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1. Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems

   https://doi.org/10.1088/1361-6544/ac50bb  

   Chen, Qinbo; de la Llave, Rafael (March 2022, Nonlinearity)

   Abstract The genericity of Arnold diffusion in the analytic category is an open problem. In this paper, we study this problem in the followinga prioriunstable Hamiltonian system with a time-periodic perturbation ${\mathcal{H}}_{\epsilon }\left(p,q,I,\phi ,t\right)=h\left(I\right)+\sum _{i=1}^n\pm \left(\frac{1}{2}{p}_i^2+V_i\left(q_i\right)\right)+\epsilon H_1\left(p,q,I,\phi ,t\right),$where $\left(p,q\right)\in {\mathbb{R}}^n\times {\mathbb{T}}^n$, $\left(I,\phi \right)\in {\mathbb{R}}^d\times {\mathbb{T}}^d$withn,d⩾ 1,V_iare Morse potentials, andɛis a small non-zero parameter. The unperturbed Hamiltonian is not necessarily convex, and the induced inner dynamics does not need to satisfy a twist condition. Using geometric methods we prove that Arnold diffusion occurs for generic analytic perturbationsH₁. Indeed, the set of admissibleH₁isC^ωdense andC³open (a fortiori,C^ωopen). Our perturbative technique for the genericity is valid in theC^ktopology for allk∈ [3, ∞) ∪ {∞,ω}. 

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2. Toward Astrometric Constraints on a Supermassive Black Hole Binary in the Early-type Galaxy NGC 4472

   https://doi.org/10.3847/1538-4357/ac680b  

   Wrobel, J. M.; Lazio, T. J. W. (May 2022, The Astrophysical Journal)

   Abstract The merger of two galaxies, each hosting a supermassive black hole (SMBH) of mass 10⁶M_⊙or more, could yield a bound SMBH binary. For the early-type galaxy NGC 4472, we study how astrometry with a next-generation Very Large Array could be used to monitor the reflex motion of the primary SMBH of massM_pri, as it is tugged on by the secondary SMBH of mass $M_{\sec }$. Casting the orbit of the putative SMBH binary in terms of its periodP, semimajor axisa_bin, and mass ratio $q=M_{\sec }/M_{\mathrm{pri}}\le 1$, we find the following: (1) Orbits with fiducial periods ofP= 4 yr and 40 yr could be spatially resolved and monitored. (2) For a 95% accuracy of 2μas per monitoring epoch, subparsec values ofa_bincould be accessed over a range of mass ratios notionally encompassing major $\left(q>\frac{1}{4}\right)$and minor $\left(q<\frac{1}{4}\right)$galaxy mergers. (3) If no reflex motion is detected forM_priafter 1 (10) yr of monitoring, an SMBH binary with periodP= 4 (40) yr and mass ratioq> 0.01 (0.003) could be excluded. This would suggest no present-day evidence for a past major merger like that recently simulated, where scouring by aq∼ 1 SMBH binary formed a stellar core with kinematic traits like those of NGC 4472. (4) Astrometric monitoring could independently check the upper limits onqfrom searches for continuous gravitational waves from NGC 4472. 

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3. Acceleration and Spectral Redistribution of Cosmic Rays in Radio-jet Shear Flows

   https://doi.org/10.3847/1538-4357/acfda9  

   Webb, G. M.; Xu, Y.; Biermann, P. L.; Al-Nussirat, S.; Mostafavi, P.; Li, G.; Barghouty, A. F.; Zank, G. P. (November 2023, The Astrophysical Journal)

   Abstract A steady-state, semi-analytical model of energetic particle acceleration in radio-jet shear flows due to cosmic-ray viscosity obtained by Webb et al. is generalized to take into account more general cosmic-ray boundary spectra. This involves solving a mixed Dirichlet–Von Neumann boundary value problem at the edge of the jet. The energetic particle distribution functionf₀(r,p) at cylindrical radiusrfrom the jet axis (assumed to lie along thez-axis) is given by convolving the particle momentum spectrum $f_0(\infty ,p\prime )$with the Green’s function $G(r,p;p\prime )$, which describes the monoenergetic spectrum solution in which $f_0\to \delta (p-p\prime )$asr→ ∞ . Previous work by Webb et al. studied only the Green’s function solution for $G(r,p;p\prime )$. In this paper, we explore for the first time, solutions for more general and realistic forms for $f_0(\infty ,p\prime )$. The flow velocityu=u(r)e_zis along the axis of the jet (thez-axis).uis independent ofz, andu(r) is a monotonic decreasing function ofr. The scattering time $\tau {(r,p)={\tau }_0\left(p/p_0\right)}^{\alpha }$in the shear flow region 0 <r<r₂, and $\tau {(r,p)={\tau }_0\left(p/p_0\right)}^{\alpha }{\left(r/r_2\right)}^s$, wheres> 0 in the regionr>r₂is outside the jet. Other original aspects of the analysis are (i) the use of cosmic ray flow lines in (r,p) space to clarify the particle spatial transport and momentum changes and (ii) the determination of the probability distribution ${\psi }_p(r,p;p\prime )$that particles observed at (r,p) originated fromr→ ∞ with momentum $p\prime$. The acceleration of ultrahigh-energy cosmic rays in active galactic nuclei jet sources is discussed. Leaky box models for electron acceleration are described. 

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4. Convolution identities for divisor sums and modular forms

   https://doi.org/10.1073/pnas.2322320121  

   Fedosova, Ksenia; Klinger-Logan, Kim; Radchenko, Danylo (October 2024, Proceedings of the National Academy of Sciences)

   Ribet, Kenneth (Ed.)

   We consider certain convolution sums that are the subject of a conjecture by Chester, Green, Pufu, Wang, and Wen in string theory. We prove a generalized form of their conjecture, explicitly evaluating absolutely convergent sums $\sum _{n_1\in \mathbb{Z}\setminus \{0,n\}}\phi (n_1,n-n_1){\sigma }_{2m_1}\left(n_1\right){\sigma }_{2m_2}(n-n_1),$ where $\phi (n_1,n_2)$is a Laurent polynomial with logarithms. Contrary to original expectations, such convolution sums, suitably extended to $n_1\in \{0,n\}$, do not vanish, but instead, they carry number theoretic meaning in the form of Fourier coefficients of holomorphic cusp forms. 

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5. The Hubble Space Telescope Survey of M31 Satellite Galaxies. II. The Star Formation Histories of Ultrafaint Dwarf Galaxies

   https://doi.org/10.3847/1538-4357/acf46f  

   Savino, Alessandro; Weisz, Daniel R.; Skillman, Evan D.; Dolphin, Andrew; Cole, Andrew A.; Kallivayalil, Nitya; Wetzel, Andrew; Anderson, Jay; Besla, Gurtina; Boylan-Kolchin, Michael; et al (October 2023, The Astrophysical Journal)

   Abstract We present the lifetime star formation histories (SFHs) for six ultrafaint dwarf (UFD;M_V> − 7.0, $4.9<{\log }_{10}\left(M_{*}\right(z=0\left)/M_{\odot }\right)<5.5$) satellite galaxies of M31 based on deep color–magnitude diagrams constructed from Hubble Space Telescope imaging. These are the first SFHs obtained from the oldest main-sequence turnoff of UFDs outside the halo of the Milky Way (MW). We find that five UFDs formed at least 50% of their stellar mass byz= 5 (12.6 Gyr ago), similar to known UFDs around the MW, but that 10%–40% of their stellar mass formed at later times. We uncover one remarkable UFD, Andxiii, which formed only 10% of its stellar mass byz= 5, and 75% in a rapid burst atz∼ 2–3, a result that is robust to choices of underlying stellar model and is consistent with its predominantly red horizontal branch. This “young” UFD is the first of its kind and indicates that not all UFDs are necessarily quenched by reionization, which is consistent with predictions from several cosmological simulations of faint dwarf galaxies. SFHs of the combined MW and M31 samples suggest reionization did not homogeneously quench UFDs. We find that the least-massive MW UFDs (M_*(z= 5) ≲ 5 × 10⁴M_⊙) are likely quenched by reionization, whereas more-massive M31 UFDs (M_*(z= 5) ≳ 10⁵M_⊙) may only have their star formation suppressed by reionization and quench at a later time. We discuss these findings in the context of the evolution and quenching of UFDs. 

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