The self-energy of the electron: a quintessential problem in the development of QED

The development of Quantum Electrodynamics (QED) is sketched from its earliest beginnings until the formulations of 1949, using the example of the divergent self-energy of the electron as a quintessential problem of the 1930's-40's. The lack of progress towards solving this problem led researchers to believe that after the conceptual revolution of quantum mechanics a new conceptual change was needed. It took a war and a new generation of algorithmically inclined physicists to pursue the conventional route of regularization and renormalization that led to the solution in 1947-1949. Some remarks on contemporary high energy physics are made.Comment: 19 pages, 4 figures; Submitted to Studies in History and Philosophy of Modern Physic

Detecting the disruption of dark-matter halos with stellar streams

Narrow stellar streams in the Milky Way halo are uniquely sensitive to dark-matter subhalos, but many of these subhalos may be tidally disrupted. I calculate the interaction between stellar and dark-matter streams using analytical and $N$-body calculations, showing that disrupting objects can be detected as low-concentration subhalos. Through this effect, we can constrain the lumpiness of the halo as well as the orbit and present position of individual dark-matter streams. This will have profound implications for the formation of halos and for direct and indirect-detection dark-matter searches.Comment: PRL in press; 4 pages; all code available at https://github.com/jobovy/stream-strea

galpy: A Python Library for Galactic Dynamics

I describe the design, implementation, and usage of galpy, a Python package for galactic-dynamics calculations. At its core, galpy consists of a general framework for representing galactic potentials both in Python and in C (for accelerated computations); galpy functions, objects, and methods can generally take arbitrary combinations of these as arguments. Numerical orbit integration is supported with a variety of Runge-Kutta-type and symplectic integrators. For planar orbits, integration of the phase-space volume is also possible. galpy supports the calculation of action-angle coordinates and orbital frequencies for a given phase-space point for general spherical potentials, using state-of-the-art numerical approximations for axisymmetric potentials, and making use of a recent general approximation for any static potential. A number of different distribution functions (DFs) are also included in the current release; currently these consist of two-dimensional axisymmetric and non-axisymmetric disk DFs, a three-dimensional disk DF, and a DF framework for tidal streams. I provide several examples to illustrate the use of the code. I present a simple model for the Milky Way's gravitational potential consistent with the latest observations. I also numerically calculate the Oort functions for different tracer populations of stars and compare it to a new analytical approximation. Additionally, I characterize the response of a kinematically-warm disk to an elliptical m=2 perturbation in detail. Overall, galpy consists of about 54,000 lines, including 23,000 lines of code in the module, 11,000 lines of test code, and about 20,000 lines of documentation. The test suite covers 99.6% of the code. galpy is available at http://github.com/jobovy/galpy with extensive documentation available at http://galpy.readthedocs.org/en/latest .Comment: ApJS, in press; 29 pages, 30 figures, including many code examples. galpy is available at http://github.com/jobovy/galpy and code to reproduce this paper's figures can be found at http://github.com/jobovy/galpy-paper-figure

Dynamical modeling of tidal streams

I present a new framework for modeling the dynamics of tidal streams. The framework consists of simple models for the initial action-angle distribution of tidal debris, which can be straightforwardly evolved forward in time. Taking advantage of the essentially one-dimensional nature of tidal streams, the transformation to position-velocity coordinates can be linearized and interpolated near a small number of points along the stream, thus allowing for efficient computations of a stream's properties in observable quantities. I illustrate how to calculate the stream's average location (its "track"') in different coordinate systems, how to quickly estimate the dispersion around its track, and how to draw mock stream data. As a generative model, this framework allows one to compute the full probability distribution function and marginalize over or condition it on certain phase-space dimensions as well as convolve it with observational uncertainties. This will be instrumental in proper data analysis of stream data. In addition to providing a computationally-efficient practical tool for modeling the dynamics of tidal streams, the action-angle nature of the framework helps elucidate how the observed width of the stream relates to the velocity dispersion or mass of the progenitor, and how the progenitors of "orphan"' streams could be located. The practical usefulness of the proposed framework crucially depends on the ability to calculate action-angle variables for any orbit in any gravitational potential. A novel method for calculating actions, frequencies, and angles in any static potential using a single orbit integration is described in an Appendix

Stellar Inventory of the Solar Neighborhood using Gaia DR1

The absolute number and the density profiles of different types of stars in the solar neighborhood are a fundamental anchor for studies of the initial mass function, stellar evolution, and galactic structure. Using data from the Gaia DR1 Tycho-Gaia Astrometric Solution, we reconstruct Gaia's selection function and we determine Gaia's volume completeness, the local number density, and the vertical profiles of different spectral types along the main sequence from early A stars to late K stars as well as along the giant branch. We clearly detect the expected flattening of the stellar density profile near the mid-plane for all stellar types: All vertical profiles are well represented by sech^2 profiles, with scale heights ranging from ~50 pc for A stars to ~150 pc for G and K dwarfs and giants. We determine the luminosity function along the main sequence for M_V ~ $0.72 M_\odot$) and along the giant branch for M_J >~ -2.5. Converting this to a mass function, we find that the high-mass (M > $1\,M_\odot$) present-day mass function along the main sequence is d n / d M = 0.016 $(M/M_\odot)^{-4.7}$ stars/pc^3/$M_\odot$. Extrapolating below M = $0.72\,M_\odot$, we find a total mid-plane stellar density of 0.040+/-0.002 $M_\odot$/pc^3. Giants contribute 0.00039+/-0.00001 stars/pc^3 or about 0.00046+/-0.00005 $M_\odot$/pc^3. The star-formation rate surface density is \Sigma(t) = 7+/-1 exp(-t/[7+/-1 Gyr]) $M_\odot$/pc^2/Gyr. Overall, we find that Gaia DR1's selection biases are manageable and allow a detailed new inventory of the solar neighborhood to be made that agrees with and extends previous studies. This bodes well for mapping the Milky Way with the full Gaia data set.Comment: TGAS selection-function code available at https://github.com/jobovy/gaia_tools, paper-specific code available at https://github.com/jobovy/tgas-completenes

The chemical homogeneity of open clusters

Determining the level of chemical homogeneity in open clusters is of fundamental importance in the study of the evolution of star-forming clouds and that of the Galactic disk. Yet limiting the initial abundance spread in clusters has been hampered by difficulties in obtaining consistent spectroscopic abundances for different stellar types. Without reference to any specific model of stellar photospheres, a model for a homogeneous cluster is that it forms a one-dimensional sequence, with any differences between members due to variations in stellar mass and observational uncertainties. I present a novel method for investigating the abundance spread in open clusters that tests this one-dimensional hypothesis at the level of observed stellar spectra, rather than constraining homogeneity using derived abundances as traditionally done. Using high-resolution APOGEE spectra for 49 giants in M67, NGC 6819, and NGC 2420 I demonstrate that these spectra form one-dimensional sequences for each cluster. With detailed forward modeling of the spectra and Approximate Bayesian Computation, I derive strong limits on the initial abundance spread of 15 elements: <0.01 (0.02) dex for C and Fe, <~0.015 (0.03) dex for N, O, Mg, Si, and Ni, <~0.02 (0.03) dex for Al, Ca, and Mn, and <~0.03 (0.05) dex for Na, S, K, Ti, and V (at 68% and 95% confidence, respectively). The strong limits on C and O imply that no pollution by massive core-collapse supernovae occurred during star formation in open clusters, which, thus, need to form within <~6 Myr. Further development of this and related techniques will bring the power of differential abundances to stars other than solar twins in large spectroscopic surveys and will help unravel the history of star formation and chemical enrichment in the Milky Way through chemical tagging

Vertical waves in the solar neighbourhood in Gaia DR2

The vertical structure and dynamics of stars in our local Galactic neighbourhood contains much information about the local distribution of visible and dark matter and of perturbations to the Milky Way disc. We use data on the positions and velocities of stars in the solar neighbourhood from \gaia\ DR2 and large spectroscopic surveys to investigate the vertical number counts and mean-velocity trend as a function of distance from the local Galactic mid-plane. We perform a detailed measurement of the wave-like North-South asymmetry in the vertical number counts, which reveals a number of deficits at heights $\approx 0.4\,\mathrm{kpc}$, $\approx 0.9\,\mathrm{kpc}$, and $\approx 1.5\,\mathrm{kpc}$, and peaks at $\approx 0.2\,\mathrm{kpc}$, $\approx 0.7\,\mathrm{kpc}$, and $\approx 1.1\,\mathrm{kpc}$. We find that the asymmetry pattern is independent of colour. The mean vertical velocity is almost constant to $<1\,\mathrm{km\,s}^{-1}$ within a few 100 pc from the mid-plane and then displays a North-South symmetric dip at $\approx0.5\,\mathrm{kpc}$ with an amplitude of $\approx 2\,\mathrm{km\,s}^{-1}$ that is a plausible velocity counterpart to the main number-count dip at a similar height. Thus, with \gaia\ DR2 we confirm at high fidelity that the local Galactic disc is undergoing a wave-like oscillation and a dynamically-consistent observational picture of the perturbed local vertical structure emerges for the first time. We also present the most precise and accurate determination of the Sun's height above the local Galactic mid-plane, correcting for any asymmetry in the vertical density: $z_\odot = 20.8 \pm 0.3\,\mathrm{pc}$.Comment: Code available at https://github.com/morganb-phys/VWaves-GaiaDR

A direct dynamical measurement of the Milky Way's disk surface density profile, disk scale length, and dark matter profile at 4 kpc < R < 9 kpc

We present and apply rigorous dynamical modeling with which we infer unprecedented constraints on the stellar and dark matter mass distribution within our Milky Way (MW), based on large sets of phase-space data on individual stars. Specifically, we model the dynamics of 16,269 G-type dwarfs from SEGUE, which sample 5 < R_GC/kpc < 12 and 0.3 < |Z|/kpc < 3. We independently fit a parameterized MW potential and a three-integral, action-based distribution function (DF) to the phase-space data of 43 separate abundance-selected sub-populations (MAPs), accounting for the complex selection effects affecting the data. We robustly measure the total surface density within 1.1 kpc of the mid-plane to 5% over 4.5 < R_GC/kpc < 9. Using metal-poor MAPs with small radial scale lengths as dynamical tracers probes 4.5 < R_GC/kpc < 7, while MAPs with longer radial scale lengths sample 7 < R_GC/kpc < 9. We measure the mass-weighted Galactic disk scale length to be R_d = 2.15+/-0.14 kpc, in agreement with the photometrically inferred spatial distribution of stellar mass. We thereby measure dynamically the mass of the Galactic stellar disk to unprecedented accuracy: M_* = 4.6+/-0.3+3.0x(R_0/kpc-8)x10^{10}Msun and a total local surface density of \Sigma_{R_0}(Z=1.1 kpc) = 68+/-4 Msun/pc^2 of which 38+/-4 Msun/pc^2 is contributed by stars and stellar remnants. By combining our surface density measurements with the terminal velocity curve, we find that the MW's disk is maximal in that V_{c,disk} / V_{c,total} = 0.83+/-0.04 at R=2.2 R_d. We also constrain for the first time the radial profile of the dark halo at such small Galactocentric radii, finding that \rho_{DM} (r;near R_0) \propto 1 / r^\alpha with \alpha < 1.53 at 95% confidence. Our results show that action-based distribution-function modeling of complex stellar data sets is now a feasible approach that will be fruitful for interpreting Gaia data.Comment: Table 3 is available electronically as an Ancillary file (added again in v3

The dimensionality of stellar chemical space using spectra from the Apache Point Observatory Galactic Evolution Experiment

Chemical tagging of stars based on their similar compositions can offer new insights about the star formation and dynamical history of the Milky Way. We investigate the feasibility of identifying groups of stars in chemical space by forgoing the use of model derived abundances in favour of direct analysis of spectra. This facilitates the propagation of measurement uncertainties and does not presuppose knowledge of which elements are important for distinguishing stars in chemical space. We use ~16,000 red-giant and red-clump H-band spectra from the Apache Point Observatory Galactic Evolution Experiment and perform polynomial fits to remove trends not due to abundance-ratio variations. Using expectation maximized principal component analysis, we find principal components with high signal in the wavelength regions most important for distinguishing between stars. Different subsamples of red-giant and red-clump stars are all consistent with needing about 10 principal components to accurately model the spectra above the level of the measurement uncertainties. The dimensionality of stellar chemical space that can be investigated in the H-band is therefore $\lesssim 10$. For APOGEE observations with typical signal-to-noise ratios of 100, the number of chemical space cells within which stars cannot be distinguished is approximately $10^{10\pm2} \times (5\pm 2)^{n-10}$ with $n$ the number of principal components. This high dimensionality and the fine-grained sampling of chemical space are a promising first step towards chemical tagging based on spectra alone.Comment: 16 pages, 12 figures, accepted to MNRAS Dec 201

Searching for the GD-1 Stream Progenitor in Gaia DR2 with Direct N-body Simulations

We perform a large suite of direct N-body simulations aimed at revealing the location of the progenitor, or its remnant, of the GD-1 stream. Data from \gaia\ DR2 reveals the GD-1 stream extends over $\approx 100^\circ$, allowing us to determine the stream's leading and trailing ends. Our models suggest the length of the stream is consistent with a dynamical age of between 2-3 Gyr and the exact length, width and location of the GD-1 stream correspond to the stream's progenitor being located between $-30^\circ < \phi_{1,\mathrm{pro}} < -45^{\circ}$ in the standard GD-1 coordinate system. The model stream density profiles reveal that intact progenitors leave a strong over-density, recently-dissolved progenitors appear as gaps in the stream as escaped stars continue to move away from the remnant progenitor's location, and long-dissolved progenitors leave no observational signature on the remaining stream. Comparing our models to the GD-1 stream yields two possible scenarios for its progenitor's history: a) the progenitor reached dissolution approximately 500 Myr ago during the cluster's previous perigalactic pass and is both located at and responsible for the observed gap at $\phi_1=-40^{\circ}$ or b) the progenitor reached dissolution over 2.5 Gyr ago, the fully-dissolved remnant is at $-30^\circ < \phi_1 < -45^{\circ}$, and an observational signature of its location no longer exists. That the dissolved progenitor is in the range $-30^\circ < \phi_1 < -45^{\circ}$ implies that density fluctuations outside of this range, e.g., a deep gap at $\phi_1 \approx -20^\circ$, are likely produced by compact baryonic or dark-matter perturbers.Comment: 9 pages, 5 figures, Accepted for publication in MNRA