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GENERAL RELATIVITY
General Relativity PHY-5-GenRel U01429 16 lectures Alan Heavens, School of Physics, University of Edinburgh afh@roe.ac.uk Mathematical references: RHB = Riley, Hobson and Bence, Mathematical Methods forPhysics
FORTRAN 90 ARRAYS
Fortran 90 Lecture 5 AS 3013 3 Arrays (cont.)• Each array element is a variable just like any scalar variable. LECTURE 5: SURFACE BRIGHTNESS AND THE SERSIC PROFILE 4 Galaxies – AS 3011 7 The Sersic profile • Empirically devised by Sersic (1963) as a good fitting fn I(r) = intensity at radius r I 0 = central intensity (intensity at centre) α = scalelength (radius at which intensity drops by e-1) n = Sersic index (shape parameter) PHOTONS (RADIATION FIELDS) THE TOTAL ENERGY DENSITY OF THE Cold gas or Cold DM ( p > mc > 0) E ≈ m c2 = const ε M ≈ N m c 2 R3 ∝ R−3 λ∝ R (wavelengths stretch) : E = h ν= h c λ ∝ R−1 ε R = N h ν R3 ∝ R−4 volume R3 N particles particle mass m momentum p energy E = hν= m2c 4 + THE COPERNICAN REVOLUTION AS1001:EXTRA-GALACTIC ASTRONOMY Abell 2218 a giant gravitational lens! Cosmology in the 1920s! • Einstein’s GR, applied to the Universe as a whole, (uniform density ρ, pressure p) gives 2 equations: • These describe a Dynamical Universe: R( t ) • Universe must be expanding or collapsing. • If ρ (density) and p (pressure) are positive, then an expanding Universedecelerates.
TOM ELSDEN | HOME
Tom Elsden. Hi! I've recently moved from the University of St Andrews to the University of Leicester, to take up an Early Career Fellowship funded by the Leverhulme trust. I'm working within the Radio and Space Plasma Physics Group in the School of Physics and Astronomy. My research involves the study of ultra low frequency (ULF) waves inEarth
WIDTHS OF SPECTRAL LINES • This effect is called natural broadening. • If the spontaneous decay of an atomic state n(to all lower energy levels n’) proceeds at a rate, it can be shown (see damped harmonic oscillator derivations in Chapter 3.6 of Rybicki& Lightman) thatthe line profile fortransitions
FORTRAN 90 INPUT AND OUTPUT (I/O) Fortran Lecture 6 Press F1 for HelpFortran Lecture 6 Pre1ss F1 forHelp
JAPANESE ORGANISATION Japanese Organisation in WW2 Type "B" Japanese Infantry Company (1944-45) Headquarters 3 Platoons Headquarters 3 Squads 1 x 6.5mm LMG 12 x 6.5mm Rifle Weapons Squad PH5011 GENERAL RELATIVITY PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013GENERAL RELATIVITY
General Relativity PHY-5-GenRel U01429 16 lectures Alan Heavens, School of Physics, University of Edinburgh afh@roe.ac.uk Mathematical references: RHB = Riley, Hobson and Bence, Mathematical Methods forPhysics
FORTRAN 90 ARRAYS
Fortran 90 Lecture 5 AS 3013 3 Arrays (cont.)• Each array element is a variable just like any scalar variable. LECTURE 5: SURFACE BRIGHTNESS AND THE SERSIC PROFILE 4 Galaxies – AS 3011 7 The Sersic profile • Empirically devised by Sersic (1963) as a good fitting fn I(r) = intensity at radius r I 0 = central intensity (intensity at centre) α = scalelength (radius at which intensity drops by e-1) n = Sersic index (shape parameter) PHOTONS (RADIATION FIELDS) THE TOTAL ENERGY DENSITY OF THE Cold gas or Cold DM ( p > mc > 0) E ≈ m c2 = const ε M ≈ N m c 2 R3 ∝ R−3 λ∝ R (wavelengths stretch) : E = h ν= h c λ ∝ R−1 ε R = N h ν R3 ∝ R−4 volume R3 N particles particle mass m momentum p energy E = hν= m2c 4 + THE COPERNICAN REVOLUTION AS1001:EXTRA-GALACTIC ASTRONOMY Abell 2218 a giant gravitational lens! Cosmology in the 1920s! • Einstein’s GR, applied to the Universe as a whole, (uniform density ρ, pressure p) gives 2 equations: • These describe a Dynamical Universe: R( t ) • Universe must be expanding or collapsing. • If ρ (density) and p (pressure) are positive, then an expanding Universedecelerates.
TOM ELSDEN | HOME
Tom Elsden. Hi! I've recently moved from the University of St Andrews to the University of Leicester, to take up an Early Career Fellowship funded by the Leverhulme trust. I'm working within the Radio and Space Plasma Physics Group in the School of Physics and Astronomy. My research involves the study of ultra low frequency (ULF) waves inEarth
WIDTHS OF SPECTRAL LINES • This effect is called natural broadening. • If the spontaneous decay of an atomic state n(to all lower energy levels n’) proceeds at a rate, it can be shown (see damped harmonic oscillator derivations in Chapter 3.6 of Rybicki& Lightman) thatthe line profile fortransitions
FORTRAN 90 INPUT AND OUTPUT (I/O) Fortran Lecture 6 Press F1 for HelpFortran Lecture 6 Pre1ss F1 forHelp
JAPANESE ORGANISATION Japanese Organisation in WW2 Type "B" Japanese Infantry Company (1944-45) Headquarters 3 Platoons Headquarters 3 Squads 1 x 6.5mm LMG 12 x 6.5mm Rifle Weapons Squad PH5011 GENERAL RELATIVITY PH5011 General Relativity Dr HongSheng Zhao shortened/expanded from notes of MD hz4@st-andrews.ac.uk Martinmas 2012/2013KENNETH FALCONER
Kenneth Falconer is a mathematician who specialises in Fractal Geometry and related topics. He is Regius Professor of Mathematics at the University of St Andrews and a member of the Analysis Research Group of the School of Mathematics and Statistics.He was an undergraduate (graduating as a Wrangler), research student and Research Fellow at Corpus Christi College, Cambridge.CONFORMAL GRAVITY
Conformal Gravity. CONFORMAL SYMMETRY: Conformal Gravity is based on a local symmetry principle, conformal stretching of the space-time metric, which is observed by the strong, weak, and electro-magnetic forces, but not by Einstein's General Relativity. Conformal symmetry eliminates all terms in the action except conformally invariant terms WELCOME TO THE TURNBULL SERVER Welcome to the TURNBULL world wide web server, named after the mathematician Herbert Westren Turnbull (1885-1961) who was the Regius Professor of Mathematics at the University of St Andrews from 1921 to 1950. To find out more about Turnbull you can read his biography. TURNBULL is run by the School of Mathematics and Statistics at the University of St Andrews, St Andrews, Fife, Scotland.CIRCA | NEWS
CIRCA members took part in the Research Day, held in the School of Mathematics and Statistics on 24th January 2019. The speakers were Rosemary Bailey, Collin Bleak, Peter Cameron, Martyn Quick, Louis Theran, and Peter Cameron. Alexander Konovalov and Michael Torpey presented a poster. Posted on January 26th, 2019.TOM ELSDEN | HOME
Tom Elsden. Hi! I've recently moved from the University of St Andrews to the University of Leicester, to take up an Early Career Fellowship funded by the Leverhulme trust. I'm working within the Radio and Space Plasma Physics Group in the School of Physics and Astronomy. My research involves the study of ultra low frequency (ULF) waves inEarth
COLVA M. RONEY-DOUGAL E-Mail: colva.roney-dougal@st-andrews.ac.uk Tel: 01334 463806 School of Mathematics and Statistics The University of St Andrews North Haugh, St Andrews Fife, KY16 9SS The views and opinions expressed in this page are my own. WHAT'S THE MOON DOING? The Moon rises and sets every day, like the Sun. But the Sun always rises in the morning and sets in the evening; the Moon does it at a different time every day. At New Moon, the Moon lies in the same direction as the Sun. But the Moon is orbiting around the Earth; every day, it moves eastwards (further left from the Sun) by about 12degrees.
LECTURE 6: JEANS MASS & LENGTH Lecture 6: Jeans mass & length Anisotropies in the CMB temperature density ripples at the time of decoupling ( z = 1100 ) . These are the seeds that evolve (gravitational collapse) INTRODUCTION TO SCIENTI C COMPUTING IN PYTHON Contents 1 Introduction to scienti c computing with Python6 1.1 The role of computing in science. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 MATH 354 SUMMER 2004 HOMEWORK #2 SOLUTIONS Math 354 Summer 2004 5 Let A be an m×n matrix and let b be a vector in Rm.Prove that the set of vectors x in Rn satisfying Ax = b is aconvex set.
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