The changing-look active galactic nucleus (CL-AGN), an extraordinary subpopulation of supermassive black holes, has attracted growing attention for understanding its nature. We present an analysis of the spectral properties of 203 low-redshift CL-AGNs (z<0.35) using two-epoch spectra from SDSS DR16 and DESI DR1 with time baseline ranging from $\sim$1000 to 8000 days, based on spectral fitting and decomposition. The sample consists of 11.3\% Type 1.0, 26.6\% Type 1.2, 43.1\% Type 1.5, and 19\% Type 1.8/2.0 AGNs. The total sample is divided into two datasets: Dataset A (110 objects) with minor spectral type variations, likely general AGN variability, and Dataset B (93 objects) showing significant type transitions and characteristic turn-on or turn-off behavior. Our results reveal clear optical continuum and emission-line variability, showing both bluer-when-brighter and redder-when-brighter trends. A strong correlation between the broad H$\beta$/[O~{\sc iii}] ratio and broad H$\alpha$ luminosity ($L_{\rm H\alpha}$), ${\rm log(H\beta/[O~III])}=(0.63\pm 0.07){\rm log}(L_{\rm H\alpha})-(26.49\pm2.96)\pm0.48$ for Dataset B, as well as the correlation between H$\beta$/[O~{\sc iii}] and Eddington ratio ($L_{\rm bol}/L_{\rm Edd}$), ${\rm log(H\beta/[O~III])}=(0.59\pm 0.08){\rm log}(L_{\rm bol}/L_{\rm Edd})+(1.02\pm0.15)\pm0.53$ for Dataset B, suggests that accretion rate variations drive changes in ionizing flux within the broad-line region, thereby triggering AGN type transitions. These findings underscore the critical role of supermassive black hole accretion processes in refining the AGN unification model. Future work should investigate potential connections between stellar evolution in outer accretion disk and the observed scatter in these correlations.

The Seyfert 1 galaxy J1626+5120 is estimated to host a $10^8 M_{\odot}$ black hole (BH) accreting at Eddington ratio $\dot{m}_{\text{Edd}} \approx 0.043$. Its long-term multi-band light curve data show flicker-like variations, but in a well-sampled $g$-band light curve, we are able to determine a $\simeq 329$\,d quasi-periodic oscillation (QPO) at a $\sim$4.53$\sigma$ significance. Six optical spectra were obtained for the source, three of which were taken by us. The spectra show that the variations were mainly because of flux changes blueward of 4000\,Å. We also analyze X-ray and ultraviolet (UV) data obtained with {\it the Neil Gehrels Swift Observatory (Swift)}, which targeted the source in the past two years. X-ray and UV emissions of the source show variations correlated with optical. Time lags of four UV bands and four optical bands are determined with respect to the X-ray emission, which are consistent with a continuum reprocessing disk model. These properties point out a disk origin for the QPO, likely due to Lense-Thirring (LT) precession of the accretion flow at $\sim$20 gravitational radii of the BH. This QPO could be a key case linking sub-year long QPOs in jets, which have more cases reported, to LT precession.

We present the results of the 2023 spectroscopic reverberation mapping (RM) campaign for active galactic nuclei (AGN) of NGC 5548, continuing our long-term monitoring program. Using the Lijiang 2.4-meter telescope, we obtained 74 spectra with a median cadence of 1.9 days. Through detailed spectral decomposition, we measured the light curves of the optical continuum at 5100~Å and the broad He~{\sc ii}, He~{\sc i}, H$\gamma$, and H$\beta$ emission lines. The time lags of these lines relative to the continuum are measured as $1.3^{+1.6}_{-0.6}$, $2.3^{+1.5}_{-2.1}$, $10.0^{+2.0}_{-1.8}$, and $15.6^{+2.6}_{-2.9}$ days (rest-frame), respectively. Velocity-resolved lag profiles for H$\gamma$ and H$\beta$ were constructed. Combined with data from previous seasons (2015$-$2021), we find that the radial ionization stratification of the broad-line region (BLR) is stable; the average virial mass of the supermassive black hole in NGC~5548 is $(2.6\pm1.1)\times 10^{8}M_{\odot}$, consistent with the $M_{\rm BH}-\sigma_*$ relation; the broad He~{\sc ii} line exhibits the largest responsivity, followed by broad He~{\sc i} (or H$\gamma$) and H$\beta$ lines; the BLR kinematics show significant temporal evolution, transitioning from virialized motions to signatures of inflow and outflow. Furthermore, an analysis of 35 years of historical data confirms a 3.5-year time lag between variations in the optical luminosity and the BLR radius, potentially implicating the role of radiation pressure or dynamical structure changes in the inner accretion disk. Long-term campaign demonstrates that the BLR in NGC 5548 is a robust yet dynamically evolving entity, providing crucial insights into AGN structure and accretion physics.

Gravitational waves (GWs) can provide crucial information about the central engines of core-collapse supernovae (CCSNe). In order to unveil the nature of GW emission in CCSNe, we apply perturbative analyses with the same underlying equations as simulations to diagnose oscillations of the proto-neutron star (PNS) during $\sim$1 s postbounce. In the pseudo-Newtonian case, we find that radial profiles of GW emission match well between the perturbative analysis with $l=2$ and simulations inside the PNS at \emph{any} frequency and time. This confirms that the GW emission of CCSNe arises from the global PNS oscillations in the perturbative regime. Based on this, we solve for the discrete eigenmodes with a free PNS surface and tentatively identify a set of $g$ modes and the $f$ mode contributing to the peak GW emission. We also offer a possible explanation for the power gap in the GW spectrum found in simulations that lies at the frequency with vanishing cumulative emission of the PNS. Our results enhance the predictive power of perturbative analyses in the GW signals of CCSNe.

We present a comprehensive photometric and spectroscopic analysis of the Algol-type binary \textit{Gaia} DR3 1892576067672499328. We identified the system as a spectroscopic binary based on medium-resolution LAMOST spectra. Combined with \textit{TESS} photometry, we determine an orbital period of $P = 2.47757 (1)$ days, a low mass ratio of $q = 0.098 \pm 0.002$, and an orbital inclination of $i = 46.934^{+2.613}_{-1.11}$ degrees. The orbit is consistent with being circular ($e = 0$). The binary comprises a $M_1 = 1.817 ^{ +0.106}_{-0.202} \,M_\odot$, $R_1 = 1.265^{+0.121}_{-0.160}\,R_\odot$ A-type primary and a Roche-lobe-filling secondary of $M_2 = 0.179 ^{ +0.011}_{-0.020} \,M_\odot$, $R_2 = 1.994 ^{ +0.041}_{-0.077} \,R_\odot$. The double-peak H$\alpha$ emission line indicates the possible existence of a Keplerian accretion disc. We established a simple standard accretion disc model and modeled the geometric and dynamical properties of the accretion disc. The obtained outer disc radius $R_{\mathrm{out}} \approx 3.36 \pm 0.43\,R_\odot$ is consistent with the values inferred from the emission velocity of H$\alpha$. Systemic velocity variations observed over time suggest the possible presence of a tertiary companion, with a minimum mass of $M_3 > 0.369 \pm 0.024 \,M_\odot$. Given the low mass ratio, the secondary may evolve into a proto-helium white dwarf, forming an \text{EL CVn}-type system in the future. This system offers valuable insights into accretion dynamics and the formation of binaries.

We present extensive photometric and spectroscopic observations of the peculiar Type Ia supernova (SN Ia) 2022vqz. It shares many similarities with the SN 2002es-like SNe Ia, such as low luminosity ($M_{B,\rm max}=-18.11\pm0.16$ mag) and moderate post-peak decline rate ($\Delta m_{15,B}=1.33\pm0.11$ mag). The nickel mass synthesised in the explosion is estimated as $0.20\pm0.04~{\rm M}_\odot$ from the bolometric light curve, which is obviously lower than that of normal SNe Ia. SN 2022vqz is also characterised by slowly expanding ejecta, with Si II velocities persisting around 7000 km s$^{-1}$ since 16 days before peak brightness, unique among all known SNe Ia. While all of these properties imply a lower-energy thermonuclear explosion that should leave a considerable amount of unburnt materials, the absent signature of unburnt carbon in spectra of SN 2022vqz is puzzling. A prominent early peak is clearly detected in the ATLAS $c$- and $o$-band light curves and in the ZTF $gr$-band data within days after the explosion. Possible mechanisms for the early peak are discussed, including the sub-Chandrasekhar-mass double-detonation model and interaction of SN ejecta with circumstellar material. We find that both models face some difficulties in replicating all aspects of the observed data. As an alternative, we propose a hybrid C-O-Ne white dwarf as the progenitor of SN 2022vqz; it can simultaneously reconcile the tension between low ejecta velocity and the absence of carbon. We further discuss the diversity of SN 2002es-like objects and their origin in the context of different scenarios.

Ultracool dwarfs consist of lowest-mass stars and brown dwarfs. Their interior is fully convective, different from that of the partly-convective Sun-like stars. Magnetic field generation process beneath the surface of ultracool dwarfs is still poorly understood and controversial. To increase samples of active ultracool dwarfs significantly, we have identified 962 ultracool dwarfs in the latest LAMOST data release, DR11. We also simulate the Chinese Space Station Survey Telescope (CSST) low-resolution slitless spectra by degrading the LAMOST spectra. A semi-supervised machine learning approach with an autoencoder model is built to identify ultracool dwarfs with the simulated CSST spectra, which demonstrates the capability of the CSST all-sky slitless spectroscopic survey on the detection of ultracool dwarfs. Magnetic activity of the ultracool dwarfs is investigated by using the H$\alpha$ line emission as a proxy. The rotational periods of 82 ultracool dwarfs are derived based on the Kepler/K2 light curves. We also derive the activity-rotation relation of the ultracool dwarfs, which is saturated around a Rossby number of 0.12.

We present the results of low-resolution spectroscopic and densely sampled multiband simultaneous optical imaging ($ugi$ and $vrz$ bands) follow-up of supernova (SN) 2024aecx. The photometric data is supplemented with $Swift$/UVOT and ATLAS survey observations. The SN was discovered in the spiral galaxy NGC 3521 (distance $\sim$11 Mpc) within a day after the explosion. The early spectra of SN 2024aecx show a weak signature of hydrogen lines, which disappeared in $\sim$30 days after the explosion. Light curves in all bands show a distinct feature of two peaks, and the first peak is likely due to the shock cooling emission. The early phase light curve evolution of SN 2024aecx has similarity with the typical Type IIb events, but the decay rate in different bands (e.g., $\rm \Delta m_{15}$ = 1.60 $\pm$ 0.05 mag, $g$-band) is significantly faster in the post-peak phase. It attained the secondary maximum in $\sim$19 days ($g$-band) with a peak absolute magnitude of M$_{g}$= -17.94 $\pm$ 0.10 mag. The color evolution of SN 2024aecx is displaying a red-blue-red trend between days $\sim$8 to 40. The analytical model fitting to the light curves reveals an envelope mass and progenitor radii in the range of $\sim$0.03 - 0.24 $M_\odot$ and $\sim$169 - 200 $R_\odot$, respectively. Modeling of the pseudo-bolometric light curve suggests that synthesized $^{56}$Ni in the explosion was $\sim$0.15 M$_{\odot}$ with ejecta mass and kinetic energy of $\sim$0.7 M$_{\odot}$ and $\sim$0.16 x 10$^{51}$ erg, respectively. The observational properties and modeling indicate that the SN 2024aecx progenitor belongs to the extended progenitor category.

For the first time, we obtain the analytical form of black hole space-time metric in dark matter halo for the stationary situation. Using the relation between the rotation velocity (in the equatorial plane) and the spherical symmetric space-time metric coefficient, we obtain the space-time metric for pure dark matter. By considering the dark matter halo in spherical symmetric space-time as part of the energy-momentum tensors in the Einstein field equation, we then obtain the spherical symmetric black hole solutions in dark matter halo. Utilizing Newman-Jains method, we further generalize spherical symmetric black holes to rotational black holes. As examples, we obtain the space-time metric of black holes surrounded by Cold Dark Matter and Scalar Field Dark Matter halos, respectively. Our main results regarding the interaction between black hole and dark matter halo are as follows: (i) For both dark matter models, the density profile always produces "cusp" phenomenon in small scale in the relativity situation; (ii) Dark matter halo makes the black hole horizon to increase but the ergosphere to decrease, while the magnitude is small; (iii) Dark matter does not change the singularity of black holes. These results are useful to study the interaction of black hole and dark matter halo in stationary situation. Particularly, the "cusp" produced in the $0\sim 1$ kpc scale would be observable in the Milky Way. Perspectives on future work regarding the applications of our results in astrophysics are also briefly discussed.

As mixed with real pulsations, the reflection of super-Nyquist frequencies (SNFs) pose a threat to asteroseismic properties. Although SNFs have been studied in several pulsating stars, a systematic survey remains scarcely explored. Here we propose a method to identify SNFs from Kepler and TESS photometry by characterizing their periodic frequency modulations using a sliding Fourier transform. After analyzing long cadence photometry in the Kepler legacy, we have identified 304 SNFs in 56 stars from 45607 frequencies in $\sim600$ $\gamma$ Doradus stars, corresponding to a fraction of approximately $0.67\%$ and $9.2\%$, respectively. Most SNFs are detected in the frequency range of pressure mode over 120 $\mu$Hz and the fraction of SNF detection increases as frequency up to $\sim7\%$. We barely found two potential SNFs mixed with gravity modes in two $\gamma$ Doradus stars. These findings indicate that SNFs have a negligible impact on the global seismic properties, such as those derived from period spacing in $\gamma$ Doradus stars. However, we stress that SNFs must be carefully and systematically examined by this method in other pulsating stars, particularly $\delta$ Scuti and hot B subdwarf stars, to establish a solid foundation for precise asteroseismolgy of various types of pulsators.

We present preliminary results of the quasar survey in Large Sky Area Multi- Object Fiber Spectroscopic Telescope (LAMOST) first data release (DR1), which includes pilot survey and the first year regular survey. There are 3921 quasars identified with reliability, among which 1180 are new quasars discovered in the survey. These quasars are at low to median redshifts, with highest z of 4.83. We compile emission line measurements around the H{\alpha}, H{\beta}, Mg II, and C IV regions for the new quasars. The continuum luminosities are inferred from SDSS photo- metric data with model fitting as the spectra in DR1 are non-flux-calibrated. We also compile the virial black hole mass estimates, and flags indicating the selec- tion methods, broad absorption line quasars. The catalog and spectra for these quasars are available online. 28% of the 3921 quasars are selected with optical- infrared colours independently, indicating that the method is quite promising in completeness of quasar survey. LAMOST DR1 and the on-going quasar survey will provide valuable data in the studies of quasars.

The Seyfert 1 galaxy J1626+5120 is estimated to host a $10^8 M_{\odot}$ black hole (BH) accreting at Eddington ratio $\dot{m}_{\text{Edd}} \approx 0.043$. Its long-term multi-band light curve data show flicker-like variations, but in a well-sampled $g$-band light curve, we are able to determine a $\simeq 329$\,d quasi-periodic oscillation (QPO) at a $\sim$4.53$\sigma$ significance. Six optical spectra were obtained for the source, three of which were taken by us. The spectra show that the variations were mainly because of flux changes blueward of 4000\,Å. We also analyze X-ray and ultraviolet (UV) data obtained with {\it the Neil Gehrels Swift Observatory (Swift)}, which targeted the source in the past two years. X-ray and UV emissions of the source show variations correlated with optical. Time lags of four UV bands and four optical bands are determined with respect to the X-ray emission, which are consistent with a continuum reprocessing disk model. These properties point out a disk origin for the QPO, likely due to Lense-Thirring (LT) precession of the accretion flow at $\sim$20 gravitational radii of the BH. This QPO could be a key case linking sub-year long QPOs in jets, which have more cases reported, to LT precession.

Core-collapse supernova (CCSN) explosions powered by rotation and magnetic fields present an interesting astrophysical site for nucleosynthesis that potentially contributes to the production of $r$-process elements. Here we present yields of the innermost ejecta in 3D magnetorotational CCSN models simulated using the CoCoNuT-FMT code. Strong magnetic fields tap the rotational energy of the proto-neutron star and lead to earlier and more energetic ($\sim 3\times 10^{51}$ erg) explosions than typical neutrino-driven CCSNe. Compared to a reference non-magnetic model, the ejecta in the magnetorotational models have much more neutron-rich components with Ye down to $\sim$0.25. Our post-processing calculations with the reaction network SkyNet show significant production of weak $r$-process elements up to mass number $\sim$130. We find negligible differences in the synthesis of heavy elements between two magnetorotational models with different initial field strength of 10$^{10}$ and 10$^{12}$ G, in accord with their similar explosion dynamics. The magnetorotational models produce about $\sim$0.19 and 0.14 Msun of radioactive $^{56}$Ni, on the low end of inferred hypernova nickel masses. The yields are publicly available at Zenodo: doi:https://doi.org/10.5281/zenodo.10578981 for comparison with stellar abundance patterns, inclusion in modelling galactic chemical evolution, and comparison with other yield calculations. Our results add to the yet restricted corpus of nucleosynthesis yields from 3D magnetorotational supernova simulations and will help quantify yield uncertainties.

We propose a flexible method for estimating luminosity functions (LFs) based on kernel density estimation (KDE), the most popular nonparametric density estimation approach developed in modern statistics, to overcome issues surrounding binning of LFs. One challenge in applying KDE to LFs is how to treat the boundary bias problem, since astronomical surveys usually obtain truncated samples predominantly due to the flux-density limits of surveys. We use two solutions, the transformation KDE method ($\hat{\phi}_{\mathrm{t}}$), and the transformation-reflection KDE method ($\hat{\phi}_{\mathrm{tr}}$) to reduce the boundary bias. We develop a new likelihood cross-validation criterion for selecting optimal bandwidths, based on which, the posterior probability distribution of bandwidth and transformation parameters for $\hat{\phi}_{\mathrm{t}}$ and $\hat{\phi}_{\mathrm{tr}}$ are derived within a Markov chain Monte Carlo (MCMC) sampling procedure. The simulation result shows that $\hat{\phi}_{\mathrm{t}}$ and $\hat{\phi}_{\mathrm{tr}}$ perform better than the traditional binned method, especially in the sparse data regime around the flux-limit of a survey or at the bright-end of the LF. To further improve the performance of our KDE methods, we develop the transformation-reflection adaptive KDE approach ($\hat{\phi}_{\mathrm{tra}}$). Monte Carlo simulations suggest that it has a good stability and reliability in performance, and is around an order of magnitude more accurate than using the binned method. By applying our adaptive KDE method to a quasar sample, we find that it achieves estimates comparable to the rigorous determination by a previous work, while making far fewer assumptions about the LF. The KDE method we develop has the advantages of both parametric and non-parametric methods.

We present a newly developed version of BayeSED, a general Bayesian approach to the spectral energy distribution (SED) fitting of galaxies. The new BayeSED code has been systematically tested on a mock sample of galaxies. The comparison between estimated and inputted value of the parameters show that BayeSED can recover the physical parameters of galaxies reasonably well. We then applied BayeSED to interpret the SEDs of a large Ks-selected sample of galaxies in the COSMOS/UltraVISTA field with stellar population synthesis models. With the new BayeSED code, a Bayesian model comparison of stellar population synthesis models has been done for the first time. We found that the model by Bruzual & Charlot (2003), statistically speaking, has larger Bayesian evidence than the model by Maraston (2005) for the Ks-selected sample. Besides, while setting the stellar metallicity as a free parameter obviously increases the Bayesian evidence of both models, varying the IMF has a notable effect only on the Maraston (2005) model. Meanwhile, the physical parameters estimated with BayeSED are found to be generally consistent with those obtained with the popular grid-based FAST code, while the former exhibits more natural distributions. Based on the estimated physical parameters of galaxies in the sample, we qualitatively classified the galaxies in the sample into five populations that may represent galaxies at different evolution stages or in different environments. We conclude that BayeSED could be a reliable and powerful tool for investigating the formation and evolution of galaxies from the rich multi-wavelength observations currently available. A binary version of MPI parallelized BayeSED code is publicly available at this https URL.

We present absorption variability results for 134 bona fide \mgii\ broad absorption line (BAL) quasars at 0.46~$\lesssim z \lesssim$~2.3 covering days to $\sim$ 10 yr in the rest frame. We use multiple-epoch spectra from the Sloan Digital Sky Survey, which has delivered the largest such BAL-variability sample ever studied. \mgii-BAL identifications and related measurements are compiled and presented in a catalog. We find a remarkable time-dependent asymmetry in EW variation from the sample, such that weakening troughs outnumber strengthening troughs, the first report of such a phenomenon in BAL variability. Our investigations of the sample further reveal that (i) the frequency of BAL variability is significantly lower (typically by a factor of 2) than that from high-ionization BALQSO samples; (ii) \mgii\ BAL absorbers tend to have relatively high optical depths and small covering factors along our line of sight; (iii) there is no significant EW-variability correlation between \mgii\ troughs at different velocities in the same quasar; and (iv) the EW-variability correlation between \mgii\ and \aliii\ BALs is significantly stronger than that between \mgii\ and \civ\ BALs at the same velocities. These observational results can be explained by a combined transverse-motion/ionization-change scenario, where transverse motions likely dominate the strengthening BALs while ionization changes and/or other mechanisms dominate the weakening BALs.

Although type Ia supernovae (SNe Ia) are very useful in many astrophysical fields, their exact progenitor nature is still unclear. A basic method to distinguish the different progenitor models is to search the signal from the single degenerate (SD) model, e.g. the signal for the existence of a non-degenerate companion before or after supernova explosion. Observationally, some SNe Ia show such signal, while the others do not. Here, we propose a universal model to explain these observations based on the spin-up/spin-down model, in which a white dwarf (WD) will experience a spin-down phase before supernova explosion, and the spin-down timescale is determined by its initial mass, i.e. the more massive the initial WD, the shorter the spin-down timescale and then the more likely the SN Ia to show the SD signature. Therefore, our model predicts that the SNe Ia from hybrid carbon-oxygen-neon WDs are more likely to show the SD signature observationally, as some peculiar SNe Ia showed.

We measure the total mass-density profiles out to three effective radii for a sample of 63 $z \sim 0.5$, massive early-type galaxies (ETGs) acting as strong gravitational lenses through a joint analysis of lensing and stellar dynamics. The compilation is selected from three galaxy-scale strong-lens samples including the Baryon Oscillation Spectroscopic Survey (BOSS) Emission-Line Lens Survey (BELLS), BELLS for GALaxy-Ly$\alpha$ EmitteR sYstems Survey, and Strong Lensing Legacy Survey (SL2S). Utilizing the wide source-redshift coverage (0.8--3.5) provided by these three samples, we build a statistically significant ensemble of massive ETGs for which robust mass measurements can be achieved within a broad range of Einstein radii up to three effective radii. Characterizing the three-dimensional total mass-density distribution by a power-law profile as $\rho \propto r^{-\gamma}$, we find that the average logarithmic density slope for the entire sample is $\langle\gamma\rangle=2.000_{-0.032}^{+0.033}$ ($68\%$CL) with an intrinsic scatter $\delta=0.180_{-0.028}^{+0.032}$. Further parameterizing $\langle\gamma\rangle$ as a function of redshift $z$ and ratio of Einstein radius to effective radius $R_{ein}/R_{eff}$, we find the average density distributions of these massive ETGs become steeper at larger radii and later cosmic times with magnitudes $\mathrm{d} \langle\gamma\rangle / \mathrm{d}z = -0.309_{-0.160}^{+0.166}$ and $\mathrm{d} \langle\gamma\rangle / \mathrm{d} \log_{10} \frac{R_{ein}}{R_{eff}} = 0.194_{-0.083}^{+0.092}$.

Spherically symmetric accretion incorporating self-gravity constitutes a three-point boundary value problem (TPBVP) governed by constraints at the outer boundary, sonic point, and accretor surface. Previous studies have two limitations: either employing an incorrect formula for self-gravity potential in analytical treatments, or introducing additional input parameters in numerical implementations to circumvent solving the full TPBVP. To address these issues, we present a self-consistent TPBVP formulation, solved using the relaxation method. We also derive approximate analytical formulae that enable rapid estimates of self-gravity effects. Our analysis identifies a dimensionless parameter $\beta \equiv 2G \bar{\rho} r_\mathrm{out}^2/a_\mathrm{out}^2$ that characterizes the strength of self-gravity, where $\bar{\rho}$ and $r_\mathrm{out}$ are the mean density and outer radius of the flow, respectively, and $a_\mathrm{out}$ is the adiabatic sound speed of the external medium. For practical estimation, $\bar{\rho}$ may be approximated by the external medium density $\rho_\mathrm{out}$. We identify an upper limit for $\beta$, beyond which steady accretion becomes unsustainable -- a behavior consistent with classical gravitational instability that previous studies failed to capture. The accretion rate enhancement decreases monotonically as the adiabatic index $\gamma$ increases. For $\gamma=5/3$, self-gravity ceases to augment the accretion rate. These theoretical predictions are validated by our numerical solutions. We further apply our results to two astrophysical scenarios: hyper-Eddington accretion onto supermassive black hole seeds in the early Universe, where self-gravity is significant; and accretion onto stellar-mass objects embedded in active galactic nuclei (AGN) disks, where self-gravity is non-negligible under certain conditions and should be evaluated using $\beta$.

Hot dust-obscured galaxies (Hot DOGs) are a rare population of hyperluminous dust-obscured quasars discovered by the Wide-field Infrared Survey Explorer (WISE) all-sky survey. The heavy circumnuclear dust obscuration allows only a small amount of scattered light from the obscured quasar to escape, enabling the decomposition of the stellar component from the total flux. The presence of scattered light enables the redshift of the source and the properties of the black hole to be obtained from SDSS and SDSS-related literature. From WISE and SDSS data, we select 11 hyperluminous Hot DOGs at $z=1.5-3.7$ with bolometric luminosities $L_{\rm bol} \gtrsim 10^{47}\,\mathrm{erg \ s^{-1}}$. We investigate the $M_{\rm BH}-M_\star$ relation in these sources using Bayesian spectral energy distribution (SED) fitting or with extra constraints from \textit{Hubble Space Telescope} (HST) image decomposition. Stellar masses are successfully derived for eight Hot DOGs. We find high Eddington ratios $\lambda_{\rm Edd}$ in these Hot DOGs, with the median value of 1.05 and the maximum value close to 3. The super-Eddington accretion may be associated with the overdense environments of Hot DOGs. We find no significant differences in the $M_{\rm BH}/M_\star$ of these Hot DOGs compared to the local relation, suggesting that these dust-obscured quasars are the progenitors of massive early-type galaxies. We speculate that the subsequent evolution of Hot DOGs may be significantly influenced by AGN feedback and remain on the local relation.