|
|
- Charleen Henderson
- 5 years ago
- Views:
Transcription
1 Synchrotron Power Cosmic rays are astrophysical particles (electrons, protons, and heavier nuclei) with extremely high energies. Cosmic-ray electrons in the galactic magnetic field emit the synchrotron radiation that accounts for most of the continuum emission from our Galaxy at frequencies below about 3 GHz. We can use Larmor's formula to calculate the synchro tron power and synchrotron spectrum of a single electron in an inertial frame in which the electron is instantaneously at rest, but we need the Lorentz transform of special relativity to transform these results to the frame of an observer at rest in the Galaxy. (x; y; z; t) (x ; y ; z ; t ) v x The Lorentz transforms relating the event coordinates in the unprimed frame and the coordinates in the primed frame moving with velocity in the direction are: x = (x + vt ) y = y z = z t = (t + Ìx =c) (5B1) x = (x À vt) y = y z = z t = (t À Ì x=c) (5B) where Ì Ñ v =c (5B3) and Ñ ( 1 À Ì ) À1= (5B4) is called the Lorentz factor. (Áx; Áy; Áz; Á t) (Áx ; Áy ; Áz ; Át ) If and are the coordinate differences between two 1 of 6 1/16/8 11:49 AM
2 events, the differential form of the (linear) Lorentz transforms is: Áx = (Áx + vát ) Á y = Áy Á z = Áz Á t = (Át + ÌÁx =c) (5B Áx = (Áx À vát) Áy = Áy Áz = Áz Át = (Át À Ì Áx=c) (5B6 Here is a derivation of these results that you should review before proceeding. Using the famous equation m e of an electron: E = mc we can calculate the energy equivalent to the rest mass E = m c e = 9:1  1 À8 g  ( 3  1 1 À1 cm s ) = 8:  1 À7 erg 8:  1 À7 erg E = :1 ev :51 MeV 1:6  1 À1 À1 erg (ev) = 5  1 5 = 14 Cosmic-ray electrons with energies in the range to ev have and such cosmic-ray electrons are called ultrarelativistic. These electrons still move on spiral paths along magnetic field lines, but the angular frequencies of their orbits are lower because the inertial masses of the electrons are higher by a factor of :! to 1 Ù Ù 1 3 to 1 8 µ 1 :51  1 6 B = eb! = G m e c = 1 5 B Ù 5  1 À6 Example: A cosmic-ray electron with in the Galactic magnetic field G will have an orbital frequency B Ñ! B Ù 14  1 À5 Hz Ù 1 cycle in two hours: Ù v Ù c µ 1 R Since whenever, the orbital radius of an ultrarelativistic electron is quite large: À1 R Ù c Ù 3  1 1 cm s Ù 3:4  1 13 cm Ù AU! B Ù Â 14  1 À5 Hz At first glance, these results are not very promising for the production of radio radiation: the high relativistic masses of cosmic-ray electrons reduce their orbital frequencies and accelerations to extremely low values. However, the Larmor radiation formula is only valid at of 6 1/16/8 11:49 AM
3 low velocities; that is, in inertial frames in which the electron is nearly at rest. In the observer's frame, two relativistic effects account for the strong radio radiation: (1) the total power is multiplied by and () beaming turns the slow sinusoidal radiation into a series of sharp pulses containing power at much higher frequencies. We proceed to calculate these relativistic corrections. Synchrotron Power From a Single Electron Nonrelativistic equations such as Larmor's equation describing the electromagnetic radiation from an accelerated charge are correct only in inertial frames where the electron velocity v Ü c, but the results can be transformed to any other inertial frame by the Lorentz transform. In this way, it is possible to calculate the total power radiated by an ultrarelativistic electron in a magnetic field parallel to the x -axis. We use primed coordinates to describe an inertial frame in which the electron is (temporarily) nearly at rest. Then Larmor's equation correctly gives What is e (a ) P? = : 3c 3?, the magnetic acceleration of the electron in the galaxy frame? The differential form of the Lorentz transform yields so a ; vy v y = : This factor of is a consequence of relativistic time dilation clocks in moving frames appear to run slow by a factor. Consequently, Similarly, a z = a z= so a? = : Ø 3 B = G dy dy dt dy dt v y Ñ = = = v y dt dt dt dt dt dt dt dt = 1 dt dv y dv y dt a y Ñ = = 1 dt = : dt dt dt dt dt a? dv y a y Thus 3 of 6 1/16/8 11:49 AM
4 P = How do we transform P to P, the power measured by an observer at rest in the Galaxy? The following argument is from Rindler's Essential Relativity, p. 98. Imagine two identical electrons of rest mass m e, one at rest in the unprimed frame and the other at rest in the primed frame. If one electron is slightly displaced from the other along the y -axis, they will interact as they pass each other and be accelerated in the Æy direction. Observers at rest in each frame see "their" electron move with some small, but the "other" electron will appear to move in y the opposite direction by a factor more slowly because of time dilation recall the result above. Invoking momentum conservation, observers in each frame conclude that v = y = v y the "other" electron has inertial mass. Thus? e (a ) 3c 3 v y Ü c = e a? 3c 3 that is, power is the same in all frames. Consequently, m e and hence its energy is greater by the same factor de de dt P Ñ = = de de dt = P 1 ; dt dt dt de dt dt = P P = e a? 4 ( a ) 3c 3 k = 4 Recall that and, by force balance,! B = eb mc so Ë a? Ñ dv? v dt =! B? a? = ebv? ebv sin Ë = ; mc mc ~ v B ~ Ë where the angle between and is called the pitch angle. For a given pitch angle, the time-averaged radiated power of a single electron is e P = eb v sin Ë 3c 3 mc 4 of 6 1/16/8 11:49 AM
5 Ë B ~ ~ v The pitch angle between the directions of the magnetic field and the electron velocity. We can express this power in terms of the Thomson cross section of an electron, T. The Thomson cross section is the classical scattering cross section for electromagnetic radiation. If a plane wave of electromagnetic radiation is incident on a charge at rest, the electric field of that radiation will accelerate the charge, which in turn will radiate power in other directions according to Larmor's equation. This process is called scattering, not absorption, because the total power in electromagnetic radiation is unchanged: all of the power lost from the incident plane wave is reradiated in other directions. In one of the problem sets, you show that the geometric area that would intercept this amount of incident power is Û Û T Ñ 8Ù Ò e Ó 3 m e c (5B7) Numerically, Û T = 8Ù Ô (4:8 Â 1 À1 statcoul) Õ Ù 6:65 Â 1 À5 cm 3 9:1 Â 1 À8 g (3 Â 1 1 À1 cm s ) The reason for using the Thomson cross section will become clear when we discuss inverse- Compton scattering of radiation by the same cosmic rays that are producing synchrotro n radiation. Also, we can replace B by the magnetic energy density B UB = 8Ù (5B8) to get 5 of 6 1/16/8 11:49 AM
6 Ô 8Ù Ò e Ó Õ Ò B Ó v P = c sin Ë 3 mc 8Ù c P = Û T Ì c U B sin Ë (5B9) Ì Ñ v=c where. The radiated power depends only on physical constants, the square of the electron energy (via ; Ì Ù 1 for all µ 1), the magnetic energy density, and the pitch angle. The relativistic electrons in radio sources can have lifetimes of thousands to millions of years before losing their ultrarelativistic energies via synchrotron radiation or other pro cesses, so they are scattered repeatedly by magnetic-field fluctuations and charged particles in their environment, and the distribution of their pitch angles Ë gradually becomes random. The average synchrotron power per electron in an ensemble of electrons with the same Lorentz factor hp i but random pitch angles is therefore hp i = Û Ì T c U hsin B Ëi : Z Z hsin Ëi Ñ sin ËdÊ d Ê = 1 Z sin ËdÊ 4Ù Z Ù Z Ù hsin Ë i = 1 sin Ë sin Ë dë d 4Ù = Ë= hsin Ë i = 1 4 Ù = 4Ù hp i = Û Ì cu 3 T B (5B1) µ 1 This is the average synchrotron power emitted by a relativistic electron. When, Ì Ù 1 and the Ì factor may be ignored. Relativistic effects make the synchrotron power a factor larger than in the limit v Ü c, so for electrons with Ø 1 4, the power radiated by 8 each electron is multiplied by, a huge amount. 1 6 of 6 1/16/8 11:49 AM
- Synchrotron emission: A brief history. - Examples. - Cyclotron radiation. - Synchrotron radiation. - Synchrotron power from a single electron
- Synchrotron emission: A brief history - Examples - Cyclotron radiation - Synchrotron radiation - Synchrotron power from a single electron - Relativistic beaming - Relativistic Doppler effect - Spectrum
More informationLorentz Force. Acceleration of electrons due to the magnetic field gives rise to synchrotron radiation Lorentz force.
Set 10: Synchrotron Lorentz Force Acceleration of electrons due to the magnetic field gives rise to synchrotron radiation Lorentz force 0 E x E y E z dp µ dτ = e c F µ νu ν, F µ E x 0 B z B y ν = E y B
More information1 Monday, November 7: Synchrotron Radiation for Beginners
1 Monday, November 7: Synchrotron Radiation for Beginners An accelerated electron emits electromagnetic radiation. The most effective way to accelerate an electron is to use electromagnetic forces. Since
More informationSpecial relativity and light RL 4.1, 4.9, 5.4, (6.7)
Special relativity and light RL 4.1, 4.9, 5.4, (6.7) First: Bremsstrahlung recap Braking radiation, free-free emission Important in hot plasma (e.g. coronae) Most relevant: thermal Bremsstrahlung What
More informationRadiative Processes in Astrophysics
Radiative Processes in Astrophysics 11. Synchrotron Radiation & Compton Scattering Eline Tolstoy http://www.astro.rug.nl/~etolstoy/astroa07/ Synchrotron Self-Absorption synchrotron emission is accompanied
More information1 Monday, November 21: Inverse Compton Scattering
1 Monday, November 21: Inverse Compton Scattering When I did the calculations for the scattering of photons from electrons, I chose (for the sake of simplicity) the inertial frame of reference in which
More informationNotes on x-ray scattering - M. Le Tacon, B. Keimer (06/2015)
Notes on x-ray scattering - M. Le Tacon, B. Keimer (06/2015) Interaction of x-ray with matter: - Photoelectric absorption - Elastic (coherent) scattering (Thomson Scattering) - Inelastic (incoherent) scattering
More informationRadiative processes from energetic particles II: Gyromagnetic radiation
Hale COLLAGE 2017 Lecture 21 Radiative processes from energetic particles II: Gyromagnetic radiation Bin Chen (New Jersey Institute of Technology) e - Shibata et al. 1995 e - magnetic reconnection Previous
More informationno incoming fields c D r
A x 4 D r xx ' J x ' d 4 x ' no incoming fields c D r xx ' : the retarded Green function e U x 0 r 0 xr d J e c U 4 x ' r d xr 0 0 x r x x xr x r xr U f x x x i d f d x x xi A x e U Ux r 0 Lienard - Wiechert
More informationPropagation in the Galaxy 2: electrons, positrons, antiprotons
Propagation in the Galaxy 2: electrons, positrons, antiprotons As we mentioned in the previous lecture the results of the propagation in the Galaxy depend on the particle interaction cross section. If
More informationRadiative Processes in Astrophysics
Radiative Processes in Astrophysics 9. Synchrotron Radiation Eline Tolstoy http://www.astro.rug.nl/~etolstoy/astroa07/ Useful reminders relativistic terms, and simplifications for very high velocities
More information- Potentials. - Liénard-Wiechart Potentials. - Larmor s Formula. - Dipole Approximation. - Beginning of Cyclotron & Synchrotron
- Potentials - Liénard-Wiechart Potentials - Larmor s Formula - Dipole Approximation - Beginning of Cyclotron & Synchrotron Maxwell s equations in a vacuum become A basic feature of these eqns is the existence
More information5. SYNCHROTRON RADIATION 1
5. SYNCHROTRON RADIATION 1 5.1 Charge motions in a static magnetic field Charged particles moving inside a static magnetic field continuously accelerate due to the Lorentz force and continuously emit radiation.
More informationFinal Exam Sample Problems
UNIVERSITY OF ALABAMA Department of Physics and Astronomy PH 253 / LeClair Spring 2010 Final Exam Sample Problems 1. The orbital speed of the Earth around the Sun is 30 km/s. In one year, how many seconds
More informationRadiation processes and mechanisms in astrophysics I. R Subrahmanyan Notes on ATA lectures at UWA, Perth 18 May 2009
Radiation processes and mechanisms in astrophysics I R Subrahmanyan Notes on ATA lectures at UWA, Perth 18 May 009 Light of the night sky We learn of the universe around us from EM radiation, neutrinos,
More informationAstrophysical Radiation Processes
PHY3145 Topics in Theoretical Physics Astrophysical Radiation Processes 3: Relativistic effects I Dr. J. Hatchell, Physics 407, J.Hatchell@exeter.ac.uk Course structure 1. Radiation basics. Radiative transfer.
More informationimin...
Pulsar Timing For a detailed look at pulsar timing and other pulsar observing techniques, see the Handbook of Pulsar Astronomy by Duncan Lorimer and Michael Kramer. Pulsars are intrinsically interesting
More informationCompton Scattering I. 1 Introduction
1 Introduction Compton Scattering I Compton scattering is the process whereby photons gain or lose energy from collisions with electrons. It is an important source of radiation at high energies, particularly
More informationRadiative Processes in Astrophysics
Radiative Processes in Astrophysics 6. Relativistic Covariance & Kinematics Eline Tolstoy http://www.astro.rug.nl/~etolstoy/astroa07/ Practise, practise, practise... mid-term, 31st may, 9.15-11am As we
More information4 Relativistic kinematics
4 Relativistic kinematics In astrophysics, we are often dealing with relativistic particles that are being accelerated by electric or magnetic forces. This produces radiation, typically in the form of
More informationHomework 11. Relativity Problems PH3110 Fall 2006 Due 12/6/06
Homework 11. Relativity Problems PH3110 Fall 006 Due 1/6/06 1. F&C 5.13. Complete the time dilation derivation we started in class based on the light reflecting off of the mirror eperiment. Show that t
More informationNeutrinos, nonzero rest mass particles, and production of high energy photons Particle interactions
Neutrinos, nonzero rest mass particles, and production of high energy photons Particle interactions Previously we considered interactions from the standpoint of photons: a photon travels along, what happens
More informationSynchrotron Radiation II
Synchrotron Radiation II 1 Synchrotron radiation from Astrophysical Sources. 1.1 Distributions of electrons In this chapter we shall deal with synchrotron radiation from two different types of distribution
More informationPARTICLE ACCELERATORS
VISUAL PHYSICS ONLINE PARTICLE ACCELERATORS Particle accelerators are used to accelerate elementary particles to very high energies for: Production of radioisotopes Probing the structure of matter There
More informationCosmic Rays: I. General Phenomenology, Energy Loss, and Electromagnetic Signatures Friday, March 4, 2011
Cosmic Rays: I. General Phenomenology, Energy Loss, and Electromagnetic Signatures Friday, March 4, 2011 CONTENTS: 1. Introduction 2. General Phenomenology 3. Energy Loss Mechanisms A. Electromagnetic
More informationPH 253 Exam I Solutions
PH 253 Exam I Solutions. An electron and a proton are each accelerated starting from rest through a potential difference of 0.0 million volts (0 7 V). Find the momentum (in MeV/c) and kinetic energy (in
More informationSingle particle motion
Single particle motion Plasma is a collection of a very large number of charged particles moving in, and giving rise to, electromagnetic fields. Before going to the statistical descriptions, let us learn
More informationPhysics 111 Homework Solutions Week #9 - Thursday
Physics 111 Homework Solutions Week #9 - Thursday Monday, March 1, 2010 Chapter 24 241 Based on special relativity we know that as a particle with mass travels near the speed of light its mass increases
More informationHIGH ENERGY ASTROPHYSICS - Lecture 7. PD Frank Rieger ITA & MPIK Heidelberg Wednesday
HIGH ENERGY ASTROPHYSICS - Lecture 7 PD Frank Rieger ITA & MPIK Heidelberg Wednesday 1 (Inverse) Compton Scattering 1 Overview Compton Scattering, polarised and unpolarised light Di erential cross-section
More informationGeneral Physics (PHY 2140) Lecture 14
General Physics (PHY 2140) Lecture 14 Modern Physics 1. Relativity Einstein s General Relativity 2. Quantum Physics Blackbody Radiation Photoelectric Effect X-Rays Diffraction by Crystals The Compton Effect
More informationNuclear Fusion and Radiation
Nuclear Fusion and Radiation Lecture 2 (Meetings 3 & 4) Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Nuclear Fusion and Radiation p. 1/41 Modern Physics Concepts
More informationPHY313 - CEI544 The Mystery of Matter From Quarks to the Cosmos Fall 2005
PHY313 - CEI544 The Mystery of Matter From Quarks to the Cosmos Fall 2005 Peter Paul Office Physics D-143 www.physics.sunysb.edu PHY313 Peter Paul 09/8/05 PHY313-CEI544 Fall-05 1 The Energy Scales of Matter
More informationRetarded Potentials and Radiation
Retarded Potentials and Radiation No, this isn t about potentials that were held back a grade :). Retarded potentials are needed because at a given location in space, a particle feels the fields or potentials
More informationThe incident energy per unit area on the electron is given by the Poynting vector, '
' ' # Thompson Scattering Consider a beam of radiation moving in the direction, and being scattered by an electron through an angle in the plane. The electron experiences the following electric fields
More information8.04 Spring 2013 February 13, 2013 Problem 1. (15 points) Radiative collapse of a classical atom
Problem Set 1 Solutions 8.04 Spring 01 February 1, 01 Problem 1. (15 points) Radiative collapse of a classical atom (a) (5 points) We begin by assuming that the orbit is circular. This seems like circular
More informationCompton Scattering II
Compton Scattering II 1 Introduction In the previous chapter we considered the total power produced by a single electron from inverse Compton scattering. This is useful but limited information. Here we
More informationRecap Lecture + Thomson Scattering. Thermal radiation Blackbody radiation Bremsstrahlung radiation
Recap Lecture + Thomson Scattering Thermal radiation Blackbody radiation Bremsstrahlung radiation LECTURE 1: Constancy of Brightness in Free Space We use now energy conservation: de=i ν 1 da1 d Ω1 dt d
More informationWe start with a reminder of a few basic concepts in probability. Let x be a discrete random variable with some probability function p(x).
1 Probability We start with a reminder of a few basic concepts in probability. 1.1 discrete random variables Let x be a discrete random variable with some probability function p(x). 1. The Expectation
More informationThe Bohr Model of Hydrogen
The Bohr Model of Hydrogen Suppose you wanted to identify and measure the energy high energy photons. One way to do this is to make a calorimeter. The CMS experiment s electromagnetic calorimeter is made
More informationChapter 2 Radiation of an Accelerated Charge
Chapter 2 Radiation of an Accelerated Charge Whatever the energy source and whatever the object, (but with the notable exception of neutrino emission that we will not consider further, and that of gravitational
More informationRelativistic Effects. 1 Introduction
Relativistic Effects 1 Introduction 10 2 6 The radio-emitting plasma in AGN contains electrons with relativistic energies. The Lorentz factors of the emitting electrons are of order. We now know that the
More informationSynchrotron Radiation I
Synchrotron Radiation I Examples of synchrotron emitting plasma Following are some examples of astrophysical objects that are emitting synchrotron radiation. These include radio galaxies, quasars and supernova
More informationPhysics 111 Homework Solutions Week #9 - Friday
Physics 111 Homework Solutions Week #9 - Friday Tuesday, March 1, 2011 Chapter 24 Questions 246 The Compton shift in wavelength for the proton and the electron are given by Δλ p = h ( 1 cosφ) and Δλ e
More informationAnnouncements. Muon Lifetime. Lecture 4 Chapter. 2 Special Relativity. SUMMARY Einstein s Postulates of Relativity: EXPERIMENT
Announcements HW1: Ch.2-20, 26, 36, 41, 46, 50, 51, 55, 58, 63, 65 Lab start-up meeting with TA tomorrow (1/26) at 2:00pm at room 301 Lab manual is posted on the course web *** Course Web Page *** http://highenergy.phys.ttu.edu/~slee/2402/
More informationModern Physics. Third Edition RAYMOND A. SERWAY CLEMENT J. MOSES CURT A. MOYER
Modern Physics Third Edition RAYMOND A. SERWAY CLEMENT J. MOSES CURT A. MOYER 1 RELATIVITY 1.1 Special Relativity 1.2 The Principle of Relativity, The Speed of Light 1.3 The Michelson Morley Experiment,
More informationCrab Pulsar. Chandra Image of the Crab Nebula. Crab is the most famous pulsar, which is studied in detail across the entire energy spectrum
Crab Pulsar Chandra Image of the Crab Nebula Crab is the most famous pulsar, which is studied in detail across the entire energy spectrum Conventional view on the Crab Pulsar Related Emitting Zones Pulsar(Massaro+)
More informationSynchrotron Radiation: II. Spectrum
Synchrotron Radiation: II. Spectrum Massimo Ricotti ricotti@astro.umd.edu University of Maryland Synchrotron Radiation: II. Spectrum p.1/18 ds=v dt_em dt=ds cos(theta)/c=v/c cos(theta)dt_em Synchrotron
More informationCHAPTER 27. Continuum Emission Mechanisms
CHAPTER 27 Continuum Emission Mechanisms Continuum radiation is any radiation that forms a continuous spectrum and is not restricted to a narrow frequency range. In what follows we briefly describe five
More informationSynchrotron Radiation II
Synchrotron Radiation II Summary of Radiation Properties Thermal Blackbody Bremsstrahlung Synchrotron Optically thick YES NO Maxwellian distribution of velocities YES YES NO Relativistic speeds YES Main
More informationShort Course on High Energy Astrophysics. Exploring the Nonthermal Universe with High Energy Gamma Rays
Short Course on High Energy Astrophysics Exploring the Nonthermal Universe with High Energy Gamma Rays Lecture 1: Introduction Felix Aharonian Dublin Institute for Advanced Studies, Dublin Max-Planck Institut
More informationPhysics 504, Lecture 22 April 19, Frequency and Angular Distribution
Last Latexed: April 16, 010 at 11:56 1 Physics 504, Lecture April 19, 010 Copyright c 009 by Joel A Shapiro 1 Freuency and Angular Distribution We have found the expression for the power radiated in a
More informationAtom Model and Relativity
Atom Model and Relativity Kimmo Rouvari March, 204 Abstract What is the theoretical explanation for fine structure? What is the mechanism behind relativity? These questions have bothered numerous physicists
More information1. Why photons? 2. Photons in a vacuum
Photons and Other Messengers 1. Why photons? Ask class: most of our information about the universe comes from photons. What are the reasons for this? Let s compare them with other possible messengers,
More informationNuclear Physics and Astrophysics
Nuclear Physics and Astrophysics PHY-30 Dr. E. Rizvi Lecture 4 - Detectors Binding Energy Nuclear mass MN less than sum of nucleon masses Shows nucleus is a bound (lower energy) state for this configuration
More informationUltrahigh Energy Cosmic Rays propagation I
Ultrahigh Energy Cosmic Rays propagation I Microwave background Energy loss processes for protons: - photoproduction interactions - pair production interactions - adiabatic loss due to the expansion of
More informationLecture 9 - Applications of 4 vectors, and some examples
Lecture 9 - Applications of 4 vectors, and some examples E. Daw April 4, 211 1 Review of invariants and 4 vectors Last time we learned the formulae for the total energy and the momentum of a particle in
More informationr,t r R Z j ³ 0 1 4π² 0 r,t) = 4π
5.4 Lienard-Wiechert Potential and Consequent Fields 5.4.1 Potential and Fields (chapter 10) Lienard-Wiechert potential In the previous section, we studied the radiation from an electric dipole, a λ/2
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department. Problem Set 5 Solutions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.033 October, 003 Problem Set 5 Solutions Problem A Flying Brick, Resnick & Halliday, #, page 7. (a) The length contraction factor along
More informationIntroduction. Classical vs Modern Physics. Classical Physics: High speeds Small (or very large) distances
Introduction Classical vs Modern Physics High speeds Small (or very large) distances Classical Physics: Conservation laws: energy, momentum (linear & angular), charge Mechanics Newton s laws Electromagnetism
More informationChapter 1. From Classical to Quantum Mechanics
Chapter 1. From Classical to Quantum Mechanics Classical Mechanics (Newton): It describes the motion of a classical particle (discrete object). dp F ma, p = m = dt dx m dt F: force (N) a: acceleration
More informationFinal Exam - Solutions PHYS/ECE Fall 2011
Final Exam - Solutions PHYS/ECE 34 - Fall 211 Problem 1 Cosmic Rays The telescope array project in Millard County, UT can detect cosmic rays with energies up to E 1 2 ev. The cosmic rays are of unknown
More informationRelativity. An explanation of Brownian motion in terms of atoms. An explanation of the photoelectric effect ==> Quantum Theory
Relativity Relativity In 1905 Albert Einstein published five articles in Annalen Der Physik that had a major effect upon our understanding of physics. They included:- An explanation of Brownian motion
More informationRotational Mechanics and Relativity --- Summary sheet 1
Rotational Mechanics and Relativity --- Summary sheet 1 Centre of Mass 1 1 For discrete masses: R m r For continuous bodies: R dm i i M M r body i Static equilibrium: the two conditions for a body in static
More informationShock Waves. = 0 (momentum conservation)
PH27: Aug-Dec 2003 Shock Waves A shock wave is a surface of discontinuity moving through a medium at a speed larger than the speed of sound upstream. The change in the fluid properties upon passing the
More informationChapter V: Interactions of neutrons with matter
Chapter V: Interactions of neutrons with matter 1 Content of the chapter Introduction Interaction processes Interaction cross sections Moderation and neutrons path For more details see «Physique des Réacteurs
More information1240 ev nm nm. f < f 0 (5)
Chapter 4 Example of Bragg Law The spacing of one set of crystal planes in NaCl (table salt) is d = 0.282 nm. A monochromatic beam of X-rays produces a Bragg maximum when its glancing angle with these
More information3145 Topics in Theoretical Physics - radiation processes - Dr J Hatchell. Multiwavelength Milky Way
Multiwavelength Milky Way PHY3145 Topics in Theoretical Physics Astrophysical Radiation Processes Dr. J. Hatchell, Physics 406, J.Hatchell@exeter.ac.uk Textbooks Main texts Rybicki & Lightman Radiative
More informationLecture 14 (11/1/06) Charged-Particle Interactions: Stopping Power, Collisions and Ionization
22.101 Applied Nuclear Physics (Fall 2006) Lecture 14 (11/1/06) Charged-Particle Interactions: Stopping Power, Collisions and Ionization References: R. D. Evans, The Atomic Nucleus (McGraw-Hill, New York,
More informationUnit- 1 Theory of Relativity
Unit- 1 Theory of Relativity Frame of Reference The Michelson-Morley Experiment Einstein s Postulates The Lorentz Transformation Time Dilation and Length Contraction Addition of Velocities Experimental
More informationRadiation processes and mechanisms in astrophysics 3. R Subrahmanyan Notes on ATA lectures at UWA, Perth 22 May 2009
Radiation proesses and mehanisms in astrophysis R Subrahmanyan Notes on ATA letures at UWA, Perth May 009 Synhrotron radiation - 1 Synhrotron radiation emerges from eletrons moving with relativisti speeds
More information1. (16) A point charge e moves with velocity v(t) on a trajectory r(t), where t is the time in some lab frame.
Electrodynamics II Exam 3. Part A (120 pts.) Closed Book Radiation from Acceleration Name KSU 2016/05/10 14:00-15:50 Instructions: Some small derivations here, state your responses clearly, define your
More informationPHYS 5012 Radiation Physics and Dosimetry
Radiative PHYS 5012 Radiation Physics and Dosimetry Mean Tuesday 24 March 2009 Radiative Mean Radiative Mean Collisions between two particles involve a projectile and a target. Types of targets: whole
More informationColumbia University Department of Physics QUALIFYING EXAMINATION
Columbia University Department of Physics QUALIFYING EXAMINATION Wednesday, January 13, 2016 3:10PM to 5:10PM Modern Physics Section 4. Relativity and Applied Quantum Mechanics Two hours are permitted
More informationAnnouncement. Einstein s Postulates of Relativity: PHYS-3301 Lecture 3. Chapter 2. Sep. 5, Special Relativity
Announcement PHYS-3301 Lecture 3 Sep. 5, 2017 2 Einstein s Postulates of Relativity: Chapter 2 Special Relativity 1. Basic Ideas 6. Velocity Transformation 2. Consequences of Einstein s Postulates 7. Momentum
More informationPulsars. The maximum angular frequency of a spinning star can be found by equating the centripetal and gravitational acceleration M R 2 R 3 G M
Pulsars Pulsating stars were discovered in 1967 via radio dipole antennae by Jocelyn Bell and Anthony Hewish Pulse period of PSR 1919+21 is 1.337 s Most pulsars have periods between 0.25 s and 2 s The
More informationParticle acceleration and generation of high-energy photons
Particle acceleration and generation of high-energy photons For acceleration, see Chapter 21 of Longair Ask class: suppose we observe a photon with an energy of 1 TeV. How could it have been produced?
More informationHigh-Energy Astrophysics
M.Phys. & M.Math.Phys. High-Energy Astrophysics Garret Cotter garret.cotter@physics.ox.ac.uk High-Energy Astrophysics MT 2016 Lecture 2 High-Energy Astrophysics: Synopsis 1) Supernova blast waves; shocks.
More informationdt = p m, (2.1.1) dt = p
Chapter 2 Special relativity 2.1 Galilean relativity We start our discussion of symmetries by considering an important example of an invariance, i.e. an invariance of the equations of motion under a change
More informationModern Physics for Scientists and Engineers International Edition, 4th Edition
Modern Physics for Scientists and Engineers International Edition, 4th Edition http://optics.hanyang.ac.kr/~shsong Review: 1. THE BIRTH OF MODERN PHYSICS 2. SPECIAL THEORY OF RELATIVITY 3. THE EXPERIMENTAL
More informationModule II: Relativity and Electrodynamics
Module II: Relativity and Electrodynamics Lecture 2: Lorentz transformations of observables Amol Dighe TIFR, Mumbai Outline Length, time, velocity, acceleration Transformations of electric and magnetic
More informationPhysics 126 Practice Exam #4 Professor Siegel
Physics 126 Practice Exam #4 Professor Siegel Name: Lab Day: 1. Light is usually thought of as wave-like in nature and electrons as particle-like. In which one of the following instances does light behave
More informationParticle Dynamics Particle Dynamics
2 Particle Dynamics Understanding and utilizing the response of charged particles to electromagnetic forces is the basis of particle optics and accelerator theory. The goal is to find the time-dependent
More informationSolutions for Assignment of Week 06 Introduction to Astroparticle Physics
s for Assignment of Week 06 Introduction to Astroparticle Physics Georg G. Raffelt Max-Planck-Institut für Physik (Werner-Heisenberg-Institut) Föhringer Ring 6, 80805 München Email: raffelt(at)mppmu.mpg.de
More informationExamples of relativistic transformations
Examples of relativistic transformations Lecture 9 1 Field transformations In the last lecture we obtained the field transformation equations. For a boost in the 1 direction E 1 = E 1 ; B 1 = B 1 E 2 =
More informationHigh energy neutrinos from curvature pions in magnetars
High energy neutrinos from curvature pions in magnetars Tamás Herpay MTA ELTE, Statistical and Biological Physics Research Group Collaborators: Péter Mészáros, András Patkós, Soeb Razzaque Motivation Neutrino
More informationNewton s Laws of Motion, Energy and Oscillations
Prof. O. B. Wright, Autumn 007 Mechanics Lecture Newton s Laws of Motion, Energy and Oscillations Reference frames e.g. displaced frame x =x+a y =y x =z t =t e.g. moving frame (t=time) x =x+vt y =y x =z
More informationBethe-Block. Stopping power of positive muons in copper vs βγ = p/mc. The slight dependence on M at highest energies through T max
Bethe-Block Stopping power of positive muons in copper vs βγ = p/mc. The slight dependence on M at highest energies through T max can be used for PID but typically de/dx depend only on β (given a particle
More informationElectrodynamics of Radiation Processes
Electrodynamics of Radiation Processes 7. Emission from relativistic particles (contd) & Bremsstrahlung http://www.astro.rug.nl/~etolstoy/radproc/ Chapter 4: Rybicki&Lightman Sections 4.8, 4.9 Chapter
More informationThermal Radiation Studies for an Electron-Positron Annihilation Propulsion System
Thermal Radiation Studies for an Electron-Positron Annihilation Propulsion System Jonathan A. Webb Embry Riddle Aeronautical University Prescott, AZ 8631 Recent studies have shown the potential of antimatter
More information- Covered thus far. - Specific Intensity, mean intensity, flux density, momentum flux. - Emission and absorp>on coefficients, op>cal depth
- Covered thus far - Specific Intensity, mean intensity, flux density, momentum flux - Emission and absorp>on coefficients, op>cal depth - Radia>ve transfer equa>on - Planck func>on, Planck spectrum, brightness
More informationAstronomy 421. Lecture 23: End states of stars - Neutron stars
Astronomy 421 Lecture 23: End states of stars - Neutron stars 1 Outline Neutron stars Pulsars properties distribution emission mechanism evolution 2 Neutron stars Typical values: M ~ 1.4M R ~ 10 km ρ ~
More informationChapter 2 Problem Solutions
Chapter Problem Solutions 1. If Planck's constant were smaller than it is, would quantum phenomena be more or less conspicuous than they are now? Planck s constant gives a measure of the energy at which
More informationAn Introduction to Diffraction and Scattering. School of Chemistry The University of Sydney
An Introduction to Diffraction and Scattering Brendan J. Kennedy School of Chemistry The University of Sydney 1) Strong forces 2) Weak forces Types of Forces 3) Electromagnetic forces 4) Gravity Types
More informationPHYS 280 Midterm α Fall You may answer the questions in the space provided here, or if you prefer, on your own notebook paper.
PHYS 280 Midterm α Fall 2014 Name: You may answer the questions in the space provided here, or if you prefer, on your own notebook paper. Short answers 1. If you are measuring an astrophysical phenomenon,
More informationCompton Scattering. hω 1 = hω 0 / [ 1 + ( hω 0 /mc 2 )(1 cos θ) ]. (1) In terms of wavelength it s even easier: λ 1 λ 0 = λ c (1 cos θ) (2)
Compton Scattering Last time we talked about scattering in the limit where the photon energy is much smaller than the mass-energy of an electron. However, when X-rays and gamma-rays are considered, this
More informationAtom Model and Relativity
Atom Model and Relativity Kimmo Rouvari September 8, 203 Abstract What is the theoretical explanation for fine structure? What is the mechanism behind relativity? These questions have bothered numerous
More informationUltra High Energy Cosmic Rays I
Ultra High Energy Cosmic Rays I John Linsley (PRL 10 (1963) 146) reports on the detection in Vulcano Ranch of an air shower of energy above 1020 ev. Problem: the microwave background radiation is discovered
More informationAccretion Disks. 1. Accretion Efficiency. 2. Eddington Luminosity. 3. Bondi-Hoyle Accretion. 4. Temperature profile and spectrum of accretion disk
Accretion Disks Accretion Disks 1. Accretion Efficiency 2. Eddington Luminosity 3. Bondi-Hoyle Accretion 4. Temperature profile and spectrum of accretion disk 5. Spectra of AGN 5.1 Continuum 5.2 Line Emission
More informationA) n L < 1.0 B) n L > 1.1 C) n L > 1.3 D) n L < 1.1 E) n L < 1.3
1. A beam of light passes from air into water. Which is necessarily true? A) The frequency is unchanged and the wavelength increases. B) The frequency is unchanged and the wavelength decreases. C) The
More information