-Z^2]-1/2-4/3log3/2 .
            
            The expansions at the boundary and the horizon of the η_L,+^0 solution are
            
                η_L,+^0(Z)    ∼   3/2-4/3log3/2+6Z^2+⋯,
                η_L,+^0(Z)    ∼    (1-Z)^-1-4/3log(1-Z)+⋯ .
            
            Therefore, the connection coefficients of these solutions are
            
                α_-^0=1, β_-^0=0,  α_+^0=3/2-4/3log3/2, β_+^0=6.
            
            We can relate these to the k̅→ 0 limit of the connection coefficients of the non-zero momentum solutions. First, note that the coefficient j_-,n in (<ref>) go to zero in the k̅→ 0 limit, so η_L,-→η_L,-^0. On the other hand j_+,1 in (<ref>) is singular in this limit, but the singularity does not propagate to other coefficients in the expansion, one can already see that it is absent in j_+,2. In order to remove the singular term we can take the following combination of the solutions
            
                η̃_L,+(Z)=η_L,+(Z)-j_1,+η_L,-(Z) .
            
            Expanding this solution at the horizon and for k̅→ 0 one can check that it coincides with the expansion in (<ref>). We then conclude that  η̃_L,→η_L,+^0 in the k̅→ 0 limit. This determines the k̅→ 0 behaviour of the connection coefficients
            
                α_-∼α_-^0,     β_-∼β_-^0,    α_+∼α_+^0+j_+,1α_-^0,     β_+∼β_+^0+j_+,1β_-^0 .
            
            Since j_+,1→∞ and β_-→ 0, in order to completely determine β_+ we need to find the non-zero subleading contribution to β_-. We expand
            
                η_L,-(Z)= (1-Z)^-1/2(1-2ν_k)(1+(ν_k-1/2)η_-^1(Z)+O[(ν_k-1/2)^2]) .
            
            The solution for η_-^1(Z) can be found explicitly
            
                η_-^1(Z)=   1/9(log (1-Z) (9-12 log (Z))+9 log (1+Z)+π ^2+9-9 log (2)-6Li_2(-2).
                   .+6 log (Z) (2 log(1+2
                   Z^2/1+Z)+3 Z^2/1-Z^2) + 6Li_2(-Z^2)-6Li_2(Z^2)) .
            
            The boundary and horizon expansions are
            
                η_L,-^1(Z)    ∼   1/9(9 + π^2 - 9log(2)- 6Li_2(-2)) + 3 Z^2 (-1 +2 log Z)+⋯,
                η_L,-^1(Z)    ∼   log(1-Z)+ 11/6(1-Z)+⋯ .
            
            Therefore
            
                β_-∼ -3/2(2ν_k-1) .
            
            Using now that to leading order
            
                j_+,1∼4/3(2ν_k-1),
            
            We find that when k̅→ 0
                α_-∼ 1+O(ν_k-12),    β_-∼ -3/2(2ν_k-1)+O[(ν_k-1/2)^2] ,
                   α_+∼43(2ν_k-1)+O(1),    β_+∼ 4+O(ν_k-1/2) .
            utphys ] ]


 Paper Title: Shocking and Mass Loss of Compact Donor Stars in Type Ia Supernovae.








0000-0001-9195-7390]Tin Long Sunny Wong
Department of Physics, University of California, Santa Barbara, CA 93106, USA


Center for Computational Astrophysics, Flatiron Institute, Simons Foundation, New York, NY 10010, USA
Department of Astrophysical Sciences, Princeton University, Princeton, NJ, USA

0000-0001-8038-6836]Lars Bildsten
Department of Physics, University of California, Santa Barbara, CA 93106, USA
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA


Tin Long Sunny Wong
tinlongsunny@ucsb.edu


§ ABSTRACT 
Type Ia supernovae arise from thermonuclear explosions of white dwarfs accreting from a binary companion. 
Following the explosion, the surviving donor star leaves at roughly its orbital velocity. 
The discovery of the runaway helium subdwarf star US 708, and seven hypervelocity stars from Gaia data, all with spatial velocities ≳ 900 km/s, strongly support a scenario in which the donor is a low-mass helium star, or a white dwarf. 
Motivated by these discoveries, we perform three-dimensional hydrodynamical simulations with the  code modeling the hydrodynamical interaction between a helium star or helium white dwarf, and the supernova ejecta. 
We find that ≈ 0.01-0.02 M_⊙ of donor material is stripped, and explain the location of the stripped material within the expanding supernova ejecta. 
We continue the post-explosion evolution of the shocked donor stars with the  code. As a result of entropy deposition, they remain luminous and expanded for ≈ 10^5-10^6 yrs. We show that the post-explosion properties of our helium white dwarf donor agree reasonably with one of the best-studied hypervelocity stars, D6-2. 








§ INTRODUCTION


Although type Ia supernovae (SNe Ia) are important cosmological distance indicators <cit.>, the identity of their progenitor systems remains debated today. There is general agreement that SNe Ia result from the thermonuclear explosion of a carbon-oxygen white dwarf <cit.> induced by accretion from a companion donor star, but the stellar type of the donor remains elusive <cit.>. 

In the Chandrasekhar-mass scenario, the accretor grows up to Chandrasekhar mass and explodes, through accretion from a non-degenerate companion <cit.>. 
However, the non-detection of a progenitor system in pre-explosion images of nearby SNe <cit.>, the lack of uncontested surviving companions in galactic and Large Magellanic Cloud SN Ia remnants <cit.>, etc., have dimmed the prospects of this scenario. 


In the double-detonation scenario, the accretor WD accumulates 0.01 - 0.1 of helium (He) from a He-rich companion. 
The accumulated He shell detonates and sends a shock wave into the the C/O core leading to its subsequent detonation <cit.>. 
This scenario requires the donor to be a Roche-lobe-filling He star <cit.>, He WD <cit.>, or C/O white dwarf with a surface He layer <cit.>. 
The orbital period at explosion is ≲ 15 minutes, and the donor orbital velocity is v_ orb≳ 600 (≳ 1,000  for a white dwarf donor). 
When the SN explosion unbinds the binary, the surviving donor continues at its velocity v ≈ v_orb≳ 600. 

Discovery of the runaway He-burning subdwarf star US 708 <cit.>
and seven hypervelocity WDs <cit.> provides smoking gun evidence that the double-detonation scenario can account for some SNe Ia. 
The hypervelocity WDs appear to have larger radii (≈ 0.02-0.2) compared to typical white dwarfs, likely due to pre-explosion tidal-heating, or shock-heating by the SN ejecta <cit.>. Some have metal-polluted atmospheres, likely due to contamination by the SN ejecta <cit.>. 

Previous numerical studies of SN ejecta-companion interaction have largely focused on non-degenerate companions relevant to the Chandrasekhar-mass scenario, including a main sequence star, subgiant, red giant, and He star <cit.>. In the context of the double-detonation scenario, only <cit.> & <cit.> studied the response of a He star to interaction with SN ejecta and its long-term evolution, and recently <cit.> studied the long-term evolution of CO WD donors. On the other hand, <cit.>, <cit.>, <cit.>, <cit.>, <cit.> & <cit.> found that in some cases, the donor also detonates and leaves behind no surviving remnant. 


We continue the work of <cit.> and model the interaction between SN ejecta and He WD and He star donors, using the  hydrodynamic code <cit.>. 
We improve upon their work by using a new passive scalar capability to differentiate between donor and SN ejecta materials, and adopting a more realistic SN ejecta profile based on the sub- explosion simulations by <cit.>.  In addition to the two He star models first introduced in <cit.>, we also present a low-mass, high-entropy (semi-degenerate) He WD model from <cit.>. 
They modeled the stable mass transfer from a high-entropy He WD donor onto a C/O WD accretor, up to the start of a dynamical He flash on the accretor that yields a He detonation. Assuming the donor leaves with a spatial velocity ≈ v_ orb, they found good agreement with the spatial velocity of the hypervelocity WD D6-2 <cit.>. 


Our paper is organized as follows. 
In Section <ref>, we describe our  setup. In Section <ref>, we describe our SN ejecta profile based on simulations by <cit.>. We detail the outcomes for a He WD donor with a total ejecta kinetic energy = 1.2 × 10^51 erg and ejecta mass = 1.0 in Section <ref>, and discuss the observational implications for donor mass loss mixed with SN ejecta. In Section <ref> we compare the results of varying  between 0.5 and 1.5× 10^51 erg for the same He WD donor. We then describe the results with He star donors in Section <ref>. In Section <ref>, we map the post-explosion donor from  to the 1D stellar evolution code  <cit.> and study its subsequent evolution. We find a reasonable match to the properties of the hypervelocity WD D6-2 given an inferred age of ≈ 10^5 yrs. We conclude in Section <ref>. 
Our  input and output files, simulation movies, and our  inlists and outputs are uploaded to Zenodo (<https://doi.org/10.5281/zenodo.12850558>). 






§ NUMERICS



We configure  to solve Γ=5/3 ideal Eulerian hydrodynamic equations in Cartesian coordinates, using the second-order van Leer time integrator <cit.>, piecewise parabolic method for spatial reconstruction <cit.>, and HLLC Riemann solver, as was done in <cit.>. In contrast with that work's use of Fourier methods to solve Poisson's equation, here we utilize the new multigrid capabilities of  for self-gravity <cit.>. The gravitational potential is found via the full multigrid method, using sufficient iterations to reach convergence. We use isolated boundary conditions – the potential on the grid boundary agrees with the distribution of mass, assuming vacuum outside the domain, up to hexadecapole order about the center of mass. In the fluid sector, we impose diode boundary conditions (zero gradient in density, pressure, and velocity, with velocity outside the domain floored to never point inward); these are overridden with the modeled ejecta where it enters the grid. 

We consider three donor models from the stellar evolution code  <cit.> – a He WD from <cit.>, and two He star models from <cit.>. The He WD has an initial mass of 0.21 and a high central specific entropy of s/() = 4.0, where  is Avogadro's number and  is the Boltzmann constant. It is evolved with an initially 1 CO WD in a binary until a dynamical He flash occurs on the CO WD accretor. At this moment, the He WD mass is =0.126. This model is chosen since it is nearly nondegenerate and its pre-explosion orbital velocity agrees with the velocity of the runaway object D6-2 <cit.>. We refer to <cit.> for details of the He star models. However, we stress that all donor models are only semi-degenerate, which makes our assumption of Γ = 5/3 somewhat reasonable. We further explore the impact of the equation of state in Appendices <ref> & <ref>. 


Following <cit.>, for each of the three donors we adopt their initial central density  and pressure  as the respective simulation units. We also choose t_0≡√( 5 / ( 8 π G  ) ) as the time unit. The velocity and length units follow as √(/) and x_0≡√(/)× t_0. These are provided in Table <ref>, where we also provide additional details of each donor model, including the mass , radius R_ He^i and sound-crossing time t_ sound^i≡∫ ( dr / c_ s ) at the start of the  simulation, the binary separation a, orbital period  and donor orbital velocity v_ orb. 

For all simulations, we set the accretor mass  to be 1 and the binary separation such that the donor is Roche-lobe-filling <cit.>. All these donor models come from  binary simulations where the accretor (albeit with a different mass) is evolved through stable mass transfer up to the start of a dynamical He flash, and so a barely Roche-lobe-filling donor is a reasonable assumption. However, we note that if the double detonation occurs due to unstable mass transfer, the binary separation may be even smaller. For an ejecta mass of 1.0 and total kinetic energy =1.2×10^51 erg which defines a peak ram pressure velocity v_0 (see Section <ref>), we define the interaction time t_ interaction≡ a / v_0. We also provide the dimensions of the simulation box in units of x_0. 
The whole simulation lasts ≈ 150-200 t_0, until the donor becomes approximately spherical again. 
We set a density and pressure floor of 3 × 10^-7 and 3 × 10^-10 in units of  and . 


cccccccccccBcccc
1


Donor models and description of   setup

Donor model     
     
R_ He^i     
    
a     
     
v_ orb    
^i    
^i    
x_0    
t_0    

t^i_ sound    
t_ interaction    
Box size (x, y & z) 


    
[]    
[]    
[]    
[]    
[min]    
[]    
[]    
[dyne cm^-2]    
[]    
[s]    

[s]    
[s]    
[x_0]

100pt
HeWD     0.126     
0.0410     1.0    
0.186     
12.6     
954    
2.02×10^4    
6.47×10^19    
0.00988    
12.15    
81    
14    
(-16,64) × (-50,50) ×(-50,50)


HeStar1     0.348     
0.0805     1.0    
0.275     20.8    
717     3.74×10^4     2.83×10^20     
0.0112     8.93     
161    
21    
(-20,60) × (-40,40) ×(-40,40)


HeStar2     0.236     
0.0414     1.0    
0.157     
9.34     991     
6.71×10^4     4.55×10^20    
0.00788     6.66     
61    
12    
(-15,65) × (-40,40) ×(-40,40)


Here  and R^i_ He are the initial donor mass and radius,  is the accretor (SN ejecta) mass, a and  are the binary separation and orbital period, v_ orb is the orbital velocity of the donor, ρ^i_ c and P^i_ c are the initial donor central density and pressure, x_0 and t_0 are the length and time units of the simulation, t^i_ sound is the initial donor sound-crossing time, and t_ interaction≡ a/v_0 where v_0 is given by equation 5. 





 §.§ Passive scalars for ejecta and donor



One novelty of our work compared to <cit.> is that we make use of the new passive scalar capability of  <cit.>. The passive scalar is transported with the fluid without modifying fluid properties. We add two passive scalars, one for the donor and one for the SN ejecta. This allows the accurate tracking of the amount of mass lost from the donor and its velocity distribution. In Section <ref>, we rerun our fiducial simulation but with 4 passive scalars to the donor, so that we can track the original location of mass loss in the donor. This clarifies the interaction between the SN ejecta and the donor. 



 §.§ Donor relaxation to Roche potential



Prior to the injection of the SN ejecta, we interpolate the density and pressure profiles of the donor from  onto the  grid. We then relax the donor in the corotating frame, accounting for its distortion due to the combined effects of the gravity from the accretor, = - ∇Φ_acc, and the centrifugal force, = - ∇Φ_cent, but without the Coriolis force because it does not contribute to the distortion. The donor potential Φ_ d is accounted for in the  self-gravity module. The other potentials are given by 

    Φ_acc = -  G / r_a ,

where r_a = √( (x - a)^2 + y^2 + z^2) is the distance from the accretor, and 

    Φ_cent = - 1/2Ω^2√( ( x - x_COM )^2 + y^2) ,

where Ω = √( G (  +  ) / a^3) and x_COM = a [  / ( + ) ]. 
We add source terms to the momentum and energy equations as ρ_d and ρ_d· respectively, where =  + and ρ_d is the density of the donor material (enabled by the use of passive scalars) so that acceleration of the background “fluff” does not limit the simulation timestep. 

We apply velocity damping while relaxing the donor to  its Roche configuration. We find this necessary so that (1) large velocities do not severely limit the timestep, (2) the donor does not overshoot its equilibrium structure during relaxation under the Roche potential, and (3) the donor center-of-mass does not change due to imperfect numerical cancellation of the acceleration terms. During relaxation, for each time t with timestep Δ t, the momentum is reduced by a factor f_damp and the kinetic energy f^2_damp, where 

    f_damp   = 
        exp( -  dt_eff/τ_damp)  , 
        
    
        dt_eff   = 
        Δ t exp( -  t /  t_damp - t  ) ,

and t_damp and τ_damp are taken to be 20 and 1 in units of t_0. 

We confirm that after the relaxation (at 20 t_0), the isobars of the donor agree well with the equipotentials of the Roche potential Φ_acc + Φ_d + Φ_cent
, where Φ_d is the gravitational potential of the donor calculated by   since a point-mass approximation is unrealistic. This is shown in Figure <ref>. 

For the HeStar1 model, we find no difference whether we relax the donor with a Roche potential. However, for the HeWD model, the post-explosion central density drops to 10% of its initial value if we adopt a spherical donor (as opposed to 20%). We speculate that this difference stems from the detailed interaction between the SN ejecta and the upstream side of the donor especially near the inner L1 Lagrange point, and its effect increases with more extreme mass ratios due to stronger distortion of the donor. 






 §.§ Transitioning the gravitational model as supernova ejecta is introduced


After relaxation to the Roche configuration (after 20 t_0), we inject SN ejecta into the grid. At the same time, assuming that the binary is unbound immediately, we switch to the initial donor rest frame. We shut off source terms corresponding to the accretor gravity and the centrifugal force in the corotating frame. 
The assumption that the binary is unbound immediately is valid because the ejecta mass enclosed within the binary rapidly drops within 2 t_0, and this time is ≪ t^i_ sound. 

While the unshocked parts of the donor would re-adjust given the absence of the external source terms, we find that the re-adjustment timescale is longer than the shock travel time across the donor. In other words, throughout the shocking, the donor has no time to deviate greatly from its equilibrium shape under the Roche potential. 




 §.§ Bound material


We define bound material as having a Bernoulli parameter < 0. It is defined as

    = 1/2(  - _COM,bound)^2 + Γ/Γ - 1 P /ρ + Φ,

where _COM,bound is the center-of-mass of the bound material. While ideally one would iteratively find _COM,bound so that  is defined self-consistently <cit.>, we find that using the _COM,bound from the previous timestep is sufficient to achieve fractional errors < 10^-5 in integrated quantities such as the total bound mass, M_bound. 




 §.§ Stopping condition


Due to the kick from the SN ejecta, the donor moves along the +x-axis (to the right). 
In order to allow more time for the donor material to settle down, we keep the donor inside the simulation box by removing its center-of-mass momentum once it reaches the center of the box. This typically happens around ≈ 80 t_0 after explosion. We end the simulation once oscillations from the donor are nearly damped out (see the beginning of Section <ref>). 



§ SN EJECTA MODELING



We adopt an SN ejecta profile motivated by the sub- bare CO core explosion models of <cit.>, which have total masses of 0.9,1.0 and 1.1, and C/O ratios of 50/50 and 70/30. 
Prior to encountering the donor, we assume homologous and adiabatic expansion. The adiabatic assumption is because the timescale prior to the ejecta-donor interaction is so short (≲ 100 s) that radioactive heating is of no consequence. 
Following <cit.>, we exclude ejecta with initial v> 20,000 . We also account for the relative orbital motion by shifting the ejecta y-velocity. 






While <cit.> adopted a broken power-law ejecta profile <cit.>, and many other works have adopted an exponential ejecta profile <cit.>, here we introduce a Gaussian ejecta profile, defined by a total ejecta mass , total kinetic energy , and time t since explosion, 

    ρ( v, t)
        = 
        (  3 / 4 π)^3/2^5/2/^3/2exp[ - (v / v_0)^2] / t^3 ,

where the ram pressure ρ v^2 peaks at a value

    v_0 
        = 
        (  4 / 3 /)^1/2 .

The Gaussian ejecta profile shows better agreement across all total ejecta masses with the sub- CO core explosion models of <cit.>, than the broken power-law and exponential forms. 
A comparison among the 1.0, C/O=50/50 model from <cit.>, our Gaussian ejecta profile and the exponential ejecta profile <cit.> is shown in Figure <ref>. 
Our Gaussian ejecta profile also yields a constant density core shown by the models <cit.>, and agrees well in the shape and location of the peak of the ram pressure ρ v^2. For v < 20,000 , the agreement in ρ and hence ρ v^2 is within 20%, which is further reduced to 10% near the peak of the ram pressure. We also show the Chandrasekhar-mass delayed-detonation model N100 from <cit.>, scaled by a factor (100/30)^3. Despite the different explosion physics, our Gaussian ejecta profile also shows better agreement than the exponential profile. As our prime goal in this work is in assessing the impact on the donor from the ejecta, we place a premium on the agreement of the ρ v^2 profile. 



As noted earlier, at the time of injection into the  grid, the fluid has expanded adiabatically and the flow is approaching homology. Hence we are not so concerned about the explicit choice of the ejecta internal energy. 
<cit.> set the internal energy density of the ejecta to be ( 0.08 a / r_a) ρ v^2, where r_a is the distance from the accretor/explosion center, such that after undergoing homologous expansion, the internal energy density was about 7-9% of the kinetic energy density when the ejecta reaches the donor. 
Instead, we take advantage of the fact that fluid elements in an adiabatic flow with Γ=5/3 maintain a constant P/ρ^5/3 until they are shocked by the donor collision. 
Rather than fitting a profile of P/ρ^5/3 from the explosion models of <cit.>, we adopt a fixed value of ⟨ P/ρ^5/3⟩ = 0.59 × 10^14 in cgs units, which is the mass-averaged value of the =1.0, C/O=30/70 model (averaged over v < 10,000 ). The pressure is then given by P(v,t) = ⟨ P/ρ^5/3⟩( ρ(v,t) )^5/3, where ρ(v,t) is given by Eqn <ref>. Application to other models shows that the internal energy density agreement is within 50% for v ≲ 10,000 . However, in all cases the internal energy density remains ≲ 10% of the kinetic energy density. 
Hence, none of these choices have dramatic impact on the ejecta-donor interaction. 
In this paper, we choose explosion kinetic energies E_ KE of 1.5, 1.2, 0.8 & 0.5× 10^51 erg. 





§ OUTCOMES FOR A HELIUM WD DONOR WITH E_ KE=1.2×10^51 ERG







We show three different snapshots of the HeWD, =1.2×10^51 erg model in Figure <ref>. 
As others have shown, the bow shock extends to an angle that is roughly 
 twice that subtended by the star, θ_ wake≈arctan( 2  / a ) = 25^∘ <cit.>, implying that ≈10% of the ejecta is modified. 
The ejecta flows around the donor, entraining donor material mostly from the upstream side (facing the explosion center), and leaving behind a low-density conical hole in the wake behind the donor. 
Meanwhile, the donor is compressed by the ejecta and a shock propagates through it, visible in the top panel of Figure <ref>. 

The shock eventually breaks out from the downstream side of the donor (facing away from the explosion center). Unbound, expanding donor material from the shock breakout quickly fills up the conical hole behind the donor. The shocked donor then expands and moves to the right due to momentum transfer from the ejecta. This is visible in the middle panels of Figure <ref>.

As the donor adjusts to a new hydrostatic equilibrium, it contracts and expands periodically with decreasing amplitude. With each contraction, a shock wave is sent radially outwards. We find that the passing of the first shock wave greatly increases the entropy of the outermost donor material, whereas subsequent shocks do little to raise the entropy. The shock waves sent by the oscillating donor are seen as nearly concentric rings of enhanced density in the bottom panels of Figure <ref>. This snapshot is near the end of the simulation, and at this moment almost all material in the grid, consisting mostly of donor material, is bound. The flow is nearly radial around the donor in the center-of-mass frame. Material flows slightly slower at the downstream side due to the slightly higher density there. 






 §.§ Origin of Unbound Donor Mass



To clarify where most of the unbound He originates, we reran the HeWD, =1.2×10^51 erg model but with four different passive scalars for different sectors of the donor. 
As displayed in Figure <ref>, scalar 0 faces the explosion center whereas scalar 3 faces the backside. 
We record the velocity of the unbound donor material leaving the simulation box originating from each sector, and show the distribution in Figure <ref> after accounting for motion relative to the binary center-of-mass. The total unbound donor mass loss is ≈ 0.013, which is only ≈ 10% of the initial donor mass. 



There is a high-velocity tail at v ⩾ 5,000  dominated by mass loss from the upstream hemisphere, with higher contributions from sector 0. This is not surprising because the upstream hemisphere is subject to strong momentum transfer from the SN ejecta. 

Mass loss from the upstream hemisphere (sectors 0 and 1) increases towards lower velocities and peaks at ≈ 1,300 . Around the same velocity, we also find significant mass loss from sector 3, the downstream sector facing directly away from the explosion center. The mass loss there is not a consequence of direct momentum transfer from the SN ejecta, but rather due to ablation from shock breakout down the density gradient. In all velocity ranges, sector 2 contributes to little mass loss, whereas sectors 0, 1 and 3 collectively contribute to a significant peak in mass loss at  v≈ 1,300 . We note that even without accounting for orbital motion of the donor, the mass loss peak would still peak at ≈ 900 . 

We compare the donor mass loss velocity distribution to that of the ejecta scaled by 0.1, approximately the fraction of solid angle intercepted by the donor. Assuming that most of the donor mass loss is constrained within ≈ 30^∘ downstream from the explosion center (see next section), then we predict that donor material dominates over ejecta at v ≲2,500  at these angles in the wake. Our prediction that donor mass loss dominates at low velocities is robust – even without scaling down d M_ ej / d log_10 v by 0.1, the donor material dominates at v ≲1,800 . We also note that since the donor mass loss lies mostly at ≲ 10,000 , the conical hole in the ejecta cannot be filled with donor material at high velocities. 




 §.§ Angular distributions of donor material and ejecta


Figure <ref> shows the angular distribution of unbound mass leaving the simulation box. 
We define the angle θ relative to the explosion center, such that θ = 0^∘ points downstream. 
For a spherically symmetric flow, the mass per solid angle is 1 M_⊙/(4 π), and so the mass between angles θ and θ + dθ is given by 1 M_⊙/(4 π) × 2 π× [ cosθ - cos(θ + dθ) ] ∝sinθ dθ, i.e. at larger angles, the mass per θ bin increases due to a larger solid angle. This is observed for the ejecta at θ≳ 50^∘. However, for θ≲ 50^∘, the flow of the ejecta deviates from spherical symmetry because it has to divert around the donor. In particular, θ≲θ_ wake≈ 25^∘ is relatively devoid of ejecta, and at θ≈ 30^∘, approximately where a bow shock is formed, there is a bump in ejecta. 
On the other hand, we find a peak in donor mass loss at θ≈ 20^∘ extending to ≈ 40^∘. Donor material dominates over the ejecta at θ≲ 15^∘. 
However, our results should be taken with a grain of salt since we adopt a constant Γ = 5/3, but the post-shock ejecta should be radiation-pressure dominated (Γ = 4/3). 







 §.§ Origin of bound ejecta material


To identify the composition of the ejecta material that remains bound to the donor, 
we reran the HeWD, = 1.2 × 10^51 erg model with two passive scalars for the ejecta, distinguished by whether their incoming velocity is below 10,000 . This choice because the =1.0 model in <cit.> is dominated by iron group elements (IGE) below 10,000 . We find that ≈ 85% of the bound ejecta material originates from v ≲ 10,000 , thus showing that most of it is IGE in composition. 





§ HE WD DONOR WITH OTHER 



We now discuss the results of the HeWD simulations with other . 
Figure <ref> shows snapshots at t=1155 s for =1.5 & 0.5×10^51 erg. As expected, a higher  deposits more entropy into the HeWD donor, causing the donor remnant to be much more extended, so much that the entire grid is filled with bound donor material. In contrast, the low  simulation shows a more centrally concentrated donor remnant, with some bound ejecta in the grid. 
We can also see that the donor has moved further to the right for =1.5×10^51 erg, which is a result of the greater momentum transfer from the ejecta. 







Figure <ref> shows the density at the center-of-mass of the donor for different . Right before the ejecta is injected, the central density is slightly lower than at the beginning of the simulation, due to relaxation of the donor in the co-rotating frame. As a shock propagates through the donor, the central density jumps by roughly a factor of 2, and falls subsequently as the donor expands. For the rest of the simulation, the central density oscillates around some equilibrium value as the donor rings. With higher , the donor has a lower  since it is more loosely bound, and a longer oscillation period. 
The oscillation periods shown by the models roughly agree with the fundamental mode pulsation period of a polytrope <cit.>, with a polytropic index chosen to approximately match the density profile. 

Table <ref> shows the final change in bound helium mass Δ M^f_ He, bound ejecta mass M^f_ ej, final central density of the donor ^f in units of the initial central density ^i, its change in central specific entropy Δ, and the kick velocity in x- and y-directions, v_ kick,x & v_ kick,y, for different . 
We estimate the change in donor central specific entropy as follows:

    Δ = 3/2/μ m_ pln[  P_ c^f/ P_ c^i( ρ_ c^i/ρ_ c^f)^5/3] ,

where μ≈ 4/3 is the mean molecular weight, and P^f_ c is the final central pressure. 


With higher , mass loss from the donor increases, with a roughly linear scaling in agreement with <cit.>. In general, we find that the donor loses ≈ 0.01-0.025. 
Meanwhile, ∼ 10^-7-10^-4 of SN ejecta stays bound to the donor surface, which we consider as lower limits, since our finite box sizes do not allow us to track all ejecta material. 
A higher  causes greater shock deposition into the donor and hence higher Δ. In turn, the donor experiences greater expansion, and hence lower ^f. 


With higher , the donor also receives a greater kick velocity. 
We find that the x-momentum intercepted by the donor is about ≈ 18 - 24% of that expected, 
in the range of ≈ 1/3 as predicted by <cit.> and <cit.>, yielding v_ kick,x≈ 170 - 230. The donor also receives a kick in the negative y-direction ≈ few  due to the orbital motion. However, the final spatial velocity is only ≈ 2% higher than v_ orb, since the kick direction and the donor orbital motion are mostly perpendicular to each other <cit.>. 


<cit.> and <cit.> showed that a 0.4-0.45 He WD donor can undergo a detonation after being shock-heated by the supernova ejecta. Although nuclear reactions are not accounted for in our  simulations, we check for this possibility in our post-processing. For each cell, we solve for the temperature given the density and pressure, using the  equation-of-state module, and calculate the local heating timescale c_ p T / ϵ_ nuc using the  nuclear net module, where c_ p is the specific heat capacity at constant pressure and ϵ_ nuc is the specific nuclear energy generation rate. Although some cells, particularly at the upstream side of the donor, may momentarily get shock-heated to above 10^8 K, we find that the local heating timescale is always longer than 10^4 s, which is still longer than our simulation time. This suggests that helium detonation does not occur in our HeWD simulations. This differs from <cit.> and <cit.> likely because our He WD donor is of lower mass (0.13) and therefore density. 


ccccccccccc
2


Final results

Donor Model     
    
Δ M^f_ He    
M^f_ ej    
ρ^f_c    
Δ s_ c    
v_kick,x    
v_kick,y    
v_ total    


    
[10^51 erg]    
[]    
[]    
[ρ^i_c]    
[/μ m_ p]   
[]    
[]    
[] 

100pt
HeWD     1.5     0.025     10^-7     0.14     0.29     227     -5.7     975 

         1.2     0.020     10^-5     0.19     0.26     220     -10     969 

         0.8     0.015     10^-4     0.32     0.15     195     -8.7     965 

         0.5     0.010     10^-4     0.46     0.15     166     -16     952 

HeStar1     1.5     0.010     10^-4     0.80     0.04      121     -1.5     726 

            1.2     0.008     10^-4     0.83     0.03     112     -2.3     724 

            0.8     0.005     10^-3     0.88     0.02     94     -3.4     720 

HeStar2     1.5     0.016     10^-4     0.48     0.10     247     -4.9     1017 

            1.2     0.012     10^-4     0.55     0.08     238     -6.1     1013 

            0.8     0.009     10^-4     0.67     0.07     210     -9.0     1004 

 Here  is the total SN kinetic energy, Δ M^f_ He is the mass loss from the donor, M^f_ ej is a rough estimate of the bound SN ejecta mass, ρ^f_ c is the final central density of the donor, Δ s_ c is the change in central specific entropy of the donor as estimated by equation <ref>, v_ kick,x and v_ kick,y are the kick velocities received by the donor in the x and y-directions, and v_ total is the final ejection velocity of the donor after accounting for the kick and its orbital motion. 








§ HELIUM STAR MODELS



We revisited the He star models from <cit.> for comparison and to explore the effects of different numerical treatments. These are summarized in Appendix <ref>. 
We now discuss simulations with the HeStar1 and HeStar2 donor models using the fiducial setup described in Section <ref>. 
The results are summarized in Table <ref>. 

Based on the discussion in <cit.>, we expect that the ratio between the ejecta pressure at the donor location and the donor central pressure is given by:

    P_ ej/ P_ c≈7 × 10^-3/ q^2/3 (1+q)^7/3( v_ej/v_orb)^2 ,

where q =  / is the mass ratio. Although donors in wider orbits tend to suffer more entropy impact due to the v_ orb scaling, after also accounting for the mass ratio q we expect the HeWD donor to be the most impacted. Indeed, comparing the results in Table <ref> shows that the HeWD donor gains much more entropy than the He star models, and as a result of the expansion experiences a steeper drop in . 
In general, the He star donors lose ≈ 0.01, slightly less than the HeWD model. 
The velocity distribution peaks at v ≈ 1,100 & 1,600  for HeStar1 and HeStar2 respectively. 
Also, ≈ 10^-4 of SN ejecta stays bound to the He star donors. 
Like the He WD models, we find that a helium detonation is unlikely for the He stars. 




§ POST-EXPLOSION EVOLUTION IN MESA



After the oscillations in the shocked donor are damped, we continue its subsequent evolution on much longer timescales using the stellar evolution code  version 24.03.1. 
We first adjust the  donor model to the appropriate post-explosion mass. 
Since  does not evolve T as a base variable, we obtain spherical averages of P and ρ from , and then iterate to find the corresponding T profile using 's eos module. 
We then adjust the  model to this T-ρ profile using 's relax_initial_entropy routine.
We do not take into account pollution by the SN ejecta, and adopt solar metallicty (Z=0.014) for the opacity. We also do not consider rotation. Convection is treated with 's time-dependent convection formulation <cit.> and a mixing length parameter α_ MLT = 1.89. 

We show the post-explosion evolution of the donor on the Hertzsprung-Russell diagram (HRD) in Figure <ref>. 
The entropy deposited by the ejecta increases toward the outer layers of the donor. As a result, all donor models initially show a temperature inversion in their outer layers. 
The HeStar1 and HeStar2 models brighten and expand to ≈ 10^3 and 10^2 within the first ≈ 10 and 10^2 yrs, respectively, while the HeWD model does not. 
Similar brightening is shown by other works, for main sequence donors <cit.> and He star models <cit.>. 
However, we do not show this short-lived phase, corresponding to the first 10^3 yrs, in Figure <ref>. As noted by others, objects this bright are of interest to search for in young supernova remnants <cit.>. 



After ≈ 10^4-10^5 yrs, roughly corresponding to the local heat diffusion timescale of peak heating <cit.>, the donor envelope loses entropy and the temperature inversion vanishes. The donor dims and contracts. 
The HeWD and HeStar2 models keep on cooling, but the HeStar1 model eventually resumes core He burning after ≈ 10^6 yrs, as its mass is above the limit for He burning (≈ 0.3). It continues core He burning for the next ≈ 10^8 yrs. 


We now compare our models to several runaway stars. The hypervelocity star D6-2 has reliable astrometry from Gaia EDR3 and photometry, allowing a fit to the spectral energy distribution that yields T_ eff = 7500 ± 100 K and R = 0.20 ± 0.01 <cit.>. Its trajectory traces back to a known supernova remnant, allowing derivation of a kinetmatic age of ≈ 10^5 yrs <cit.>. 
The HRD evolution of our HeWD model, whose pre-explosion v_ orb agrees well with the ejection velocity of D6-2 <cit.>, agrees decently with the location of D6-2. However, it is somewhat over-luminous compared to D6-2 at an age of 10^5 yrs. 


J1332-3541 is among the 4 newly discovered hypervelocity stars <cit.>, and has an uncertain radius R = 0.017 ^+0.013_-0.007 due to its uncertain distance. We adopt a temperature of 70,000 K <cit.>. 
US 708 is a He-rich subdwarf O star, with an inferred disk-crossing ≈ 10^7 yrs ago. We adopt T_ eff = 47200 and log g = 5.69 to calculate its luminosity assuming a mass of 0.3. 
The He star models have T_ eff between D6-2, and J1332 and US 708. 
At an age of ≈ 10^7 yrs, the HeStar1 model settles near the He main sequence, but appears too dim and red compared to US 708. 
In agreement with <cit.>, we find that if US 708 is currently a He-burning star, its mass must be higher than that of HeStar1 (≈ 0.35). However, <cit.> inferred a mass of 0.34-0.37 for US 708, not much different than our HeStar1 model. Alternatively, it could still be undergoing thermal relaxation if the local thermal diffusion timescale at the location of peak heating is long. 

Interestingly, the HeWD trajectories pass near GD 492 (LP 40-365), which has an estimated T_ eff = 9800± 300 K and L = 0.20 ± 0.04 <cit.>. This star represents a class of runaway objects mostly with ejection velocities ≈ 600 - 700 <cit.> and Ne-dominated atmospheres, believed to be the surviving accretor after incomplete burning fails to unbind the star <cit.>. The estimated disk-crossing time for most of these objects is ≈ 10^6 yrs, coinciding with the age at which the HeWD trajectories intersect GD 462 on the HR diagram. 



We note that pollution by SN ejecta can greatly enhance the surface opacity. We find that adopting a solar-scaled metallicity of Z=0.03 shifts the HeWD, =1.2×10^51 erg model redwards by ≈1,000-2,000 K, and a Z=0.05 model agrees well with the present-day location of D6-2 at an age of ≈ 10^6 yrs <cit.>. Given a surface convection zone mass of ≈ 0.01 from the models, these Z values require only a few 10^-4 of ejecta be deposited and stay on the donor surface, which is within the range of bound SN ejecta mass in our hydrodynamic models (see Table <ref>). 

D6-2 has a photometric period of 15.4 hrs, inferred to be its rotation period <cit.>. 
Although our models do not account for rotation of the donor, we can estimate the rotation rate. We assume that the pre-explosion donor is tidally synchronized to its orbit <cit.>, and uniformly rotating. 
We also assume that the donor conserves its total angular momentum of the remaining bound mass immediately after explosion, J_ i. 
In the limit of solid-body rotation, we can calculate the total moment of inertia I_ f = 2/3∫^M_ f_0 r^2 dm, and hence the rotation period P_ f = 2 π I_ i / J_ f. For the HeWD, = 1.2 × 10^51 erg model, this implies a rigid rotation period of ≈ 1.7 hrs at 10^5 yrs, which is too rapid compared to D6-2. 
In the limit of no angular momentum transport, each fluid element retains its specific angular momentum j(r) = i(r) ω(r), where i(r) = 2/3 r^2 is the specific moment of inertia. 
The final surface rotation period is then P_ f = P_ i(  R_ f / R_ i)^2, where R_ f is the final donor surface radius, and R_ i is the corresponding pre-explosion radius of these outermost mass element. At an age of 10^5, we find a rotation period of 18.5 hr, in much better agreement with the photometric period of D6-2. 
This comparison suggests that D6-2 has not had sufficient time for interior angular momentum transport, or that it experienced angular momentum loss during the post-explosion evolution <cit.>. 
As the donor remnant further contracts, its rotation period decreases. For example, its rotation period would be 0.8 hrs for rigid-body rotation at 10^6 yrs, and correspondingly 4.1 hrs with no angular momentum transport. 

Similarly, the He star models start off slowly rotating but spin up as they contract. At 10^5 yrs, the HeStar1 model has a rotational period of 0.3 hrs (1.2 hrs) for solid-body rotation (no angular momentum transport), which decreases to 0.2 hrs (0.9 hrs) at 10^7 yrs. These values agree reasonably with <cit.>. 
The corresponding values for HeStar2 are 0.2 hrs (2.2 hrs) at 10^5 yrs and 0.1 hrs (0.4 hrs) at 10^7 yrs. 








§ CONCLUSIONS



We have modeled the interaction between a He WD/ He star donor and SN Ia ejecta. 
We adopt an ejecta profile based on sub- explosion models <cit.>, and vary the explosion kinetic energy. 
During the interaction, ≈ 0.01 - 0.02 of He-rich material is stripped from the donor and mixed into the SN ejecta. The stripped donor mass loss peaks at v ≈ 1,000 , and dominates over SN ejecta within a cone of half-opening angle ≈ 15^∘. 
The velocity distribution and the multi-dimensionality of the stripped material should be accounted for in future radiative transfer calculations <cit.>. 
Placing detection limits on the stripped He material from late-time SNe Ia spectrum could then constrain the donor type of the progenitor system <cit.>. 
We also find that ∼ 10^-7 - 10^-4 of ejecta material, mostly consisting of iron-group elements, remains bound to the donor. However, we cannot accurately estimate the amount of bound ejecta material due to our finite box sizes. 


Due to entropy deposition from the shock caused by the SN ejecta, the post-explosion donor appears bright and puffy for ≈ 10^5-10^6 yrs. This is in contrast to <cit.>, who find that for CO WD donors, the post-explosion donor rapidly cools in ≲ 10^3 yrs. We show that the post-explosion properties of our HeWD model, whose pre-explosion orbital velocity agrees well with the hypervelocity star D6-2 <cit.>, also agrees reasonably with D6-2. This strengthens our argument that D6-2 is a surviving low-mass (≈ 0.1-0.2) He WD donor <cit.>. However, we do not account for the surface metal pollution of D6-2, likely due to the capture of low-velocity SN ejecta. We defer this to future works. 


Our  models are limited by the use of a Γ=5/3 EOS (However, see Appendices <ref> & <ref>). We plan to use a realistic EOS accounting for radiation and degeneracy in future  simulations <cit.>. 
This will open up the opportunity to investigate other donor types, including C/O WDs which likely comprise the majority of the seven hypervelocity stars <cit.>, and allow us to clarify the post-explosion evolution of the hypervelocity stars. 





§ ACKNOWLEDGMENTS




We thank the referee for their constructive suggestions that have greatly improved our manuscript. 
We are grateful to Ken Shen for sharing his sub- explosion models and for conversations about the hypervelocity stars. 
We thank Evan Bauer for helpful conversations regarding  modelling and for sharing his He star models from <cit.>. 
We thank Logan Prust for insightful discussions regarding the  simulations. 
This research benefited from stimulating interactions with Kareem El-Badry and Jim Fuller. 
We thank Sterl Phinney for pointing out the need to account for orbital motion in the mass loss velocity distribution. 
This work made use of the Heidelberg Supernova Model Archive (HESMA), https://hesma.h-its.orghttps://hesma.h-its.org.
This work was supported, in part, by the National Science Foundation through grant PHY-2309135, and by 
the Gordon and Betty Moore Foundation through grant GBMF5076. 
The computations in this work were, in part, run at facilities supported by the Scientific Computing Core at the Flatiron Institute, a division of the Simons Foundation. 
Use was made of computational facilities purchased with funds from the National Science Foundation (CNS-1725797) and administered by the Center for Scientific Computing (CSC). The CSC is supported by the California NanoSystems Institute and the Materials Research Science and Engineering Center (MRSEC; NSF DMR 2308708) at UC Santa Barbara.






 <cit.>, 
 <cit.>, 
 <cit.>,
 <cit.>,
 <cit.>,
 <cit.>, 
 <cit.>, 
 <cit.>,
and 
 from https://www.python.orgpython.org








ccccc|ccc

3
Comparison of different numerical choices for HeStar2, = 0.7 × 10^51 erg, = 0.779

Self-gravity solver     
Resolution     
Box size    
ρ_ floor     
P_ floor     
M^f_ He     
ρ^f_ c     
v_ kick,x


    
    
[x_0]    
[ρ^i_ c]    
[P^i_ c]    
[]    
[ρ^i_ c]    
[] 

100pt
Fourier     256^3     (-10,50)     10^-5     1.65×10^-8     0.214     0.546     192 

Fourier     *512^3     (-10,50)     10^-5     1.65×10^-8     0.216     0.566     195  

Fourier     512^3     *(-15,65)     *10^-6     *10^-8     0.220     0.561     178  
multigrid     512^3     (-15,65)     *10^-5     10^-8     0.228     0.594     232  

multigrid     512^3     (-15,65)     *10^-6     10^-8     0.226     0.568     212  

multigrid     512^3     (-15,65)     *10^-7     *10^-9     0.226     0.568     208  


Asterisk (*) highlights changes relative to the previous row. The box size is for the x-axis. The y and z-axes have the same length as the x-axis, but symmetric around the origin. Here ρ_ floor and P_ floor refer to the floor density and pressure, M^f_ He and ρ^f_ c refer to the final donor mass and central density, and v_ kick,x is the kick velocity the donor receives in the x-direction. 




cccc|ccc

4
Comparison of different numerical choices for HeWD, = 1.2 × 10^51 erg, = 1.0

Resolution     
Box size    
ρ_ floor     
P_ floor     
M^f_ He     
ρ^f_ c     
v_ kick,x


    
[x_0]    
[ρ^i_ c]    
[P^i_ c]    
[]    
[ρ^i_ c]    
[] 

100pt
512^3     (-15,85)     10^-6     10^-8     0.108     0.211     218 

704^3     (-15,85)     10^-6     10^-8     0.106     0.196     230  

896^3     (-15,85)     10^-6     10^-8     0.106     0.191     240  

704^3     (-15,85)     3 × 10^-7     3 × 10^-10     0.106     0.194     219  


 Variables take the same meaning as Table <ref>. 




§ NUMERICAL CHOICES AND COMPARISON TO BAUER ET AL. 2019



We attempted to reproduce the results of <cit.> for their donor model 2 (HeStar2 here) with = 0.7 × 10^51 erg, = 0.779. 
We retain the same simulation box size, spatial resolution (256^3 cells), density and pressure floors, donor relaxation process, ejecta profile <cit.>, 
and self-gravity solver (Fourier method). However, we do not recover the same results. For example, their final donor has M^f_ He≈ 0.15 and ρ^f_ c≈ 0.25 ρ^i_ c (see their Table 2), while we find M^f_ He≈ 0.21 and ρ^f_ c≈ 0.55 ρ^i_ c. This means that we find much less mass loss, given that the pre-explosion mass of HeStar2 is 0.233, and we find a more tightly bound post-explosion donor. While the difference in M^f_ He could be due to different definitions for bound material, ρ^f_ c should only depend on the shock strength and hence numerics such as spatial resolution. We were not able to resolve these differences. 

We explored the effects of a number of numerical choices for the above simulation. 
The results are detailed in Table <ref>. 
We find that the resolution in <cit.> is likely sufficient, but the box size is likely too small, leading to the loss of some bound mass. 
We also find that a high density floor can induce a net gravitational pull given a simulation domain that is asymmetric around the x-axis. 
However, all of these choices do not change ρ^f_ c by more than 10%, and the donor mass loss only ranges from 0.01-0.02, and so cannot explain the discrepancy with <cit.>. 
For our main results in the paper, we have varied resolution, box size, and floor limits to ensure that our results are reasonably converged.  This leads us to adopt the numerical choices detailed in Section <ref>. An example for the HeWD, = 1.2 × 10^51 erg model is shown in Table <ref>. We do not show the bound ejecta mass because all choices lead to a change by factor of a few around 10^-5 and this quantity is limited by our finite box size anyways. 




§ MAPPING BETWEEN  AND 




To quantify the errors introduced by adopting a Γ=5/3 EOS, we compare the temperature and specific entropy profiles of the donor star between  and , for the HeWD, =1.2× 10^51 erg run. To obtain the  profiles, we take spherical averages of P and ρ, and solve for T using the   module. 
The T-ρ profiles are shown in Figure <ref>. 
We find good agreement between the original  profile and the  profile after the relaxation as detailed in Section <ref> (but before explosion), except near the outermost layers which is affected by interactions with floor material, though this material is most likely stripped during the explosion. 
There is also good agreement between the  profile at the end of the simulation (after explosion), and the  profile that is then evolved further (see Section <ref>). 
We note that the original  profile is only semi-degenerate in its center, which is why we adopted Γ=5/3. We also note 
the drop in center density after the explosion which indicates expansion of the donor. 




We also use the  P-ρ profiles to obtain the specific entropy profile, by adding the original specific entropy profile from  and the change in specific entropy Δ s from  (see eqn. <ref>). The resulting profiles are shown in Figure <ref>. 
Although there is a slight offset in the center between  and  profiles which reflects the error in using a Γ=5/3 EOS, the overall agreement is still good. 







§ REALISTIC EQUATION OF STATE




After our first submission, we began experimenting with using a realistic EOS using 's general EOS capability <cit.>. 
We compare two runs with the HeWD, = 1.2 × 10^51 erg model, one with Γ = 5/3 and one with the Helmholtz EOS <cit.>, which accounts for radiation, electron degeneracy and Coulomb corrections. For this initial experimentation, we assume a pure He composition for all components including the ejecta, and do not relax the donor star under a Roche potential. 
The donor central density during the simulation is shown in Figure <ref>, which shows a similar evolution between the two EOS's. Other quantities are also similar, such as the final bound mass M^f_ He which only differs by ≈ 2% between the two. We will present more details on our use of a realistic EOS in a future paper, but this initial comparison justifies our use of Γ=5/3. 
















aasjournal



 Paper Title: The extremes of AGN variability: outbursts, deep fades, changing looks, exceptional spectral states, and semi-periodicities.



S. Komossa et al.



 



1]S. Komossacor1
[cor1]Corresponding author: 
  skomossa@mpifr.de


2]D. Grupe


3]P. Marziani
4,5]L. Č. Popović
4,5]S. Marčeta-Mandić
5]E. Bon
4,6]D. Ilić
4]A. B. Kovačević
1]A. Kraus
7,8]Z. Haiman
9,10]V. Petrecca
9,10,11]D. De Cicco
5]M. S. Dimitrijević
12]V. A. Srećković
5]J. Kovačević Dojčinović
13]M. Pannikkote
5]N. Bon
14,15]K. K. Gupta
16]F. Iacob

[1]Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany






[2]Department of Physics, Geology, and Engineering Technology, Northern Kentucky University, 1 Nunn Drive, Highland Heights, KY 41099, USA

[3]National Institute for Astrophysics (INAF), Astronomical Observatory of Padua, Vicolo dell’Osservatorio 5, Padua IT, 35122, Italy

[4]University of Belgrade - Faculty of Mathematics, Department of Astronomy, Studentski trg 16, 11000 Belgrade, Serbia





[5]Astronomical Observatory, Volgina 7, 11000 Belgrade, Serbia





[6]Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany





[7]Department of Astronomy, Columbia University, New York, NY 10027, USA

[8]Department of Physics, Columbia University, New York, NY 10027, USA

[9]Dipartimento di Fisica "Ettore Pancini", Università di Napoli Federico II, via Cinthia 9, 80126 Napoli, Italy

[10]INAF - Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, 80131 Napoli, Italy





[11]Millennium Institute of Astrophysics, Nuncio Monseñor Sótero Sanz 100, Of 104, Providencia, Santiago, Chile



[12]Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia 



[13]Physics Department, Tor Vergata University of Rome, Via della Ricerca Scientifica 1, 00133 Rome, Italy



[14]STAR Institute, University of Liège, Quartier Agora, Allée du Six Auôt 19c, 4000, Liège, Belgium

[15]Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281 S9, 9000, Gent, Belgium

[16]West University of Timisoara, 300232 Parvan Blvd. no. 4, Timisoara, Timis, Romania

11 July 2024
X X 2024
X X 2024
2024
S. Sarkar


§ ABSTRACT 
The extremes of Active Galactic Nuclei (AGN) variability offer valuable new insights into the drivers and physics of AGN.  We discuss some of the most extreme cases of AGN variability; the highest amplitudes, deep minima states, extreme spectral states, Seyfert-type changes, and semi-periodic signals, including new X-ray observations. 
The properties of changing-look (CL) AGN are briefly reviewed and a classification scheme is proposed which encompasses the variety of CL phenomena; distinguishing slow and fast events, repeat events, and frozen-look AGN which do not show any emission-line response. 
Long-term light curves that are densely covered over multiple years, along with follow-up spectroscopy, 
are utilized to gain insight into the
underlying variability mechanisms including accretion disk and broad-line region physics.
Remarkable differences are seen, for instance, in the optical spectral response to
extreme outbursts, implying distinct intrinsic variability mechanisms.
Furthermore, we discuss methods for distinguishing between CL AGN and CL look-alike events (tidal disruption events or supernovae in dense media). Finally, semi-periodic light curve  variability is addressed and the latest multiwavelength (MWL) light curve of
the binary supermassive black hole (SMBH) candidate OJ 287 from the MOMO project is presented. 
Recent results from that project have clearly established the need for new binary SMBH modelling
matching the tight new constraints from observations, including the measurement of a low (primary) SMBH mass of ∼ 10^8 M_⊙ which also implies that OJ 287 is no longer in the regime of near-future pulsar timing arrays. 







active galactic nucleiquasarsSeyfert galaxieschanging-look AGNsupermassive binary black holesaccretion disksMWL variability and spectra 










§ INTRODUCTION









Active Galactic Nuclei (AGN) are thought to be powered by accretion of matter onto the supermassive black holes (SMBHs) at their centers. The central region is surrounded by two line-emitting systems, the broad-line region (BLR) closer to the nucleus and of higher density, and the narrow-line region (NLR) at larger distances up to kiloparsecs and of lower density <cit.>. 
The unified model of AGN has been successful in explaining the major differences between type 1 and type 2 AGN by viewing angle differences into the central engine <cit.>. In addition, other parameters are known to play a role in explaining variations across the Seyfert and quasar populations and within each population, including the influence of the host galaxy, evolutionary effects, accretion rate, and/or metal abundances, for instance. BLR and NLR emission-line properties as well as continuum (spectral energy distribution; SED) shape have been used to evaluate differences across the AGN population using Principal Component Analyses <cit.>, establishing the accretion rate as a main driver. Another approach involves the  investigation of continuum and emission-line variability. This is the topic of this contribution.  

The extremes of AGN variability provide valuable insights into the underlying drivers and physics of the central engine. 
In X-rays, extreme flux and spectral states can reveal the nature of the inner accretion disk, probe the physics of matter under strong gravity, may offer a way of measuring SMBH spin, and can uncover the mechanisms responsible for the expulsion of matter from the SMBH in form of strong outflows. 

Amplitudes of X-ray variability exceeding a factor 30-50 are still rare, and only a few systems are known to exceed a factor of 100 <cit.>.  
The highest amplitudes of X-ray variability to date from the cores of galaxies, exceeding factors of 1000, have nearly exclusively been observed in non-active galaxies in the form of temporary flares from stars tidally disrupted by dormant SMBHs (stellar tidal disruption events; TDEs). These latter will not be discussed much further in this contribution <cit.>. 

In the optical regime, changing-look AGN (CL AGN hereafter) with highly variable emission lines which change the spectroscopic Seyfert type, provide us with tight new constraints on the physics of the outer accretion disk, the BLR and the coronal-line region (CLR). 

The term CL AGN was initially coined to describe systems which changed their X-ray spectrum from mildly or unabsorbed to heavily absorbed or Compton-thick  <cit.>. However, in recent years the term has been widely used to refer to optical spectroscopic Seyfert-type changes, from type 1 to type 2 (or 1.8, 1.9) and vice versa <cit.>. We use this latter definition throughout this publication. 

Single cases of CL AGN were noticed already decades ago. One prominent example is NGC 4151.  
Traditionally known as intermediate-type Seyfert galaxy <cit.>, its broad Balmer lines became very faint in 1984 <cit.>, barely detectable but still
present <cit.>. They were already back to their bright state in January 1985 <cit.>. 
Two recent examples are shown in Fig. <ref>, initially selected either through their high-amplitude continuum flaring and the CL event was then found in triggered follow-up optical spectroscopy (HE 1136–2304;  results in Fig. <ref> taken from <cit.>), 
or targeted in the course of reverberation mapping and long-term spectroscopy over decades 
(NGC 3516; results in Fig. <ref> taken from <cit.>).  While HE 1136–2304  only changed its Seyfert type from 1.5 to 1.8, NGC 3516 changed from type 1 to type 1.9. 


  


The topic of CL AGN  received particular attention in recent years, because larger samples were selected from optical surveys, more extreme cases were identified, events were detected not only in Seyfert galaxies but also in quasars where the emitting regions are much more extended so that longer variability timescales are expected, and higher amplitudes of variability were detected not only in X-rays but also in the optical band <cit.>. 
The rapid and high-amplitude changes pose challenges to accretion-disk models <cit.>, and many new scenarios for rapid changes in the disk emission have been developed in recent years. 

The first part of this review will focus on extreme cases of variability of radio-quiet AGN and the second part will briefly address semi-periodic variability of candidate binary SMBHs.  In Sect. 2 we review the frequency of CL AGN. In Sect. 3 we present a new scheme which sorts the CL AGN into different phenomenological categories based on their continuum and line variability. 
Sect. 4 briefly summarizes the different theoretical approaches suggested to understand CL AGN, and in Sect. 5 representative  cases of extreme flux and spectral variability are discussed, including new X-ray observations. Sect. 6 provides a critical summary of observational effects which can lead to false CL detection and elaborates on how to distinguish true CL events from CL look-alikes. 
Sect. 7 addresses (semi-)periodic variability of radio-loud systems, which has been suggested to be associated with the presence of binary SMBHs. A focus is on the blazar OJ 287. The article concludes with an outlook into future time domain surveys (Sect. 8).

Throughout this publication, we use a cosmology with 
H_ 0=70 km s^-1 Mpc^-1, Ω_ M=0.3 and 
Ω_Λ=0.7
and the cosmology calculator of <cit.>. 



§ THE FREQUENCY OF CL AGN


Large-area spectroscopic surveys offer excellent data bases for selecting samples of CL AGN. Ultimately, these will provide us with the frequency, timescales, and disk and host properties of CL events in Seyfert galaxies and quasars.
A number of selection effects have to be kept in mind: For instance, if the starting sample consists of type 1 AGN, then turn-offs (decreases in line and continuum luminosity) will be preferentially found in the follow-up spectroscopy. If the starting sample consists only of type 2 AGN, then only turn-ons, if any, will be found at the second epoch of spectroscopy. 
Further, the amplitude of the type change (into type 1.8, 1.9, or 2) will depend on telescope sensitivity, with faint Balmer lines more difficult to detect in spectra of lower signal-to-noise ratio (see also Sect. 6). 
Further, AGN with already faint Balmer lines (lower Eddington accretion rates at fixed SMBH mass) will be preferentially detected as CL AGN (as turn-offs), because only a small change in the luminosity is needed to make the lines undetectable, and therefore make the AGN appear as CL. If, instead the broad lines are luminous, a much larger luminosity change is required to identify a CL event. An alternative could be therefore to require a minimum factor (10) of variability, independent of exact AGN type, to search for CL AGN. 

Finally, the selection criteria used to identify CL AGN (e.g., the required amplitude of the type change) so far have varied between different studies which needs to be taken into account as well when directly comparing results. 

Larger-sample searches for CL AGN have been carried out recently with, for instance, LAMOST <cit.>, 6dFGS/ Skymapper <cit.>, and DESI <cit.>. Otherwise, mostly 
the exceptional data base of the Sloan Digital Sky Survey <cit.> has been employed
for subselecting larger samples of AGN which have been observed at least twice with SDSS or once with SDSS and followed-up at other telescopes <cit.>. 
These have started to quantify the frequency and properties of CL events and types in Seyfert galaxies and quasars. 

<cit.> found that the frequency of Seyfert galaxies significantly changing their spectroscopic type was 10% in a sample of nearby type 1 Seyferts initially selected from SDSS and then followed up with Keck. The same frequency in an SDSS selected sample of quasars was lower at ∼1% <cit.>. Approximately equal numbers of events got brighter and changed towards type 1, or got fainter and changed towards type 2. 
The largest sample of CL AGN selected recently from SDSS V <cit.> contained 67% CL AGN which showed  dimming and 33% which showed brightening. The CL AGN have a median accretion rate of L/L_Edd = 0.025, slightly lower than in a control sample with L/L_Edd = 0.043.    

Archival searches on large spectroscopic samples are excellent for statistical inferences on the CL AGN population. An alternative approach has been to carry out rapid MWL follow-up observations within days to weeks, once new CL AGN candidates were identified by the onset of flaring or dipping in the optical or X-ray band <cit.>. That way, dense monitoring of the ongoing, evolving event is possible. Representatives of such AGN are discussed in Sect. 5.



§ DEFINITION OF DIFFERENT CATEGORIES OF CL AGN


In recent years, many different cases of high-amplitude BLR and continuum variability have been reported, making CL AGN quite a mixed bag, with possibly a variety of different underlying physical processes operating in different systems and at different timescales. 

Therefore, here, we first introduce several different phenomenological CL AGN categories, based on the type of continuum variability and/or line response: 

∙ Category I: Slow switch-on AGN. These are CL AGN which slowly turn on, on a timescale of years to decades.  An example for this pattern may be AT2017bcc <cit.>, even though the event is unusual and was initially interpreted as superluminous supernova or a candidate tidal disruption event (see also our Sect. 6). 

∙ Category II: Slow switch-off AGN. 
In this category are CL AGN which slowly turn off, on a timescale of years to decades.
A prime example for this category is Mrk 590. Once known as a classic Seyfert 1 galaxy, its optical continuum emission diminished by a factor of 100 over two decades <cit.>. In low-state, the only broad emission line still visible in the optical spectrum is a weak component of Hα, implying a change from type 1 to type 1.9. 

∙ Category III: Rapid outbursts. In this category are CL AGN which spend most of their time in fainter states but then show a rapid (timescales of weeks to months) strong increase in their optical–X-ray continuum luminosity, followed by a response in the broad emission lines.  Prime examples are IC 3599 <cit.> and NGC 1566 <cit.> which showed dramatic increases of their continuum flux on a timescale of weeks to months.  

∙ Category IV: Rapid drops. In this category are CL AGN which spend most of their time in brighter states but then show a  rapid (timescales of weeks to months) decrease in their optical–X-ray continuum  and broad-line flux. 

∙ Category V: Regular, repeating brightening and dimming events. In this category are CL AGN which show recurrent medium-amplitude increases and decreases in their line and continuum brightness. Many of the reverberation-mapped AGN are in this category, for instance NGC 5548 <cit.>. 

∙ Category VI: Frozen-look AGN.  
In this category are AGN which do not show any emission-line changes in response to strong continuum variability. 
This category is rare. Examples include 
saturation effects in the BLR ionization in case of particularly high continuum luminosity <cit.>.

A much stronger example is the lack of HeII response despite high-amplitude soft X-ray continuum variability of Mrk 335 <cit.>, explained in this case by changes of (dust-free) absorption occurring only along our line-of-sight, not seen in other directions (i.e., not seen by the bulk of the emission-line gas).    



Several CL events are known to repeat. Repetition may occur in all categories, but it depends on the cadence and duration of the observational coverage how easily recurrence can be detected. Single AGN may even cross categories in the long term.   
Some processes are more common than others. 
Tab. <ref> provides a summary of the different categories of CL AGN. 

The strongest cases of CL AGN are in categories I–IV since they exhibit the largest and most unexpected changes and place most tight constraints on variability models. 

Finally, we note that CL AGN were first identified based on observations, and therefore the proposed categories closely match what has been observed so far. They do not yet include any underlying theoretical scenarios, because it is still unknown how to explain CL AGN, and many different scenarios have been proposed in the literature. Not all of them make predictions for the timing and spectral evolution, yet. 
In the next Section, we provide a short overview over the different models for CL AGN  which have been proposed in recent years. 



§ (NEW) MODELS FOR EXTREME AGN VARIABILITY


There are two fundamentally different ways to explain high-amplitude variability. One of them is intrinsic variability where the emission of the accretion disk and BLR itself changes.
The other is apparent variability where the intrinsic emission remains constant, but is then affected on its way to the observer.  Such extrinsic variability can be caused by the effects of gas absorption and dust extinction, or by the effect of gravitational lensing.

We discuss the different mechanisms in turn.



 §.§ Extrinsic variability: gas absorption and dust extinction


Gas absorption affects the X-ray continuum emission through Hydrogen and metal K-shell absorption. For gas column densities exceeding  ∼10^23-24 cm^-2, the material becomes Compton-thick and no X-rays are detected at all below 10 keV where most X-ray satellites operate.

If an absorber contains dust, it also significantly affects the optical continuum through dust extinction, causing a characteristic reddening of the continuum shape and a deviation of the BLR Balmer decrement from its (case B) recombination value.

Dust extinction and/or gas absorption is therefore an efficient mechanism to change the observed luminosity of an AGN, and if at work the mechanism can be relatively easily identified  through its characteristic effects on the optical and X-ray spectra.

However, while partial obscuration from outflowing dusty clouds in the outer BLR can significantly affect line profile shapes <cit.>, and while single CL AGN do show effects of significant extinction <cit.>, this mechanism has generally not been favored for the majority of CL systems, because the expected BLR reddening was not observed <cit.>, the optical-UV continuum shape was inconsistent with extinction <cit.>, the associated X-ray absorption was not observed <cit.>, the timescale of variability was too fast <cit.>, and/or the level of polarization was very low <cit.>. 



 §.§ Extrinsic variability: gravitational lensing


Lensing or microlensing can magnify (or demagnify) the accretion disk and/or the BLR emission.  Depending on the geometry, it can give rise to achromatic light curve variability and/or it can affect different parts of the emitting region, causing complex line profile shapes and changes.

Microlensing modelling of BLR profile variations has not yet been much applied to changing-look AGN, but studied in a number of known lensed quasars <cit.>. It is rare and is therefore not expected to explain the population of nearby CL AGN as a whole.



 §.§ Intrinsic variability: Modified lampposts, disk instabilities, disk precession, influence of magnetic fields, BLR stability, binaries


CL events challenge our ideas of the properties of the accretion disk in AGN. 
Given the rapid and high-amplitude variability of some CL AGN, a large number of new theoretical studies in the last few years have explored new mechanisms for accretion disk structural and emission changes. Some of these addressed 
mechanisms of a more rapid transfer of changes across the disk (e.g., involving magnetic fields), but left open the origin of the high-amplitude outburst in the first place; others addressed the outburst mechanism directly; and yet others explored changes of the BLR structure itself.

First, a modified lamppost model has been suggested <cit.>, assuming that the X-ray emission which arises from the lamppost is not directly reprocessed by the accretion disk, but rather by a system of dense gas clumps above the disk. This explains a rapid MWL response, but leaves open the mechanism(s) of the intrinsic high-amplitude of the X-ray variability itself. 

Second, different types of accretion disk instabilities have been considered to explain high-amplitude continuum changes, including the radiation-pressure instability of the inner disk <cit.>, a Hydrogen ionization instability <cit.>, or instabilities in warped disks at large radii <cit.>. An instability in the circumnuclear gas supply has been discussed as well 
<cit.>. Tidal effects on the mini-disks surrounding both SMBHs of a binary SMBH system <cit.> have seen suggested as well.  

Third, the influence of magnetic fields, or a combination of several effects, has been considered in different ways <cit.>. 

Fourth, effects of two types of shocks developing in highly tilted, highly precessing disk were discussed <cit.>. 

Finally, some models were suggested where the BLR gas 
itself is directly affected, through an instability <cit.>, initially proposed to explain the absence of a BLR in low-accretion-rate systems.  



 §.§ New approaches of understanding BLR and CLR physics


Independent of the exact mechanism which causes the change in the illumination of the emission-line gas, the reprocessing of the radiation on BLR and CLR spatial scales provides us with a wealth of new applications regarding BLR and CLR physics, and the underlying atomic processes. 
We just give a few examples here. For instance,
some of the highest-amplitude outbursts are known to produce a strong response not only in the BLR, but also in the CLR. Strong high-ionization lines in the CLR temporary excited by an outburst provide tight constraints on the intrinsic, unobservable EUV shape of the SED because their ionization potentials are located in the EUV to soft X-ray regime. Further, atomic parameters of many coronal lines are still poorly known with uncertainties in atomic parameters and collision strengths as large as a factor of 10 <cit.>
given the difficulties of producing them in laboratory plasmas. In addition, rare coronal line transitions can be excited that way, and their ratios provide us with valuable new diagnostics of the density and temperature <cit.> 
of the CLR in AGN.  

A second example regards the BLR physics. The nature of the outer BLR is still not well understood. Once gas temperatures drop sufficiently below ∼2000 K  the gas conditions change significantly, Hydrogen molecules can be present <cit.>, and new ionization/recombination processes start to dominate (see <cit.> for a detailed discussion of the relevant physical processes, including chemi-ionization/recombination processes, the importance of H^*(n) + H(1s) collisional ionization, the influence of inverse recombination processes, and 
in <cit.>
the  importance of excitation/de-excitation processes with (n-n') mixing in such collisions).  Turn-off CL AGN may be ideally suited to probe the presence and physics of such an outer BLR component, as the inner parts of the BLR will fade first in response to a deep drop in the ionizing continuum luminosity, systematically leaving the outer parts of the BLR detectable in high-cadence, time-resolved spectroscopy of the BLR response.  



§ EXTREME CASES OF AGN VARIABILITY AND THEIR INTERPRETATION


Given the different types of extreme continuum and spectral variability of AGN and the relatively low number of sources, we discuss these on an individual basis, focussing on systems for which we have obtained long-term observations, and adding new data from the Swift archive as well. The object properties are listed in Tab. <ref>.













 §.§ High-amplitude outbursts and the cases of IC 3599 and NGC 1566




  §.§.§ IC 3599


The Seyfert galaxy IC 3599 underwent a giant outburst detected in the course of the ROSAT all-sky survey. Its X-ray flux increased by more than a factor 100. Optical spectra taken by chance during the ongoing outburst revealed bright broad Balmer lines and high-ionization iron coronal lines <cit.>. These emission lines then faded strongly in subsequent years <cit.>. In low-state, among the Balmer lines, only broad Hα is still significantly detected, leading to a Seyfert 1.9 classification <cit.>. Faint coronal lines are still present in low-state. Photoionization modelling of the optical outburst spectrum revealed that the high-ionization lines are well explained with gas typical of a CLR, with a gas density of ∼10^9 cm^-3
<cit.>. 
The cause of the CL event of IC 3599 was not discussed in more detail at that time, but Narrow-line Seyfert 1 (NLS1) variability, a TDE, or an accretion-disk instability were briefly mentioned as possibilities; the latter favored by <cit.>. 

In ongoing Swift monitoring, a second X-ray outburst of similar amplitude was detected in 2010, 19.5 yr after the first one <cit.>. The outbursts are consistent with a Lightman-Eardley disk instability, using the observed flare repeat time of 19.5 yrs 
and a SMBH mass range of 10^6-7 M_⊙ inferred from observations, 
which then implies a truncation radius between inner and outer disk of r_ trunc = 5–45 r_ g <cit.>.  
MWL monitoring is ongoing in order to search for new high-amplitude outbursts. No third one has been detected so far <cit.>, but significant lower-amplitude X-ray variability is ongoing (Fig. <ref>). During outbursts, the X-ray spectrum of IC 3599 is exceptionally steep (Γ_ x∼ 4). The cause for the steepness remains unknown, but it seems to point towards the absence of a disk corona. Even in low-states, the X-ray spectrum remains very soft (Fig. <ref>). 

Our Swift monitoring of this extreme optical CL AGN is ongoing. Rapid MWL follow-ups including optical spectroscopy at dense cadence will provide us with a unique method of reverberation-mapping the BLR and CLR following the next giant outburst.


  



  




  §.§.§ Recurrent changing-look activity of NGC 1566
 

NGC 1566 is similar in several respects to IC 3599 including the detection of strong iron coronal lines during outburst, but with a lower amplitude in X-rays, particularly hard (not soft) X-ray spectra, and better optical spectroscopic coverage, so that one can tell that it went through repeated CL events. 

NGC 1566 is a very nearby NLS1 galaxy <cit.> at redshift z=0.005. The widths of the broad Balmer lines of NGC 1566 are FWHM(Hβ) = 1950 km/s and FWHM(Hα) = 1970 km/s <cit.>. 

Fe II emission is detected in the type 1 and 1.5 spectral state <cit.>.
NGC 1566 was known as a bright type 1 AGN in the 1960s <cit.>, but became much fainter in subsequent years.
Spectroscopic observations taken between 1980 and 1982 found it to be a type 2 in 1980, then gradually changing back to a type 1. Another CL event was detected in 1985 <cit.>. Most of the changes occurred rapidly, within only 4 months. The BLR Balmer decrement did not change significantly during the CL event <cit.>.

The most recent outburst of NGC 1566 in 2018 was first detected with INTEGRAL in the hard X-ray band <cit.>. MWL follow-up observations were rapidly triggered <cit.> which resulted in the detection of a new CL event. 
During the outburst, the optical spectrum changed from type 1.8 to type 1, accompanied by strong variations in the helium lines and iron coronal lines <cit.>. 
The FWHM of the Balmer lines remains relatively constant during all observations. The emission lines OIλ8446 and CaIIλλ8498,8542,8662 are double-peaked <cit.>. 

The X-ray spectrum of NGC 1566, measured in rapid X-ray follow-up observations carried out with XMM-Newton, NuSTAR and Swift <cit.>, is a typical type 1 X-ray spectrum at high state, extending to high energies 

and modified by ionized absorption and mild reflection. With the reflection grating spectrometer (RGS) an outflow with velocity of 500 km/s, possibly launched during the outburst, was detected. The amount of X-ray cold absorption does not change between low-state and high-state, implying that variable absorption is not the cause of the outburst. <cit.> concluded that a heating front propagating through the disk caused by the Hydrogen instability provides a consistent explanation of the 2018 outburst. 

An earlier hard X-ray flare is present in the Swift BAT light curve <cit.> in 2010, but was not followed up spectroscopically.

The Swift XRT long-term X-ray light curve of NGC 1566, based on our own monitoring observations and additional archival data, is displayed in Fig. <ref>. The X-rays continue to be at low emission levels in 2024. 

NGC 1566 is the nearest of the CL cases detected so far in NLS1 galaxies <cit.>, and among the nearest repeating CL AGN known. 


  




 §.§ Regular intermediate-amplitude dimmings and brightenings of reverberation-mapped AGN and the cases of NGC 5548 and NGC 3516 

  


 §.§ NGC 5548

As a nearby, bright, intermediate-type Seyfert galaxy, NGC 5548 has been the target of several reverberation mapping campaigns and many optical spectroscopic studies covering several decades <cit.>.  It is variable in the optical and X-ray regime, densely monitored with Swift <cit.>. In low-states its BLR becomes faint, but 
Hα and Hβ are always detected. NGC 5548 can therefore be regarded as a CL AGN, changing between Seyfert type 1.5 and type 1.8. 
In the same sense, many reverberation-mapped AGN would represent CL AGN, since pre-selected to show significant continuum variability. 

The  amplitude of variability of NGC 5548 in X-rays is much lower than in the other AGN discussed in this contribution (Fig. <ref>, based on all archival Swift XRT observations of NGC 5548).

Based on evidence for periodicity of the long-term optical light curve between 1972 and 2015 and accompanying changes in the broad lines with a period of 5700 d, it was suggested that NGC 5548 may host a compact binary SMBH <cit.>.


  




 §.§ NGC 3516


The Seyfert galaxy NGC 3516 has long been known to be highly variable in its continuum and broad emission lines. Part of the initial sample of <cit.>, strong broad-line variability was first reported by <cit.>. 
NGC 3516 stands out as highly variable in the X-ray band as well. For instance, it was one of the most highly variable AGN in a dedicated search for such AGN in the ROSAT data base <cit.>.  
Its long-term Swift XRT light curve <cit.>, demonstrating the high-amplitude X-ray variability of this AGN in recent years, based on data retrieved from the Swift archive, is shown in Fig. <ref> in comparison with the other AGN discussed in this contribution.  

NGC 3516 has been the target of many spectroscopic monitoring campaigns spanning decades  including reverberation mapping   <cit.>, showing that NGC 3516 is a CL AGN which varies between type 1 and type 1.8 or even 1.9 at selected epochs.  

 Based on a detailed study of the Hβ flux and profile variability, <cit.> concluded that an intrinsic mechanism, rather than large-scale obscuration in the outer BLR, is required to explain the CL events in NGC 3516, even though a contribution of dust within the BLR may still play a role at low activity phases. 



 §.§ Frozen-look AGN and the case of Mrk 335


The NLS1 galaxy Mrk 335 was once among the X-ray brightest AGN. However, when first observed with Swift, its X-ray flux had strongly declined (Fig. <ref>). When followed up with Swift in dense, long-lasting monitoring campaigns <cit.>, Mrk 335 did exhibit episodes of high-amplitude flaring back into high-states. However, these episodes became less frequent, less long-lasting and less bright, and since 2018, Mrk 335 has remained in deep X-ray low-states most of the time (Fig. <ref>). This is still the case in our most recent observations from 2024.  


Since bright and nearby, Mrk 335 still provides us with good-quality X-ray spectra even in low-state and this is the reason why it was observed multiple times in deep spectroscopic follow-ups, e.g. with XMM-Newton and NuSTAR <cit.>. However, two very different spectral models still provided successful fits to the complex X-ray low-state spectrum, with no way to distinguish based on the quality of the spectral fit alone: The first one consists of an absorption model where the absorber partially covers our line-of-sight to the continuum source (the accretion disk),  the second one is a reflection model where the intrinsic photons are subject to blurred reflection from the inner accretion disk.  

The solution came from combining X-ray with optical–UV data. This approach showed that X-ray and optical–UV emission are uncorrelated most of the time <cit.> and that the high-ionization, broad HeII emission line did not follow the simultaneous high-amplitude X-ray variability even though its ionization potential is in the soft X-ray regime <cit.>.
These results can be understood if the X-ray variability is predominantly caused by a (partial-covering, dust-free) absorber located along our line-of-sight, but not covering a full 4π sphere shielding the BLR. 
The presence of line-of-sight absorption is independently detected with HST <cit.> and with XMM RGS <cit.> spectroscopy of Mrk 335, revealing UV and X-ray absorption lines.  This result does not exclude a reflection contribution to the X-ray spectrum, but requires a strong absorption component, which drives at least a large fraction of the observed variability. 

In summary, Mrk 335 is a good example for (dust-free) absorption-driven X-ray continuum variability. It is a frozen-look AGN as far as its broad HeII emission is concerned <cit.>, but its Balmer lines still do show a response to the mild optical continuum variability during the epoch it was the target of reverberation mapping <cit.>.  Swift monitoring of Mrk 335 is still ongoing in order to trace the duration of the deep low-state, follow the decoupling between X-ray and UV–optical continuum variability, and catch its rise back to high-state. 



§ APPARENT CL VARIABILITY AND IMPOSTORS


Not all observed cases of CL events are real. Some observational setups can mimic false BLR variability, instrumental sensitivity needs to be taken into account when assessing the presence or absence of faint Balmer-line emission, and some phenomena look spectroscopically similar to CL AGN but are due to very different physical processes. We discuss these in turn. 



 §.§ Artificial variability, biases and challenges
  

(1) Slit or fiber not on the nucleus. 
Nearby galaxies appear largely extended on the sky, occasionally making it difficult to position a (narrow) slit or fiber exactly on the nucleus. Since BLR and continuum are pointlike (only broadened by atmospheric seeing) whereas the NLR is extended, a mis-positioned slit will provide a spectrum with apparently faint BLR and continuum. That way, dimming events and turn-off CLs can be mimicked.  

(2) Telescope sensitivity. Faint broad Balmer lines will increasingly merge with the noise, and it then depends on the exposure time and telescope sensitivity, whether faint Hβ and/or Hα can still be detected in a given spectrum. Accordingly, the turn-off state will be classified as type 1.8, type 1.9, or type 2, even if both Balmer lines are still intrinsically present. It is possible that some of the historic CL events reported to reach type 2 states in the 1970s and 1980s are due to this effect. Complete turn-offs into type 2 states have been rarely observed in recent years. 



 §.§ CL look-alikes: TDEs and SNe (type IIn and/or in dense medium)


The optical spectra of both, TDEs and supernovae (SNe) IIn can show temporary broad (and/or narrow) optical emission lines, and will thus mimic a changing-look AGN. In fact, there is a parameter space, where all three outburst types (AGN, SNe, and TDEs)  look alike, and are challenging to distinguish from each other.  Here, we describe three scenarios, and discuss ways how to distinguish them from bona fide CL AGN variability <cit.>.[Further, there is a new class of  long-duration ambiguous transients <cit.>, even though most or all of them are characteristically different from CL AGN.] 

(1) SN of type IIn: These type of SNe are known to show temporary broad and narrow emission lines in their optical spectra which look very similar to AGN <cit.>. The narrow lines arise from strong interactions of the SN ejecta
with dense circum-stellar matter. Strong ionizing radiation may originate from the
shock break-out, Compton-upscattering, as well as in shocks when the SN ejecta collide with the precursor wind and/or the interstellar medium. 


(2) SN in dense medium, in a quiescent galaxy or an AGN: Supernovae in a dense medium would particulary efficiently reprocess continuum into emission lines, and can even produce temporary low-level hard X-ray emission (e.g., SNe in the outskirts of the dense torus environment could be partially obscured or extincted, and their decline light curves and continuum emission would look different from other SNe). The situation is especially complicated, if the SN does not happen in a quiescent galaxy but in a long-lived AGN, and all of the SN features would superpose on the long-lived AGN spectrum, which can itself be (mildly) variable.

(3a) TDE in a quiescent host or in an AGN. TDEs are most reliably identified in quiescent galaxies (i.e., systems with dormant SMBHs without a long-lived accretion disk and therefore without a NLR), because TDEs are then the only known mechanism to produce giant X-ray flares of quasar-like peak luminosity <cit.>.
In such a scenario, broad emission lines can arise from the temporary accretion disk formed from the disrupted star, and the flung-out stellar debris. If the environment is gas-rich, narrow emission lines can be excited as well, first detected from gas closest to the line of sight (little light travel-time delays).
If a TDE happens in an AGN, all the TDE features overlapp with the long-lived AGN features. 

How can we distinguish all of these impostors from true CL AGN ? One crucial parameter is the X-ray luminosity and X-ray light curve. Another key observation  is repeat optical spectroscopic coverage, ideally including a spectrum before the outburst.
The X-rays are important, because longer-duration (weeks to months) X-rays exceeding  10^41-42 erg/s have never been observed from SNe, thus allowing to exclude an SN interpretation of the spectrum.  
Pre-outburst optical spectroscopy is important, because it tells if the host is active or quiescent, based on the presence or absence of a NLR. Similarly, pre-outburst X-ray detections imply a long-lived AGN if L_ x > 10^42 erg/s and exclude a TDE in a quiescent host galaxy. 
Finally, post-outburst spectroscopy will tell, if the narrow lines fade (SN IIn or TDE) or are permanent.
The most challenging situation is a TDE or SN in an AGN, in which case the long-term spectroscopic evolution of multiple emission lines provides the best diagnostics to distinguish between all three scenarios. Nevertheless, individual sources could be challenging to classify. PS16dtm (SN 2016ezh) is a good example. It shows properties of all three, a superluminous SN, a TDE, and a CL NLS1, and all three classifications have been proposed and discussed in the literature <cit.>, (Ilić, these proceedings).

Finally, TDEs are extremely rare, therefore in a statistical sense we know that at most a tiny fraction of CL AGN  could be mimicked by TDEs, because the observed frequency of CL events in nearby galaxies <cit.> is far too high.



§ (SEMI)-PERIODIC VARIABILITY, CANDIDATE BINARY SMBHS, AND THE CASE OF OJ 287



  




 §.§ Candidate binary SMBH identification from light curves


Supermassive binary SMBHs form in the course of galaxy mergers <cit.>. The evolved binary SMBHs with small separations, of much less than a parsec, can no longer be spatially resolved with current techniques, and we rely on indirect methods to detect them. A large variety of different detection methods has been explored in the literature <cit.>. The most common one makes use of (semi-)periodic signals in long-term MWL light curves. These can be induced as a consequence of orbital motion of one or both SMBHs, or precession of the accretion disk or jet. 
At least several periods should be covered in any light curve to be considered a reliable periodicity candidate; otherwise noise can mimic the periodic signal <cit.>.  

Many recent searches for binary SMBH candidates have been conducted on ongoing optical photometric surveys such as the Catalina Real-time Transient Survey <cit.>, the Palomar Transient Factory <cit.>, and Pan-STARRS <cit.>. Recent examples of systems with sinusoidal light curve variations include PG 1302–102 <cit.>, and PKS 2131–021 <cit.> (see Sect. 5.3 for the case of NGC 5548, and <cit.> for a recent discussion how to search for binary SMBHs via BLR line profiles). 

A different semi-periodic light curve pattern, not sinusoidal, is shown by the candidate binary SMBH OJ 287. Its long-term optical light curve exhibits recurrent bright double-peaks repeating every 11±1 yrs, most clearly present in the 1970–1990 light curve of this blazar <cit.>. The most recent double-peak has been identified in 2016–2017 <cit.>.
OJ 287 has the most densely covered MWL lightcurve of all binary SMBH candidates <cit.> and is discussed in more detail in the next Section. 



 §.§ OJ 287 and constraints on binary SMBH models from the MOMO project


Several different binary SMBH scenarios for OJ 287 have been explored in the literature; involving either disk or jet precession, orbital motion of one or two jet-emitting SMBHs, disk impacts of the secondary SMBH during its orbit around the primary, or the presence of mini-disks around two SMBHs fed by a circum-binary disk <cit.>. 
According to other suggestions, a binary is not required, or its absence is preferred <cit.>. All of the binary models have offered explanations for the  double-peaked nature of the outbursts and are based on excellent theoretical frame works, but none has so far been able to predict correctly the timing (and spectra) of the outbursts of the recent light curve of OJ 287 since 2000 <cit.>. 

Tight new model constraints came from ongoing dense MWL monitoring in the course of the MOMO project. It was initiated in 2015 and is designed to distinguish between different binary SMBH scenario predictions, and it independently sheds new light on blazar disk-jet physics
<cit.>. Dense monitoring of OJ 287 at a cadence of a day to weeks at multiple frequencies in the radio, optical, UV, X-ray and gamma-ray bands, along with dedicated deeper MWL follow-up spectroscopy at selected epochs, provides timing, spectra, and broadband SEDs at all activity states of OJ 287 <cit.>. This is the densest and longest MWL monitoring so far carried of OJ 287 at > 12 frequencies, and among the densest of any blazar (Fig. <ref>). 

Recent MOMO results <cit.> have clearly established that (1) no outburst occurred in October 2022 <cit.> 
and instead the last double-peaked outburst was observed in 2016–2017 (our Fig. <ref>); (2) the observed upper limit on the accretion-disk luminosity of OJ 287 (based on 3 different estimates) is a factor >10-100 fainter than required by a ∼10^10 M_⊙ SMBH accreting at 10% Eddington; 
instead 
(3) the (primary) SMBH mass derived from BLR scaling relations is modest and of order 10^8 M_⊙ consistent with the faintness <cit.> of the host galaxy;  and (4) a thermal Bremsstrahlung spectrum with a predicted spectral index of α_ν,opt ≈ -0.2 has never been observed in the densely covered Swift lightcurve between 2016–2024.

The identification of the last double-peaked outburst in 2016 – 2017 implies that these outbursts are relatively regular spaced at 11±1 yr intervals, but are not exactly periodic. The low (primary) SMBH mass implies that OJ 287 is no longer a candidate for detection by near-future pulsar timing arrays. Instead, it would be in the sensitivity regime of LISA-type space-based gravitational-wave interferometers <cit.> upon its final coalescence. 

In summary, successful future binary SMBH models of OJ 287 should reproduce a regular double peak every ∼ 11 yr but allow for ±1 yr variations, should be based on a moderate primary SMBH mass of ∼10^8 M_⊙ (similar to the approach previously followed by <cit.>, but now without assuming strict periodicity), and should reproduce synchrotron outbursts. One  alternative binary modelling avenue to follow will be  the application of recent binary merger simulations which predict SMBH binaries with mini-disks fed by a larger circum-binary disk <cit.>. <cit.> loosely suggested that such a scenario could apply to OJ 287, but modelling of OJ 287 within that framework has not yet been carried out.   

The long-term light curve of OJ 287 at representative MOMO frequencies is shown in Fig. <ref>, including previously unpublished X-ray, optical, UV, and radio measurements from 2023–2024. It shows that OJ 287 is at a deep low-state in June 2024, particularly obvious in the optical, UV, and radio bands.[A previous deep optical  low-state of OJ 287 was once speculated to possibly represent an epoch in the binary SMBH orbit evolution where the jet of the primary SMBH is briefly misaligned due to the perturbation of a secondary SMBH when it passes near the jet <cit.>, but the deep low-states have happened too often in recent years <cit.> to be consistent with such an interpretation.]  
The next double-peak is expected in 2026–2028 <cit.>, and dense monitoring in the course of the MOMO project is planned.




§ UPCOMING TIME DOMAIN SURVEYS IN THE OPTICAL AND X-RAY REGIME


Upcoming time domain surveys in the optical and X-ray bands will not only detect large numbers of extremely variable AGN and CL AGN, but will also facilitate a search for the characteristic periodic signals of binary SMBHs based on the high-cadence light curves coverage every one to few days. In the optical bands, the Vera C. Rubin Observatory Legacy Survey of Space and Time  <cit.> is scheduled to start operation in the year 2025.
LSST will monitor the entire Southern sky for 10 years in 6 filters (ugrizy) with a 9.6 deg^2 field-of-view and a 5 σ g-band magnitude depth of ∼24.5 in a single 30s exposure [<https://www.lsst.org/scientists/keynumbers>]. 
LSST will detect millions of variability events per night, issuing an alert within 60s from the detection 
to enable prompt follow-ups with other instruments <cit.>.
The combination of joint photometry data sets from LSST and spectroscopy from other facilities will enable accurate classification for a plethora of events, including outbursts, flares and changing states in AGN. The extent of the sample will further increase the possibility to detect rare events, while the implementation of the Difference Image Analysis <cit.> on the entire dataset will allow us to identify variability in obscured or host-dominated AGN.
In the X-ray regime, two new transient missions will carry out high-cadence all-sky surveys. The Einstein Probe <cit.> was launched in January 2024 and the Space Variable Objects Monitor <cit.>) was launched in June 2024. 
Several studies have already highlighted how LSST is expected to produce important results in relation to accretion disc studies, reverberation mapping campaigns, and binary SMBH candidate detections in AGN <cit.>. 
In particular, binary SMBH candidates with periods of months to years, similar to candidates selected from ongoing optical surveys (Sect. 7), are expected to be detected in large numbers in radio-loud and radio-quiet AGN with LSST (longer periods) and in X-rays. 
Further, an advantage of LSST is its high cadence[the exact cadence is still being discussed and may change during its 10 year mission; the higher it is, the more short-period systems will be well covered] so that it can find shorter periods, and it is very deep, therefore it covers AGN with lower-mass SMBHs, of which there are many more, and also they are longer-lived at periods of weeks compared to more massive SMBHs.    This implies that the LSST catalog will contain large numbers of fainter short-period binaries.

Evolved, ultra-compact SMBH binaries of much shorter periods of only days which then coalesce years later in the LISA time window
can be detected in the optical band with LSST through the periodic light-curve modulation from the circum-binary disk feeding the two SMBH's mini-disks <cit.>,
while the orbital modulation from the two mini-disks themselves is best searched for in the X-ray regime, given the compactness of the mini-disks at very advanced stages of merging. 

Other future surveys with a strong time-domain component and expected to start by the late 2020s are the Cosmological Advanced Survey Telescope for Optical and UV Research (CASTOR), and the Nancy Grace Roman Space Telescope, in the NIR range. These two spacecrafts will add further MWL coverage to the variable Universe, and will have a strong synergy with contemporary surveys such as LSST and the EUCLID satellite (even though the latter does not have an explicit time-domain focus).































§ ACKNOWLEDGMENTS

 We would like to thank the Swift team for carrying out the observations we proposed.
 In addition to our own data, we have also used the Swift archive at <https://swift.gsfc.nasa.gov/archive/>. 
 We would like to thank all participants of the “VI. Conference on Active Galactic Nuclei & Gravitational Lensing” (Serbia, June 2024) for enlightening discussions. S.K. and Z.H. would also like to thank the participants of the conference on “Multi-messenger observations of supermassive black hole binaries” which was supported by the Lorentz Center (Leiden, June 2024) for very stimulating discussions. 
 S.K. would like to thank R. Antonucci, D. Hutsemékers, F. Marin, and M. Ochmann for very useful discussions. 
 E.B., N.B., D.I., A.B.K, J.K.D., S.M.M., and L.Č.P.  
 acknowledge funding provided by the University of Belgrade—Faculty of Mathematics (contract 451-03-66/2024-03/200104) and/or Astronomical Observatory Belgrade (contract 451-03-66/2024-03/200002), through grants by the Ministry of Education, Science, and Technological Development of the Republic of Serbia. D.I. acknowledges the support of the Alexander von Humboldt Foundation. A.B.K. and L.Č.P. thank the support by the Chinese Academy of Sciences President’s International Fellowship Initiative (PIFI) for visiting scientists. 
 V.P. and D.D. acknowledge the financial contribution from PRIN-MIUR 2022 and from the Timedomes grant within the “INAF 2023 Finanziamento della Ricerca Fondamentale”. D.D. also acknowledges PON R&I 2021, CUP E65F21002880003. 
 Z.H. acknowledges support by NSF grant AST-2006176 and NASA grants 80NSSC22K0822 and 80NSSC24K0440.
 K.K.G. acknowledges the Belgian Federal Science Policy Office (BELSPO) for the provision of financial support in the framework of the PRODEX Programme of the
 European Space Agency (ESA).
 This work is partly based on data obtained with the 100 m telescope of the Max-Planck-Institut für Radioastronomie at Effelsberg.   
This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester <cit.>. This work has made use of public Fermi-LAT data <cit.>.  
This research has made use of the XRT Data Analysis Software (XRTDAS) developed under the responsibility of the ASI Science Data Center (ASDC), Italy.
This work has also made use of the NASA Astrophysics Data System Abstract Service (ADS), and the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.







167
natexlab#1#1[#1],#1

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