An Extension of a Congruence by Tauraso

For a positive integer $n$ let $H_n=\sum_{k=1}^{n}1/n$ be the $n$th harmonic number. In this note we prove that for any prime $p\ge 7$, $$\sum_{k=1}^{p-1}\frac{H_k}{k^2}\equiv \sum_{k=1}^{p-1}\frac{H_k^2}{k} \equiv\frac{3}{2p}\sum_{k=1}^{p-1}\frac{1}{k^2}\pmod{p^2}.$$ Notice that the first part of this congruence is recently proposed by R. Tauraso as a problem in Amer. Math. Monthly. In our elementary proof of the second part of the above congruence we use certain classical congruences modulo a prime and the square of a prime, some congruences involving harmonic numbers and a combinatorial identity due to V. Hern\'{a}ndez.Comment: 13 page

Stellar feedback and bulge formation in clumpy discs

We use numerical simulations of isolated galaxies to study the effects of stellar feedback on the formation and evolution of giant star-forming gas ‘clumps’ in high-redshift, gas-rich galaxies. Such galactic discs are unstable to the formation of bound gas-rich clumps whose properties initially depend only on global disc properties, not the microphysics of feedback. In simulations without stellar feedback, clumps turn an order-unity fraction of their mass into stars and sink to the centre, forming a large bulge and kicking most of the stars out into a much more extended stellar envelope. By contrast, strong radiative stellar feedback disrupts even the most massive clumps after they turn ∼10–20 per cent of their mass into stars, in a time-scale of ∼10–100 Myr, ejecting some material into a superwind and recycling the rest of the gas into the diffuse interstellar medium (ISM). This suppresses the bulge formation rate by direct ‘clump coalescence’ by a factor of several. However, the galactic discs do undergo significant internal evolution in the absence of mergers: clumps form and disrupt continuously and torque gas to the galactic centre. The resulting evolution is qualitatively similar to bar/spiral evolution in simulations with a more homogeneous ISM

A p-adic Eisenstein measure for unitary groups

We construct a p-adic Eisenstein measure with values in the space of p-adic automorphic forms on certain unitary groups. Using this measure, we p-adically interpolate certain special values of both holomorphic and non-holomorphic Eisenstein series, as both the archimedean and the p-adic weights of the Eisenstein series vary.Comment: Accepted for publication in the Journal f\"ur die reine und angewandte Mathematik (Crelle's Journal

Moving mesh cosmology: tracing cosmological gas accretion

We investigate the nature of gas accretion onto haloes and galaxies at z=2 using cosmological hydrodynamic simulations run with the moving mesh code AREPO. Implementing a Monte Carlo tracer particle scheme to determine the origin and thermodynamic history of accreting gas, we make quantitative comparisons to an otherwise identical simulation run with the smoothed particle hydrodynamics (SPH) code GADGET-3. Contrasting these two numerical approaches, we find significant physical differences in the thermodynamic history of accreted gas in haloes above 10^10.5 solar masses. In agreement with previous work, GADGET simulations show a cold fraction near unity for galaxies forming in massive haloes, implying that only a small percentage of accreted gas heats to an appreciable fraction of the virial temperature during accretion. The same galaxies in AREPO show a much lower cold fraction, <20% in haloes above 10^11 solar masses. This results from a hot gas accretion rate which, at this same halo mass, is an order of magnitude larger than with GADGET, while the cold accretion rate is also lower. These discrepancies increase for more massive systems, and we explain both as due to numerical inaccuracies in the standard formulation of SPH. We also observe that the relatively sharp transition from cold to hot mode dominated accretion, at a halo mass of ~10^11, is a consequence of comparing past gas temperatures to a constant threshold value independent of virial temperature. Examining the spatial distribution of accreting gas, we find that gas filaments in GADGET tend to remain collimated and flow coherently to small radii, or artificially fragment and form a large number of purely numerical "blobs". Similar gas streams in AREPO show increased heating and disruption at 0.25-0.5 virial radii and contribute to the hot gas accretion rate in a manner distinct from classical cooling flows.Comment: 21 pages, 12 figures. MNRAS accepted (in press). High-resolution images can be found at http://www.cfa.harvard.edu/itc/research/movingmeshcosmology

A general class of Lagrangian smoothed particle hydrodynamics methods and implications for fluid mixing problems

Various formulations of smoothed particle hydrodynamics (SPH) have been proposed, intended to resolve certain difficulties in the treatment of fluid mixing instabilities. Most have involved changes to the algorithm which either introduces artificial correction terms or violates what is arguably the greatest advantage of SPH over other methods: manifest conservation of energy, entropy, momentum and angular momentum. Here, we show how a class of alternative SPH equations of motion (EOM) can be derived self-consistently from a discrete particle Lagrangian – guaranteeing manifest conservation – in a manner which tremendously improves treatment of these instabilities and contact discontinuities. Saitoh & Makino recently noted that the volume element used to discretize the EOM does not need to explicitly invoke the mass density (as in the ‘standard’ approach); we show how this insight, and the resulting degree of freedom, can be incorporated into the rigorous Lagrangian formulation that retains ideal conservation properties and includes the ‘∇h’ terms that account for variable smoothing lengths. We derive a general EOM for any choice of volume element (particle ‘weights’) and method of determining smoothing lengths. We then specify this to a ‘pressure–entropy formulation’ which resolves problems in the traditional treatment of fluid interfaces. Implementing this in a new version of the GADGET code, we show it leads to good performance in mixing experiments (e.g. Kelvin–Helmholtz and ‘blob’ tests). And conservation is maintained even in strong shock/blastwave tests, where formulations without manifest conservation produce large errors. This also improves the treatment of subsonic turbulence and lessens the need for large kernel particle numbers. The code changes are trivial and entail no additional numerical expense. This provides a general framework for self-consistent derivation of different ‘flavours’ of SPH

Following the flow: tracer particles in astrophysical fluid simulations

We present two numerical schemes for passive tracer particles in the hydrodynamical moving-mesh code AREPO, and compare their performance for various problems, from simple setups to cosmological simulations. The purpose of tracer particles is to allow the flow to be followed in a Lagrangian way, tracing the evolution of the fluid with time, and allowing the thermodynamical history of individual fluid parcels to be recorded. We find that the commonly-used `velocity field tracers', which are advected using the fluid velocity field, do not in general follow the mass flow correctly, and explain why this is the case. This method can result in orders-of-magnitude biases in simulations of driven turbulence and in cosmological simulations, rendering the velocity field tracers inappropriate for following these flows. We then discuss a novel implementation of `Monte Carlo tracers', which are moved along with fluid cells, and are exchanged probabilistically between them following the mass flux. This method reproduces the mass distribution of the fluid correctly. The main limitation of this approach is that it is more diffusive than the fluid itself. Nonetheless, we show that this novel approach is more reliable than what has been employed previously and demonstrate that it is appropriate for following hydrodynamical flows in mesh-based codes. The Monte Carlo tracers can also naturally be transferred between fluid cells and other types of particles, such as stellar particles, so that the mass flow in cosmological simulations can be followed in its entirety.Comment: Accepted for publication in MNRAS, minor updates to match accepted version. 19 pages, 14 figure

Star formation in galaxy mergers with realistic models of stellar feedback and the interstellar medium

We use hydrodynamic simulations with detailed, explicit models for stellar feedback to study galaxy mergers. These high-resolution (∼1 pc) simulations follow the formation and destruction of individual giant molecular clouds (GMC) and star clusters. We find that the final starburst is dominated by in situ star formation, fuelled by gas which flows inwards due to global torques. The resulting high gas density results in rapid star formation. The gas is self-gravitating, and forms massive (≲10¹⁰ M_⊙) GMC and subsequently super star clusters (with masses up to 10⁸ M_⊙). However, in contrast to some recent simulations, the bulk of new stars which eventually form the central bulge are not born in super-clusters which then sink to the centre of the galaxy. This is because feedback efficiently disperses GMC after they turn several per cent of their mass into stars. In other words, most of the mass that reaches the nucleus does so in the form of gas. The Kennicutt–Schmidt law emerges naturally as a consequence of feedback balancing gravitational collapse, independent of the small-scale star formation microphysics. The same mechanisms that drive this relation in isolated galaxies, in particular radiation pressure from infrared photons, extend, with no fine-tuning, over seven decades in star formation rate (SFR) to regulate star formation in the most extreme starburst systems with densities ≳10⁴ M_⊙ pc⁻². This feedback also drives super-winds with large mass-loss rates; however, a significant fraction of the wind material falls back on to the discs at later times, leading to higher post-starburst SFRs in the presence of stellar feedback. This suggests that strong active galactic nucleus feedback may be required to explain the sharp cut-offs in SFR that are observed in post-merger galaxies. We compare the results to those from simulations with no explicit resolution of GMC or feedback [‘effective equation-of-state’ (EOS) models]. We find that global galaxy properties are similar between EOS and resolved-feedback models. The relic structure and mass profile, and the total mass of stars formed in the nuclear starburst are quite similar, as is the morphological structure during and after mergers (tails, bridges, etc.). Disc survival in sufficiently gas rich mergers is similar in the two cases, and the new models follow the same scalings as derived for the efficiency of disc re-formation after a merger as derived from previous work with the simplified EOS models. While the global galaxy properties are similar between EOS and feedback models, subgalaxy-scale properties and the SFRs can be quite different: the more detailed models exhibit significantly higher star formation in tails and bridges (especially in shocks), and allow us to resolve the formation of super star clusters. In the new models, the star formation is more strongly time-variable and drops more sharply between close passages. The instantaneous burst enhancement can be higher or lower, depending on the details of the orbit and initial structural properties of the galaxies; first-passage bursts are more sensitive to these details than those at the final coalescence

Star Formation in Galaxy Mergers with Realistic Models of Stellar Feedback & the Interstellar Medium

We use simulations with realistic models for stellar feedback to study galaxy mergers. These high resolution (1 pc) simulations follow formation and destruction of individual GMCs and star clusters. The final starburst is dominated by in situ star formation, fueled by gas which flows inwards due to global torques. The resulting high gas density results in rapid star formation. The gas is self gravitating, and forms massive (~10^10 M_sun) GMCs and subsequent super-starclusters (masses up to 10^8 M_sun). However, in contrast to some recent simulations, the bulk of new stars which eventually form the central bulge are not born in superclusters which then sink to the center of the galaxy, because feedback efficiently disperses GMCs after they turn several percent of their mass into stars. Most of the mass that reaches the nucleus does so in the form of gas. The Kennicutt-Schmidt law emerges naturally as a consequence of feedback balancing gravitational collapse, independent of the small-scale star formation microphysics. The same mechanisms that drive this relation in isolated galaxies, in particular radiation pressure from IR photons, extend over seven decades in SFR to regulate star formation in the most extreme starbursts (densities >10^4 M_sun/pc^2). Feedback also drives super-winds with large mass loss rates; but a significant fraction of the wind material falls back onto the disks at later times, leading to higher post-starburst SFRs in the presence of stellar feedback. Strong AGN feedback is required to explain sharp cutoffs in star formation rate. We compare the predicted relic structure, mass profile, morphology, and efficiency of disk survival to simulations which do not explicitly resolve GMCs or feedback. Global galaxy properties are similar, but sub-galactic properties and star formation rates can differ significantly.Comment: 17 pages, 13 figures (+appendices), MNRAS accepted (matches published). Movies of the simulations are available at http://www.tapir.caltech.edu/~phopkins/Site/Movies_sbw_mgr.htm