Terry D Oswalt Editor-in-Chief Howard E Bond Volume Editor Planets, Stars and Stellar Systems volume Astronomical Techniques, Software, and Data Planets, Stars and Stellar Systems Astronomical Techniques, Software, and Data Terry D Oswalt (Editor-in-Chief ) Howard E Bond (Volume Editor) Planets, Stars and Stellar Systems Volume 2: Astronomical Techniques, Software, and Data With 121 Figures and 32 Tables Editor-in-Chief Terry D Oswalt Department of Physics & Space Sciences Florida Institute of Technology University Boulevard Melbourne, FL, USA Volume Editor Howard E Bond Labrador Lane Cockeysville, MD, USA ISBN 978-94-007-5617-5 ISBN 978-94-007-5618-2 (eBook) ISBN 978-94-007-5619-9 (print and electronic bundle) DOI 10.1007/978-94-007-5618-2 This title is part of a set with Set ISBN 978-90-481-8817-8 Set ISBN 978-90-481-8818-5 (eBook) Set ISBN 978-90-481-8852-9 (print and electronic bundle) Springer Dordrecht Heidelberg New York London Library of Congress Control Number: 2012953926 © Springer Science+Business Media Dordrecht 2013 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer Permissions for use may be obtained through RightsLink at the Copyright Clearance Center Violations are liable to prosecution under the respective Copyright Law The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made The publisher makes no warranty, express or implied, with respect to the material contained herein Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Series Preface It is my great pleasure to introduce “Planets, Stars, and Stellar Systems” (PSSS) As a “Springer Reference”, PSSS is intended for graduate students to professionals in astronomy, astrophysics and planetary science, but it will also be useful to scientists in other fields whose research interests overlap with astronomy Our aim is to capture the spirit of 21st century astronomy – an empirical physical science whose almost explosive progress is enabled by new instrumentation, observational discoveries, guided by theory and simulation Each volume, edited by internationally recognized expert(s), introduces the reader to a well-defined area within astronomy and can be used as a text or recommended reading for an advanced undergraduate or postgraduate course Volume 1, edited by Ian McLean, is an essential primer on the tools of an astronomer, i.e., the telescopes, instrumentation and detectors used to query the entire electromagnetic spectrum Volume 2, edited by Howard Bond, is a compendium of the techniques and analysis methods that enable the interpretation of data collected with these tools Volume 3, co-edited by Linda French and Paul Kalas, provides a crash course in the rapidly converging fields of stellar, solar system and extrasolar planetary science Volume 4, edited by Martin Barstow, is one of the most complete references on stellar structure and evolution available today Volume 5, edited by Gerard Gilmore, bridges the gap between our understanding of stellar systems and populations seen in great detail within the Galaxy and those seen in distant galaxies Volume 6, edited by Bill Keel, nicely captures our current understanding of the origin and evolution of local galaxies to the large scale structure of the universe The chapters have been written by practicing professionals within the appropriate subdisciplines Available in both traditional paper and electronic form, they include extensive bibliographic and hyperlink references to the current literature that will help readers to acquire a solid historical and technical foundation in that area Each can also serve as a valuable reference for a course or refresher for practicing professional astronomers Those familiar with the “Stars and Stellar Systems” series from several decades ago will recognize some of the inspiration for the approach we have taken Very many people have contributed to this project I would like to thank Harry Blom and Sonja Guerts (Sonja Japenga at the time) of Springer, who originally encouraged me to pursue this project several years ago Special thanks to our outstanding Springer editors Ramon Khanna (Astronomy) and Lydia Mueller (Major Reference Works) and their hard-working editorial team Jennifer Carlson, Elizabeth Ferrell, Jutta Jaeger-Hamers, Julia Koerting, and Tamara Schineller Their continuous enthusiasm, friendly prodding and unwavering support made this series possible Needless to say (but I’m saying it anyway), it was not an easy task shepherding a project this big through to completion! Most of all, it has been a privilege to work with each of the volume Editors listed above and over 100 contributing authors on this project I’ve learned a lot of astronomy from them, and I hope you will, too! January 2013 Terry D Oswalt General Editor Preface to Volume Volume of Planets, Stars, and Stellar Systems is entitled “Astronomical Techniques, Software, and Data.” When I began my astronomical career in the 1960s, astronomical techniques, at least for optical observers, consisted mostly of exposing and developing photographic plates, or obtaining single-channel photometry with a photomultiplier tube Software, if used at all, was run using punched cards that you took over to the computer center and came back 24 hours later to pick up the output (which often consisted of pointing out the typographical errors in your FORTRAN code) Computations were done with a slide rule or a mechanical Friden calculator (the most advanced model could actually calculate a square root, although it took about 15 seconds to so!) Data consisted of photographic plates or strip-chart recordings of your photometry or hand-written columns of numbers or plots prepared with a Leroy lettering set If you needed data from a collaborator, the plates had to be shipped to you in a sturdy wooden box or you had to travel to your colleague’s institution or the tables of numbers had to be mailed to you The advances in astronomical methods in recent decades have come in steady steps, but as I look back from our contemporary viewpoint to 40 or 50 years ago, they are all but inconceivable I hold in my hand a computing device orders of magnitude more powerful than the room-sized computer of 1965 (and it can even make telephone calls!) What’s that bright thing next to the moon? Just start up the app, aim the phone at the sky, and it will tell you In the old days you made a finding chart by laboriously pulling out a Palomar Sky Survey print and photographing it with a Polaroid camera; now, in a few seconds, and from any mountaintop observatory in the world, I can display a Digitized Sky Survey image of any point in the sky, and read off the coordinates just by moving the cursor Do you want the spectral-energy distribution of your source from the far-UV, through the optical, to the near- and mid-IR? You can find all of that in a few moments now The venerable Chicago Stars and Stellar Systems had two volumes dedicated to “Astronomical Techniques” and “Basic Astronomical Data.” The new volume captures the basic spirit of those SSS volumes, in terms of introducing the reader to some of the basic observing and dataanalysis techniques, but of course many of the actual topics are vastly different from, or didn’t exist at all, five decades ago Volume starts with two articles (Stetson, and Massey & Hanson) describing modern techniques of astronomical photometry and spectroscopy, primarily at optical wavelengths The next two articles (Tokunaga, Vacca, & Young, and Snik & Leller) move to the realms of techniques of infrared astronomy and polarimetry of astrophysical sources As I have already mentioned, the availability of multi-wavelength sky surveys has transformed modern observational astronomy, and the amazing breadth of survey data available now (or in the near future) is comprehensively reviewed by Djorgovski, Mahabal, Drake, Graham, & Donalek Moving to still longer wavelengths, Wilson reviews the techniques of radio astronomy, and then Monnier & Allen reveal the methods of interferometry at both radio and optical frequencies To understand your data, you usually have to calibrate them to absolute physical units, and these techniques are explained by Deustua, Kent, & Smith viii Preface to Volume The new science of astroinformatics is reviewed by Borne Statistical methods of particular utility in astronomy are discussed by Feigelson & Babu, and the volume closes with a review of modern numerical techniques in astronomy by Wood This volume would not have been possible without the contributions of the authors, the guiding influence of the other volume editors and the editor-in-chief, and the staff at Springer I thank all of them, and I hope that the new PSSS volumes will have as much influence on contemporary astronomy as the old and still cherished Chicago SSS volumes did Howard E Bond Cockeysville, MD USA Editor-in-Chief Dr Terry D Oswalt Department Physics & Space Sciences Florida Institute of Technology 150 W University Boulevard Melbourne, Florida 32901 USA E-mail: toswalt@fit.edu Dr Oswalt has been a member of the Florida Tech faculty since 1982 and was the first professional astronomer in the Department of Physics and Space Sciences He serves on a number of professional society and advisory committees each year From 1998 to 2000, Dr Oswalt served as Program Director for Stellar Astronomy and Astrophysics at the National Science Foundation After returning to Florida Tech in 2000, he served as Associate Dean for Research for the College of Science (2000–2005) and interim Vice Provost for Research (2005–2006) He is now Head of the Department of Physics & Space Sciences Dr Oswalt has written over 200 scientific articles and has edited three astronomy books, in addition to serving as Editor-in-Chief for the six-volume Planets, Stars, and Stellar Systems series Dr Oswalt is the founding chairman of the Southeast Association for Research in Astronomy (SARA), a consortium of ten southeastern universities that operates automated 1-meter class telescopes at Kitt Peak National Observatory in Arizona and Cerro Tololo Interamerican Observatory in Chile (see the website www.saraobservatory.org for details) These facilities, which are remotely accessible on the Internet, are used for a variety of research projects by faculty and students They also support the SARA Research Experiences for Undergraduates (REU) program, which brings students from all over the U.S each summer to participate oneon-one with SARA faculty mentors in astronomical research projects In addition, Dr Oswalt secured funding for the 0.8-meter Ortega telescope on the Florida Tech campus It is the largest research telescope in the State of Florida Dr Oswalt’s primary research focuses on spectroscopic and photometric investigations of very wide binaries that contain known or suspected white dwarf stars These pairs of stars, whose separations are so large that orbital motion is undetectable, provide a unique opportunity to explore the low luminosity ends of both the white dwarf cooling track and the main sequence; to test competing models of white dwarf spectral evolution; to determine the space motions, masses, and luminosities for the largest single sample of white dwarfs known; and to set a lower limit to the age and dark matter content of the Galactic disk 494 11 Numerical Techniques in Astrophysics The internal energy of each particle is typically integrated using Pj Pi du i  v i j ⋅ ∇ i Wi j = ∑m j (  +  + Π i j ) ⃗ dt  j ρi ρj (11.38) SPH as detailed above – including constant h particles and using the same time step for all particles – is relatively straightforward to code, but may run inefficiently and/or may under-resolve some regions Because the method provides an interpolated estimate of the field quantities, there must be a sufficient number of particles in the sum in order for the estimate to be useful Typically for 3D problems, some 30–70 neighbors are desired Much less than 30 and the field is under sampled, and more than ∼ neighbors and the incremental change in the estimate per additional particle is negligible Because SPH is often used to model systems which may have strong shocks as well as tenuous regions, it was quickly realized that the smoothing length should be variable Several authors began using smoothing lengths that varied both in space and time (Benz et al 1990; Evrard 1988; Hernquist and Katz 1989, and see the recent reviews noted above) So what is needed is a method to calculate h on a per-particle basis, where h will be small in regions where many particles are tightly clustered, and h will be large in regions that have few particles per unit volume (Price 2010) An obvious choice for constant-mass particles is thus to let h scale with the inverse of the local number density of particles h(⃗r) ∝ n(⃗r)/ , which then feeds back into the equation for the density summation We then have at the location of particle i two simultaneous equations for the local density and smoothing length ρ(⃗r i ) = ∑ m j W(⃗r i − ⃗r j , h i ); j h(⃗r i ) = ζ ( mi ) ρi / , (11.39) where ζ is a parameter specifying the scaling between the smoothing length and the mean particle spacing (m/ρ)/ In most recent codes, these equations are solved simultaneously using, for example, a Newton-Raphson root-finding method (Price 2010) As noted in, for example, Springel (2010a), this is equivalent to keeping the mass in the kernel volume constant for all particles As discussed more fully in Springel (2010a), SPH has been modified in recent years to include self-gravity (as discussed earlier), magnetic fields (Dolag and Stasyszyn 2009; Dolag et al 1999; Price 2010; Price and Monaghan 2004a, b, 2005; Rosswog 2007), and radiation in the flux-limited-diffusion approximation (Forgan et al 2009; Whitehouse and Bate 2006; Whitehouse et al 2005) Alternative approaches to implementing radiative transfer into the SPH scheme include (1) that of Petkova and Springel (2009) who simplify the transfer equation such that it can be computed in terms of the local energy density of the radiation, (2) Pawlik and Schaye (2008) who compute radiative transfer by using emission and absorption cones between particles, and (3) Nayakshin et al (2009) who use a Monte-Carlo technique wherein the photons are treated as virtual particles Although not SPH, it deserves mention that recently Springel (2010b) has introduced a novel Lagrangian scheme which goes a long way toward eliminating the weaknesses of both the Eulerian grid and Lagrangian SPH techniques The new method is based on Voronoi tessellation of a set of discrete points The mesh is used to solve via a finite volume approach and exact Riemann solver the hyperbolic conservation laws of ideal hydrodynamics The mesh-generating points can be moved arbitrarily to follow the local motion of the fluid, but if held stationary the method is equivalent to an ordinary Eulerian method The new method can also adjust the Numerical Techniques in Astrophysics 11 spatial resolution as required by the local dynamics, which is one of the principle advantages of the SPH method A similar, first-order method was originally introduced by Whitehurst (1995) in the form of his FLAME code based on Delaunay and Voronoi tessellations, but no application publications followed Eulerian Hydrodynamics: Hydrodynamics on a Grid 5.1 Introduction In standard grid-based approaches to solving hydrodynamical problems, a volume of space is subdivided into finite-sized cells, and the physical quantities such as temperature, density, fluid velocity, etc., are computed inside those cells or at cell faces using finite difference or finite volume techniques as discussed above (see, e.g., Bodenheimer et al 2007; Mignone et al 2007; O’Shea et al 2004; Ryu et al 1993; Stone and Norman 1992a) As discussed in Bodenheimer et al (2007), grid methods as a rule are preferable when shocks in the fluid are an important component of the problem being simulated However, for many kinds of astrophysical problems SPH codes that implement adaptive particle sizes may be optimal, as SPH naturally increases spatial resolution where needed 5.2 Shocks and the Rankine-Hugoniot Conditions Shocks are discontinuities in the flow of the fluid A shock front is a surface that is characterized by a nearly discontinuous change in P, ρ, v⊥ , and T, where v⊥ is the component of the velocity perpendicular to the shock front Related to shocks but generally less problematic to simulate are contact discontinuities A contact discontinuity is characterized by a nearly discontinuous change in ρ, T, or v∥ , but a continuous P and v⊥ Shocks often form naturally in physical systems when sound waves propagate – because the speed of sound increases with density, a compression wave which is initially symmetric about the maximum density will have its leading edge steepen to a shock front as it propagates through the fluid The shock front narrows to a scale of just a few mean free paths, but we can use conservation laws to relate the physical properties on the two sides of the shock The resulting relations are called the Rankine-Hugoniot (“jump”) conditions (Landau and Lifshitz 1959) Let us consider a shock front in a 1D inviscid polytropic flow in the reference frame where the front is stationary and there are no external forces In this case, the 1D Euler equations can be written as ∂ρ ∂ = − (ρu), ∂t ∂x ∂(ρu) ∂  = − (ρu + P), ∂t ∂x ∂E ∂ = − [u(E + P)] , ∂t ∂x where  E ≡ ρu  + e  (11.40) (11.41) (11.42) (11.43) 495 496 11 Numerical Techniques in Astrophysics is the total energy density, e is the internal energy density, and u is the fluid velocity relative to the stationary shock front, with u  > c s, supersonic in the pre-shock region Conservation of fluxes of mass ρu, momentum ρu  + P, and total energy u(  ρu  + e + P) across the shock front results in ρ u = ρu , (11.44) ρ  u  + P = ρ  u  + P , (11.45)     u  ( ρ  u  + e + P ) = u  ( ρ  u  + e + P )   (11.46) Using these equations and assuming a polytropic equation of state P = (γ − )e, it is easy to show that the ratio of densities in the post- and pre-shock regions is given by (e.g., Bodenheimer et al 2007) ρ  P + m  P = , (11.47) ρ  P + m  P where m = γ− γ+ (11.48) Even if the shock is very strong (P ≫ P ), the density contrast is only ρ  /ρ  =  for the case of γ = /, but for an isothermal shock able to radiate instantaneously the energy dissipated in the shock front, m  →  and the density contrast may be arbitrarily high Alternatively, the shock may be characterized by the value of the Mach number M = ∣u  ∣/c s, of the pre-shock flow, in which case it can be shown (Balachandran 2007) that ρ (γ + )M = , ρ (γ − )M +  (11.49) γM − (γ − ) P = , P γ+ (11.50) and that in the frame of reference of the shock, the post-shock medium is subsonic: M =  + M γ− γ M −  γ− (11.51) √ In the limit as M → ∞, we find M → (γ − )/ There are two ways of treating shocks in Eulerian codes The first is to implement a Riemann solver which makes use of the results obtained in this section, and the second is to implement an artificial viscosity We discuss each in turn 5.3 The Riemann Problem A Riemann problem consists of a conservation law applied to the case of piecewise constant data that includes a single discontinuity Methods to solve the Riemann problem are called Riemann solvers For a detailed treatment of this subject, see Toro (1999) or Knight (2006) The simplest application is the shock tube Consider a 1D system of two volumes of fluid initially separated by a diaphragm located at x  Let the initial pressures and densities of the 11 Numerical Techniques in Astrophysics Q Q1 Qi+1 Qi−1 Qi i−1 QM i i+1 M X Xi– Xi– Xi+ Xi+ 2 ⊡ Fig 11-4 A schematic representation of component Qi at time tn fluids to the left and right of the diaphragm be initially (P , ρ  ) and (P , ρ  ), respectively, with P > P , and the fluids originally at rest At t =  the diaphragm is ruptured, resulting in the left fluid expanding into the right Five regions subsequently develop: (1) the original undisturbed left fluid, (2) rarefaction wave, (3) uncompressed left fluid with constant velocity and pressure and bounded at the right by a contact discontinuity, (4) compressed right fluid bounded at the right by a shock, and (5) undisturbed right fluid (Bodenheimer et al 2007; Hawley et al 1984) Given the initial conditions for pressure and densities in the two initial regions, it is possible to solve for the subsequent pressures, densities, and velocities of the fluid in the middle three regions This result can then be applied to a typical piecewise constant array representing some physical variable in a simulation Godunov (1959) was the first to implement the conservative upwind method to nonlinear systems of conservation laws Godunov’s first-order upwind method computes interface numerical fluxes F i+  using solutions of local Riemann problems, under the assumption that at time  step n in the simulation the values Q i on the grid are piecewise constant (see > Fig 11-4), where Q is the vector of dependent variables Q = (ρ, ρu, E)T , and F is the vector of fluxes F = (ρu, ρu  + P, (E + P)u)T This allows the values of Q i to be updated from one time step to the next using the integral form of the Euler equations n+ Qi n = Qi − t  ∫ Δx t n n+ (F i+  − F i−  ) dt   (11.52) For example, following Knight (2006), assume the piecewise constant function for some variable Q i is known for i = , , M at time t n In general, there will be a discontinuity in Q i at 497 498 11 Numerical Techniques in Astrophysics t n+1 5Ј 4Ј 3Ј 2Ј i +1 i i−1 tn 1Ј Δt X Δxi ⊡ Fig 11-5 A schematic flow structure for Godunov’s Second Method The contact discontinuities are represented as dotted lines each interface The flux F i+  can be determined from the solution of the local general Riemann  problem at interface x i+  , and similarly for flux F i−  at the left bound of cell i (see > Fig 11-5)   In the solution the states for regions and can be taken as Qi+  = Q i , (11.53) Qi+   = Q i+ , (11.54) Qi−  = Q i− , (11.55) Qi−   (11.56)  and  = Qi , where we use superscripts to denote the region, and where we assume that as shown in > Fig 11-4 Q i− > Q i < Q i+ The solution to the Riemann problem for Q at the interface is dependent upon (x − x i+  )/(t − t n ) and not on x and t separately, and therefore the  solution for Q at the interface is independent of time We then have ∫ t n+ tn F i+  dt = F i+  Δt = F (QRP i+  )Δt,    (11.57)  where QRP i+  is the solution of the Riemann problem at interface x i+  Using a similar approach for F i−  , we have  Δt RP RP Qn+ = Q in − (11.58) (F (Q i+  ) − F (Q i i−  ))  δx This is the essence of Godunov’s Second Method, which is most often used in practice 5.4 Artificial Viscosity The second approach to handling shocks is to implement an artificial viscosity Shocks are nearly discontinuous physically, with a characteristic width of a few particle mean free paths This in general is many times smaller than the typical grid spacing in simulations By introducing an artificial viscosity, the shock front may be artificially spread over a few grid cells, as required to maintain a stable solution First introduced by von Neumann and Richtmyer (1950), artificial viscosities are now widely used to help make flows more diffusive near shock fronts so the simulation maintains stability Numerical Techniques in Astrophysics 11 The von Neumann-Richtmyer artificial viscosity is a bulk viscosity which acts as an additional pressure:  ⎧ ⎪  ∂v ⎪ ⎪ ⎪ qρ(Δx) ( ) ∂x Π= ⎨ ⎪ ⎪ ⎪ ⎪  ⎩ if ∂v ∂x < ; if ∂v ∂x > ; (11.59) where q is a parameter varied to produce the desired shock width Note that the artificial viscosity only acts during compression Throughout the equations describing the system, one replaces P → P + Π The artificial viscosity so defined has the desirable properties that it only operates at a significant level in the vicinity of shock fronts and can be tuned so as to be only as diffusive as needed Additional details can be found in Bodenheimer et al (2007) 5.5 Available Codes Stone and Norman (1992a) introduced the code ZEUS-2D to the community ZEUS-2D is a finite difference hydrodynamics code implemented in a covariant formalism In addition, ZEUS-2D incorporated magnetohydrodynamics (Stone and Norman 1992b), and radiation hydrodynamics (Stone et al 1992) ZEUS is in fact several different numerical codes, and each version is freely available ZEUS-2D and the MPI-enabled version ZEUS-MP can be obtained from the Laboratory for Computational Astrophysics website,4 and ZEUS-3D is make available by David Clarke5 (Clarke 1996, 2010) With over 700 referred citations to Stone and Norman (1992a), the ZEUS codes are arguably the best-tested astrophysical HD/MHD codes in existence Recently, Stone and collaborators have been developing a new code for astrophysical magnetohydrodynamics (MHD) called Athena (Stone et al 2008) The authors designed Athena primarily for solving problems in the areas of the interstellar medium, star formation, and accretion flows, and it is freely available to the community.6 The version at the time of this writing includes, for example, compressible hydrodynamics and MHD in 1D, 2D, and 3D, special relativistic dynamics, self-gravity or a defined static gravitational field, both Navier-Stokes and anisotropic viscosity, and optically thin radiative cooling In addition, several choices are available for the Riemann solvers and spatial reconstruction methods Also freely available to the community is the PLUTO7 code (Mignone et al 2007) PLUTO is a modular Godunov-type code designed to be used in astrophysical problems characterized by high Mach numbers PLUTO offers several choices of hydrodynamic solvers, and can be used to simulate Newtonian, relativistic, MHD, or relativistic MHD flows in Cartesian, cylindrical, or spherical coordinates The code can run on single-processor machines, but can also run on large parallel machines using MPI, if available Computational grids may be static or adaptive The FLASH code is available from the Flash Center at the University of Chicago.8 It is a multiphysics multiscale simulation code with a large base of users The FLASH code is intended primarily for applications involving high-energy density physics (e.g., supernovae) A large selection of physics solvers is available, including various hydrodynamics, MHD, equation lca.ucsd.edu/portal/software/ www.ap.smu.ca/~dclarke/zeus3d/ trac.princeton.edu/Athena plutocode.ph.unito.it/ See flash.uchicago.edu/site/ 499 500 11 Numerical Techniques in Astrophysics of state, radiative transfer, diffusion and conduction, nuclear burning, gravity, and magnetic resistivity and conductivity solvers The code is in its fourth version Conclusion Numerical Astrophysics is one of the youngest scientific fields, with a history spanning only a few decades, but one which is currently vibrant and growing Numerical experimentation allows us to study the physics of astronomical processes at a level of detail undreamed of a century ago In this short manuscript, only a few of the many areas of numerical astrophysics have been discussed The interested reader has available a vast literature, including above-cited texts and conference proceedings, as well as many public domain codes that are suitable for projects ranging from self-education to cutting-edge research Many if not most code authors are generous with their codes and time, and approachable with suggestions of collaborative efforts Roughly half a century ago, researchers were computing stellar structure and evolution by hand with a slide rule (Schwarzschild 1958), computing one time step per several hours of clock time Today we can use hundreds of CPU centuries to complete a multimillion particle simulation in a matter of calendar weeks, and by the end of the decade will likely be able to simulate galactic dynamics with a particle representation of each of the ∼ stars present in a large spiral galaxy like our own Acknowledgments Thanks to Terry Oswalt and Howard Bond for suggesting that I write this manuscript, to Dean Townsley for commenting on a draft of the manuscript, and to my wife Jane E Wood for her encouragement and patience over the many hours spent in the office References Abel, T 2011, MNRAS, 413, 271 Allen, M P., & Tildesley, D J 1989, Computer Simulat ion of Liquids (Oxford: Oxford University Press) Appel, A W 1985, SIAM J Sci Stat Comp, 6, 85 Balachandran, P 2007, Fundamentals of Compressible Fluid Dynamics (New Delhi: Prentice-Hall) Balsara, D S 1995, J Comput Phys, 131, 357 Barnes, J., & Hut, P 1986, Nature, 324, 44 Barnes, J E., & Hut, P 1989, ApJS, 70, 389 Belleman, R G., Bédorf, J., & Portegies Zwart, S F 2008, New Astron, 13, 103 Benz, W., Bowers, R L., Cameron, A G W., & Press W H 1990, ApJ, 348, 647 Bodenheimer, P., Laughlin, G P., Ró˙zyczka, M., & Yorke, H W 2007, Numerical Methods in Astrophysics: An Introduction (Boca Raton, FL: Taylor & Francis) Bowers, R L., & Wilson, J R 1991, Numerical Modeling in Applied Physics and Astrophysics (Boston: Jones and Bartlett) Burnett, D S 1987, Finite Element Analysis From Concepts to Applications (Reading, MA: Addison Wesley) Chandrasekhar, S 1939, An Introduction to the Study of Stellar Structure (Chicago: University of Chicago Press) Numerical Techniques in Astrophysics Cheng, H., Greengard, L., & Rokhlin, V 1999, J Comput Phys, 155, 468 Clarke, D A 1996, ApJ, 457, 291 Clarke, D A 2010, ApJS, 187, 119 Couchman, H M P 1991, ApJ, 368, L23 Cramer, C J 2004, Essentials of Computational Chemistry: Theories and Models (2nd ed.; Chichester, UK: Wiley) Cromer, A 1981, AJP, 49, 455 Dehnen, W 2000, ApJ, 536, L39 DeVries, P L., & Hasbun, J E 2010, A First Course in Computational Physics (2nd ed.; Sudbury, MA: Jones & Bartlett) Dolag, K., & Stasyszyn, F 2009, MNRAS, 398, 1678 Dolag, K., Bartelmann, M., & Lesch, H 1999, A&A, 348, 351 Eddington, A 1926, The Internal Constitution of the Stars (Cambridge: Cambridge University Press) Efstathiou, G., Davis, M., White, S D M., & Frenk, C S 1985, ApJS, 57, 241 Evrard, A E 1988, MNRAS, 235, 911 Forgan, D., Rice, K., Stamatellos, D., & Whitworth, A 2009, MNRAS, 394, 882 Fortin, P., Athanassoula, E., & Lambert, J.-C 2011, A&A, 531, A120 Garcia, A 2000, Numerical Methods for Physics (2nd ed.; Englewood Cliffs, NJ: Prentice-Hall) Gingold, R A., & Monaghan, J J 1977, MNRAS, 181, 375 Gingold, R A., & Monaghan, J J 1983, MNRAS, 204, 715 Giordano, N J., & Nakanishi, H 2005, Computational Physics (2nd ed.; New Jersey: Prentice Hall) Godunov, S K 1959, Math Sbornik, 47, 357 Gould, H., Tobochnik, J., & Christian, W 1988, An Introduction to Computer Simulation Methods (3rd ed.; Reading, MA: Addison Wesley) Greengard, L., & Rokhlin, V 1987, J Comp Phys, 73, 325 Hamming, R W 1962, Numerical Methods for Scientists and Engineers (New York, NY: McGraw-Hill) Harrison, P 2001, Computational Methods in Physics, Chemsitry and Biology: An Introduction (West Susses, UK: Wiley) Hawley, J F., Smarr, L L., & Wilson, J R 1984, ApJ, 277, 296 Hernquist, L., & Katz, N 1989, ApJS, 70, 419 Hockney, R W., & Eastwood, J W 1988, Computer Simulation Using Particles (Bristol, UK: Adam Hilger) Holmberg, E 1941, ApJ, 94, 385 Jernigan, J G., & Porter, D H 1989, ApJS, 71, 871 Klypin, A A., & Shandarin, S F 1983, MNRAS, 204, 891 11 Knebe, A., Green, A., & Binney, J 2001, MNRAS, 325, 845 Knight, D D 2006, Elements of Numerical Methods for Compressible Flows (New York: Cambridge University Press) Konstantinidis, S., & Kokkotas, K D 2010, A&A, 522, A70 Kravtsov, A V., Klypin, A A., & Khokhlov, A M 1997, ApJS, 111, 73 Landau, L D., & Lifshitz, E M 1959, Fluid Mechanics (London: Pergamon Press) Landau, R H., Páez, M J., & Bordeianu, C C 2007, Computational Physics: Problem Solving with Computers (2nd ed; Weinheim, Germany: Wiley-VCH) Landau, R H., Páez, M J., & Bordeianu, C C 2008, A Survey of Computational Physics: Introductory Computational Science (Princeton, NJ: Princeton University Press) Lucy, L B 1977, AJ, 82, 1013 Makino, J 2008, in Dynamical Evolution of Dense Stellar Systems, Proceedings of the International Astronomical Union, IAU Symp., 246, 457, doi:10.1017/S1743921308016165 Makino, J., Fukushige, T., Koga, M., & Namura, K 2003, PASJ, 55, 1163 Makino, J., Taiji, M., Ebisuzaki, T., & Sugimoto, D 1997, ApJ, 480, 432 Mignone, A., Bodo, G., Massaglia, S., Matsakos, T., Tesileanu, O., Zanni, C., & Ferrari, A 2007, ApJS, 170, 228 Monaghan, J J 1992, ARA&A, 30, 543 Monaghan, J J 2005, Rep Prog Phys, 68, 1703 Monaghan, J J., & Lattanzio, J C 1985, A&A, 149, 135 Nayakshin, S., Cha, S.-H., & Hobbs, A 2009, MNRAS, 397, 1314 Nelson, A F., Wetzstein, M., & Naab, T 2009, ApJS, 184, 326 O’Shea, B W., Bryan, G., Bordner, J., et al 2004, arXiv:astro-ph/0403044 Pang, T 2006, An Introduction to Computational Physics (2nd ed.; Cambridge, UK: Cambridge University Press) Pawlik, A H., & Schaye, J 2008, MNRAS, 389, 651 Petkova, M., & Springel, V 2009, MNRAS, 396, 1383 Portegies Zwart, S F., Belleman, R G., & Geldof, P M 2007, New Astron, 12, 641 Press, W H., Teukolsky, S A., Vetterling, W T., & Flannery, B P 1992, Numerical Recipes in Fortran: The Art of Scientific Computing (2nd ed.; Cambridge: Cambridge University Press) Price, D J 2010, J Comp Phys (in press) (arXiv:1012.1885) Price, D J., & Monaghan, J J 2004a, MNRAS, 348, 123 501 502 11 Numerical Techniques in Astrophysics Price, D J., & Monaghan, J J 2004b, MNRAS, 348, 139 Price, D J., & Monaghan, J J 2005, MNRAS, 364, 384 Rosswog, S 2007, Astron Nachr, 328, 663 Rosswog, S 2009, New A Rev, 53, 78 Rosswog, S 2010, J Comput Phys, 229, 8591 Ryu, D., Ostriker, J P., Kang, H., & Cen, R 1993, ApJ, 414, Schwarzschild, M 1958, Structure and Evolution of the Stars (Princeton, NJ: Princeton University Press) Shu, F H 1992, The Physics of Astrophysics, Vol II (Mill Valley, CA: University Science Books) Springel, V 2005, MNRAS, 364, 1105 Springel, V 2010a, ARA&A, 48, 391 Springel, V 2010b, MNRAS, 401, 791 Springel, V., Yoshida, N., & White, S D M 2001, New A, 6, 79 Stone, J M., & Norman, M L 1992a, ApJS, 80, 753 Stone, J M., & Norman, M L 1992b, ApJS, 80, 791 Stone, J M., Mihalas, D., & Norman, M L 1992, ApJS, 80, 819 Stone, J M., Gardiner, T A., Teuben, P., Hawley, J F., & Simon, J B 2008, ApJS, 178, 137 Strikwerda, J 2007, Finite Difference Schemes and Partial Differential Equations (2nd ed.; Pacific Grove, CA: Wadsworth & Brooks/Cole) Sugimoto, D., Chikada, Y., Makino, J., Ito, T., Ebisuzaki, T., & Umemura, M 1990, Nature, 345, 33 Teyssier, R 2002, A&A, 385, 337 Thijssen, J M 2007, Computational Physics (2nd ed.; New York: Cambridge University Press) Thomas, J W 2010, Numerical Partial Differential Equations: Finite Difference Methods (New York, NY: Springer) Toro, E F 1999, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practial Introduction (3rd ed.; New York: Springer-Verlag) Truelove, J K., Klein, R I., McKee, C F., Holliman, J H., II, Howell, L H., Greenough, J A., & Woods, D T 1998, ApJ, 495, 821 von Hoerner, S 1960, Z Astrophys, 50, 184 von Hoerner, S 1963, Z Astrophys, 57, 47 von Neumann, J., & Richtmyer, R D 1950, J Appl Phys, 21, 232 Wetzstein, M., Nelson, A F., Naab, T., & Burkert, A 2009, ApJS, 184, 298 White, S D M., Frenk, C S., & Davis, M 1983, ApJ, 274, L1 Whitehouse, S C., & Bate, M R 2006, MNRAS, 367, 32 Whitehouse, S C., Bate, M R., & Monaghan, J J 2005, MNRAS, 364, 1367 Whitehurst, R 1995, MNRAS, 277, 655 Xu, G 1995, ApJS, 98, 355 Index A Absorption, tropospheric, 306 Adaptive optics (AO), 344–346, 359, 366 Adding interferometers, 328, 336 AIPS, 354, 362 Airborne astronomy, 141–144 Airmass, 10–12, 16–22 All-sky photometry, 3, 30, 33 Amplitude, 332, 334, 338, 340, 341, 347, 351–355, 357, 362–365 Amplitude calibrator, 341, 353 Angular resolution, 327, 328, 331, 332, 341, 343, 349, 350, 357, 358, 360, 361, 365–367, 370 Angular separation, 365 Angular sizes, 327, 329, 334 Antenna, 330, 334, 335, 342, 364 baseline ripple, 306 Cassegrain, 305 error beam, 306 feed supports, 306 G/T value, 305 Hertz dipole, 292, 301 Nasmyth, 305 off-axis, 306 pattern, 335 point spread function, 301 Rayleigh distance, 301, 302 reciprocity, 300, 301 sidelobes, 301–303, 306, 310 temperature, 342 Aperture, 5, 6, 8, 16, 18, 23–25, 27–33 effective, 303 efficiency, 303, 304 geometric, 303 masking, 363 synthesis, 285, 287, 291, 311–321 Apodization, 319 Archives, 229, 246, 247, 249, 257, 263–277 Array feeds, 350 Artificial viscosity, 493, 496, 498–499 Astroinformatics, 225, 230, 235, 241, 266, 272, 274, 403–439 Astrometric precision, 365, 366 Astrometry, 328, 346, 364, 365, 368 Astrostatistics, 447, 448, 474–477 Atmospheric coherence time, 344, 345, 353, 354 Atmospheric transmission, 106–112, 114, 115, 129, 140, 141, 150, 151, 153, 160, 167, 170 Atmospheric turbulence, 327, 338–340, 344–346, 362, 363 Atmospheric windows, 338 Atomic hydrogen, 343 Autocorrelator FX, 299 XF, 300 B Backend, 335 Background emission, 112–120, 129, 132–140, 152, 164, 165 Balmer convergence, 16 Balmer jump, 16, 20 Bandpass mismatch, 15, 19, 21, 33 Bandwidth-smearing limit, 350 Baseline, 287, 306, 311–314, 318–320 curvature, 310 Basket weaving, 310 Bayes’ theorem, 361, 452, 457 Beam dirty, 319, 320 efficiency, 303, 304, 309 solid angle, 302, 303 width, 303, 309, 310, 314 Biases, 61, 65, 86, 90 frame, 7, 9, 26 offset, Bispectrum, 362–365 Blackbody radiation, 105–106, 289–292 Bolometer LABOCA, 296 MAMBO2, 296 SCUBA, 296 spectral line, 297 Bose-Einstein quantum statistics, 342 Bowl, 336 Brightness distribution, residual, 320 Brightness temperature, 290, 291, 303, 304, 309, 313, 317 Brightness temperature sensitivity, 343, 344 C Calibration, 7, 15–17, 19, 20 source, 302, 307, 313, 316, 318 strategy, 341 CASA, 354 504 Index Catalogs, 227–235, 238, 241–249, 251, 253, 255–258, 267, 270, 271, 275 CCD See Charge-coupled devices (CCD) Censoring (upper limits), 468, 469 Characteristic scale, 355 Charge-coupled devices (CCD), 2–4, 6–10, 13, 14, 18, 20–22, 24–33 Classification, 228, 238, 239, 243, 245, 267–270, 408, 409, 414, 415, 428, 432–436, 438, 439 CLEAN, 320 algorithm, 358–362, 364 loop gain, 320 Clean beam, 320 CLEAN components, 359–361 Closure amplitude, 318, 362–365 phase, 351, 354, 362–365 triangle, 364 Clustering, 417–419, 421–423, 426, 433 Coherence envelope, 350 length, 332, 344, 345 time, 332, 341, 342, 344–346, 353, 354 Coherent amplifiers, 337 Coherent integration time, 345 Coherent volume, 344, 345 Collecting area, 329, 341, 343, 344, 346, 366 Collimator, 38, 39, 41, 45, 84, 85 Combiner, 329, 332, 336, 341, 350, 356 Complex gain, 351, 352, 364 Complexity in the image, 341 Complex visibilities, 333, 334, 336, 347, 350–352, 354, 355, 357, 359, 364 Concentric aperture photometry, 27–30, 33 Convolution theorem, 359 Cooley-Tukey fast Fourier transform, 319 Correlated flux density, 336 Correlators, 329, 331, 332, 336, 341, 342, 350 Cosmic microwave background, 287, 288, 309 Coude, 51 (u,v) Coverage, 334, 347–350, 355–359, 362, 364 D Dark, 5, 7–9, 24 Dark frame, 8, Databases, 405, 407–410, 412–415, 417–419, 422, 424–428, 431–433, 435, 437, 439 Data mining, 225, 241, 266, 271–273, 403–439, 446, 450, 465 Data processing pipelines, 251, 266–268 Data science, 409, 411, 416, 425, 436–437, 439 Deep surveys, 226, 258–260, 275 Density, 289–292, 300, 302–304, 307, 309, 313, 315, 317 estimation (smoothing), 449, 464 spectral power, 290, 299 Difference equation, 484–488 Differential refraction, 65, 73–74, 84, 88 Digital correlators, 336, 342 Digital data numbers, 10 Digital image, 6, 7, 9, 22, 25 Digital imaging, 225 Direct detection, 328 Dirty beam, 358–361 Dirty map, 319, 320, 358–360 Dispersion measure, 292 Dome flats, 7–9, 26 Double beam system, 309, 310 Dynamic range, 5, 32 E Earth rotation aperture synthesis, 348 Echelle, 38, 40, 41, 48–51, 56, 76 Effective wavelengths, 15 Emission, 286–288, 290, 292, 294, 298, 306–310 extended regions, 285 Emissivity, 105–106, 114, 115, 120, 129, 131, 132 Entrance, Entropy, 360–362, 365 Error closure, 318 E-science, 408, 411 Euler equations, 483, 495, 497 Eulerian hydrodynamics, 484, 495–500 Euler’s method, 488 Exoplanets, 177, 186, 205, 218–219 Exploratory data analysis (EDA), 405, 407, 413, 419 Extinction, 10–12, 16, 18–22, 25, 27, 33 F Fabry/field lens, 5, 6, 23 Far-infrared, 102, 103, 108, 117, 119, 120, 142, 144, 148, 161 Fast Fourier transform (FFT), 319–320 Fibers, 38, 51, 54–56, 60, 69, 74, 75, 82, 83, 88–92 Field-of-view, 327, 331–333, 341, 342, 350 Figures of merit for sky surveys, 225 FITS-IDI, 354 Flat-field image, 7, 9, 20, 26 Flat fielding, 37, 50, 55, 69, 78–80, 85, 89 Floor, 336 Flux density, 289, 291, 302, 304, 307, 309, 313, 315, 317 total, 289 Fourier analysis, 336 Index Fourier component, 334, 347–349 Fourier coverage, 347–349, 358 Fourier phase, 363–365 Fourier transform, 334, 335, 355, 358–360, 362, 364 Fourth paradigm, 416, 436 Frequency response, 299 switching, 311 Fried length, 339, 340 Friis formula, 296 Fringe amplitude, 332, 334, 338, 351, 364 phase, 334, 338, 339, 345, 354, 362, 363, 365 phase fluctuations, 339, 363 spacing, 332, 350, 365 stopping, 314 tracking, 345–346, 353, 366 Frozen atmosphere model, 339 Function, autocorrelation, 300 G Gain factor, 26, 27 Galaxy clustering, 448, 472, 477 Gaussian limit, 342 Geometric models, 355 Godunov, 484, 497, 498 Grading, 311, 319 Gratings, 38–41, 44–50, 53, 55, 56, 58, 59, 64, 65, 70–72, 79, 83–85, 87–89, 91 Gregorian system, 305 Growth curve, 29, 33 Guide star, 346 Infrared photometry, 110, 150, 152 Infrared spectroscopy, 163–170 Infrared standards, 150–163 Instrumental color, 17, 18 Instrumental magnitudes, 10, 17, 19, 21–23, 26, 29, 30, 32, 33 Integral field spectroscopy, 59–60 Integration, 8–10, 18, 23, 25, 33 Interference fringe, 327, 332, 336 Interferometers, 327–330, 332–352, 354, 357–360, 363–370 ALMA, 316 fast switching, 317 noise, 300, 311–313, 317, 320 point spread function, 319 synthesized beam, 317 Interferometric spectrum, 336 Interferometry, 325–370 Interplanetary medium, 338 Interstellar medium, 338 Ionosphere, 338 Isoplanatic angle, 340, 341, 345 Isothermal medium, 289, 290 J Jansky (unit), K Kolmogorov turbulence, 339, 340, 345 L H Heisenberg uncertainty principle, 297 High-fidelity imaging, 329, 355 Hydra spectrograph, 69, 74, 88–93 I Image dynamic range, 315, 317, 320 fidelity, 315, 350, 355 frequency, 297 reconstruction, 327, 334, 365 Imaging, 327–331, 336, 341, 343, 348, 350, 352, 353, 355–365 Inamori-Magellan Areal Camera & Spectrograph (IMACS), 44, 45, 52–54, 60–63, 69, 78 Informatics, 407, 416, 436, 437 Information theory, 349 Infrared absolute calibration, 163 Infrared astronomy, 99–170 Infrared optimized telescopes, 114, 128, 130–132 Large Synoptic Survey Telescope (LSST), 405, 413–415, 420, 421, 438, 439 Laser metrology, 336 Light-emitting diodes, Limiting magnitude, 344–346 Local oscillator, 331, 335 Local standard of rest (LSR), 310 Long-slit spectroscopy, 41, 44–48, 51, 53, 58, 59, 61, 68, 71, 75–76, 78–80, 83–87, 89, 96 M Machine learning, 405, 409, 413–417, 419–421, 426–428, 430, 437–439 Magellan Echellette (MagE), 38, 48–50 Magnetic fields, 177, 185, 187–191, 213, 217 Magnitude, 2–6, 10–12, 15–33 Magnitude limit, 345 Main beam, 291, 302–304, 308, 313 Maximum coherent baseline, 339 Maximum entropy method, 321 505 506 Index Maximum entropy method (MEM), 321, 360–362, 365 Maximum likelihood estimation, 448, 452, 454, 456, 471 Measurement parameter space (MPS), 238, 239 Michelson interferometer, 339 Mid-infrared, 103, 105, 112, 115, 131, 137–140, 149, 152, 154, 156, 159, 160 Minimum detectable signal, 342 Miriad, 354 Mixer Schottky, 296, 297 superconducting, 296, 297 Model fitting, 353, 355–357 selection, 458, 460–461, 466, 470 Modulation, 196, 197, 199–208, 211, 212, 215, 217, 219 Mosaicing, 350, 362 Multivariate clustering, 465–467 Mutual coherence, 332 N Natural weighting, 360 N-body simulation, 488–492 Near infrared (NIR), 37, 38, 56–59, 61, 69–73, 82, 83, 92–96, 101–104, 110, 111, 117, 124, 131, 133–137, 149, 151, 153, 156, 159, 160, 163, 169 Noise cascaded amplifiers, 296 minimum, 293, 297–299, 315 Nonparametric statistics, 449, 461–463 Nonredundant geometry, 348 Normalized visibility, 336 Numerical techniques, 481–500 Nyquist theorem, 290–291, 299 O Observable parameterspace (OPS), 227, 228, 235–242, 249, 261–263 Observing proposal, 327, 341 Observing sequence, 18, 25 OH airglow emission, 112 Ohio state infrared imager/spectrometer (OSIRIS), 56–59 OIFITS, 355 On-the-fly mapping, 309, 311 Optical depth, 289, 307 Optical synthesis image, 329 Optical window, 338 Optimal extraction, 37, 66, 72, 76–77 Oscillator, local, 297, 299 Outer scale length, 338, 340 Outlier detection, 405, 414, 415, 417, 419, 421, 422, 428 Overscan region, P Parabolic reflector, 335 Particle-mesh methods, 490 Particle-particle methods, 489–490 Particle-particle particle-Mesh (P  M) methods, 490 Phase, 332–334, 337–342, 345–347, 351, 353, 354, 357, 362–366 calibrators, 345, 346, 351, 353 center, 332, 333 closure, 318–319 coherence, 337 referencing, 341, 345–346, 353, 354, 357, 362–365 Photocathode, 5–7, 24 Photoelectric photometer, 5, 6, 12, 18, 22–25 Photoelectric photometry, 5, 21, 23–25 Photographic emulsion, Photographic plate, 2, 4, 6, Photometric detector, 3–9 Photometry, 1–33, 342 Photomultiplier tube (PMT), 2, 4, 5, 12 Photon arrival rates, 342 Physical baseline separations, 348 Physical models, 355, 357 Physical parameter space (PPS), 238–242 Pipelines, 251, 266–269, 271 Pixel, 5–9, 18, 20, 24–28, 31–33 Planck function, 288 PMT See Photomultiplier tube (PMT) Point sources, 332, 341–343, 345, 352, 358–360 Point-source sensitivity, 343, 345 Point-spread function (PSF), 30–32 Poisson noise, 344, 345 Poisson’s equation, 484, 490 Poisson statistics, 2, 5, 6, 8, 18, 23, 25, 28, 342, 344 Polarimetric accuracy, 199, 200, 208, 209, 212 Polarimetric calibration, 211 Polarimetric sensitivity, 199, 204–208, 214–216, 218 Polarimetry, 175–219 Polarization, 177–192, 194, 195, 197–213, 215–219, 285, 288, 291–292, 295, 299, 306 Position switching, 309, 311, 316 Power, normalized pattern, 302, 303 Precipitable water, 307 Precision astrometry, 328 Primary beam, 335, 341, 350 Probability, 447, 449, 451–458, 466, 468, 472 Profile-fitting photometry, 30–33 Projected baseline separations, 348 Propagation effects, 307 PSF See Point-spread function (PSF) Index Pulsars, 287, 291, 292 Pulse, delay, 291, 292 Q Quantum efficiency, 4, 6, 12–16, 24 Quantum limits of amplifiers, 337–338 Quantum regime, 337 R R (software), 448, 475 Radial velocity, heliocentric, 310 Rankine-Hugoniot conditions, 495–496 Raster scan, 310 Rayleigh-Jeans law, 289 Readout noise, 6, 7, 18, 25, 26, 28, 30 Receiver calibration procedure, 307, 308 comparison switching, 295 heterodyne, 293, 296, 297, 299 SSB, 298, 308 stability, 295 y-factor, 294 Red leak, 7, Regularizers, 362, 365 Relative astrometry, 365 Relative photometry, 3, 30, 32, 33 Residual image, 360 Resolution, angular, 287, 291, 296, 301–304, 311, 316, 319–321 Resolved sources, 342 Riemann problem, 496–498 S Scattered-light problems, Scattering, 177, 183, 185–187, 190, 191 Seeing, 4, 5, 21, 25, 27, 31, 33 Self-calibration, 364, 365 Semantic science, 416, 439 Sensitivity, 327, 337, 341–346, 360, 366 telescope, 296, 304, 317 Separated-element interferometry, 329, 365 Serrurier truss, Shocks, 484, 486, 493–499 Short spacing problem, 336 Shutter timing error, 8–10 Signal, 6–8, 10, 22, 24, 26, 28–32 Signal-to-noise, 115, 128–151, 166 Signal-to-noise ratio (SNR), 341–345, 360, 365 Sky flats, 7, 8, 26 Sky surveys, 223–277, 405–409, 413–418, 420–421, 426–428, 438, 439 Slit, 38, 39, 41, 44–54, 56, 58–61, 64, 68, 69, 71, 73–76, 78–80, 83–87, 89, 91, 95, 96 Sloan Digital Sky Survey (SDSS), 406, 413, 418, 420, 428, 429, 432 Smoothed particle hydrodynamics, 483, 484, 492–495 Software, 235, 243, 257, 266, 267, 271–274 Solar motion, standard, 310 Space infrared calibration, 153–158 Space infrared missions, 145 Space Telescope Imaging Spectrograph (STIS), 45, 56, 57, 61, 77 Spatial coherence, 332 Spatial point processes, 448, 472–473 Spectral resolution, 336, 341, 344, 350 Spectral response, 12 Spectrographs, 37–61, 64, 71, 74, 82–91, 95, 96 Spectro-interferometry, 350, 351 Spectrometer, 295, 299–300 Spectrophotometry, 82–84, 86 Spectroscopy, 35–96 Squared visibility, 354 Standard stars, 2, 10, 13, 15, 16, 19, 21, 25, 26 Statistical studies, 227, 263 Statistics, 446–454, 457, 459, 461–475, 477 Stellar photometry, 3, 8, 32 Stellar profile, 5, 27–32 Submm, 103 Superheterodyne signal conversion, 328 Super-resolution, 362 Supervised learning, 419, 421–426 Surface brightness power spectral density, 343 Survey science, 408, 411 Synoptic surveys, 250–256, 262, 264, 276 Synthesis imaging, 327, 328, 343, 348, 353, 357–365 Synthesis imaging array, 328 Synthetic aperture, 24, 25, 28–31 Systematic errors, 341 Systematic exploration, 234–242, 249 System detection efficiency, 344 System noise, 295, 296, 299, 300, 305, 309, 313 System temperature, 342–344 T Technology, 229, 230, 234–236, 240, 245, 249, 255, 271 Temperature antenna, 23, 24, 29, 291, 303, 304, 308, 309 brightness, 290, 291, 303, 304, 309, 313, 317 fluctuations, 317 noise, 290–291, 293, 294, 296–298, 300, 305 Temperature inversion layer, 339 Temporal coherence, 332, 354 Temporal cross-correlation, 336 Temporal smearing, 350 Thermal background, 345 507 508 Index Thermal emission, 106, 109, 113–120, 131, 132, 138, 140 Thermal regime, 337 Time, 2, 3, 5, 7–12, 16–27, 29–31 delay, 332, 363 domain, 228, 236, 237, 245, 246, 249–256, 262 series analysis, 450, 469–472 Time-domain photometry, 3, 22–23 Total detected photoelectrons, 344 Total power component, 336 Total power observing, 311 Transfer equation, 289, 290 Tree methods, 491–492 Troposphere, 338, 340 Truncation, 459, 468–469, 474 Two-dimensional autocorrelation, 347 U Uniform weighting, 360 Unsupervised learning, 419, 421–423 UVFITS, 354 V van-Cittert, 333 Velocity, radial, 310 Very long baseline interferometer (VLBI), 330, 335, 350, 362, 363, 365, 367–369 Vignetting, 8, 20, 26 Virtual Observatory (VO), 230, 235, 266, 270–272, 274, 277, 403–439 Visibility, 332–334, 336, 342, 344, 345, 347, 349, 352, 354–357, 359, 360, 362 Volume absorption coefficient, 307 W Weighting natural, 319 uniform, 319 Wide-field surveys, 226, 231, 244, 258, 259, 264, 276 Wobbling scheme, 309 Y Young’s two-slit experiment, 332, 333 Z Zernike-theorem, 333 Zero optical path delay (OPD), 332 Zero points, 6, 7, 10, 16, 20–22, 26, 27, 32, 33 .. .Planets, Stars and Stellar Systems Astronomical Techniques, Software, and Data Terry D Oswalt (Editor-in-Chief ) Howard E Bond (Volume Editor) Planets, Stars and Stellar Systems Volume... to define his version of the U, B, and V photometric bandpasses of Johnson (1955; also Johnson and Morgan 1953, and Johnson and Harris 1954) and the R and I bandpasses of Cousins (1976; also Kron... complete references on stellar structure and evolution available today Volume 5, edited by Gerard Gilmore, bridges the gap between our understanding of stellar systems and populations seen in