#### Transcript Lecture 1 Coordinate Systems - Department of Physics & Astronomy

Review Ch 1-6 • The Celestial Sphere – History – Positions • Celestial Mechanics – – – – Elliptical Orbits Newtonian Mechanics Kepler’s Laws Virial Theorem • Continuous Spectrum of Light – – – – – – Stellar Parallax The Magnitude Scale The Wave Nature of light Blackbody Radiation Quantization of Energy The Color Index •Special Relativity –Lorentz Transformations –Time and Space in Relativity –Relativistic Momentum –Redshift •Interaction of Light and Matter –Spectral Lines –Photons –The Bohr Model of the Atom –Quantum Mechanics and WaveParticle Duality •Telescopes –Basic Optics –Optical Telescopes –Radio Telescopes –Infrared,UV,X-ray and GammaRay astronomy Midterm Exam 1 • The exam – Part I - in-class • Conceptual questions • “Easy” calculations (bring a calculator) • Ten sheets (back and front) of notes (no xerographic reduction) – Part II - Take-home • Application of the material to Astrophysical Problems • More of a pedagogical learning exercise • Preparation – Material from textbook ch 1-6 • Concepts • Equations (no derivations in classroom portion of exam) – Examples from textbook – Homework problems!!! See solutions on web… Possible Problem Topics • • • • • • • • • Motions of heavenly bodies Position on celestial sphere Elliptical Orbits Newtonian Mechanics and Kepler’s Laws Escape Velocity Stellar Parallax Magnitude Scale, Flux, Luminosity,… Basic Properties of Light Blackbody Radiation – – – • • • • Kirchoff’s Laws Spectrographs Photons – – • • Wien’s and Stefan/Boltzmann Planck Function (usage…not derivation) Monochromatic Flux/Luminosity Color Index, Bolometric Correction Redshift Spectral Lines – – • Photoelectric, Compton effects Energy in terms of eV-nm • Model of the Atom – Wavelengths of emitted photons Quantum Mechanics – Wave-Particle Duality – Uncertainty Principle – Quantum numbers of Atomic States – Pauli Exclusion Principle Telescopes – Basic (Snell’s Law/Reflection) – Diffraction limit/Resolution/Seeing – Brightness/Focal Ratio/Magnification – Mounts – Diameter/Resolving Power Special Relativity – – – – Time Dilation Length contraction Redshift Relativistic Momentum Positions on the Celestial Sphere The Altitude-Azimuth Coordinate System • • • • • • Coordinate system based on observers local horizon Zenith - point directly above the observer North - direction to north celestial pole NCP projected onto the plane tangent to the earth at the observer’s location h: altitude - angle measured from the horizon to the object along a great circle that passes the object and the zenith z: zenith distance - is the angle measured from the zenith to the object z+h=90 A: azimuth - is the angle measured along the horizon eastward from north to the great circle used for the measure of the altitude Equatorial Coordinate System • • • • Coordinate system that results in nearly constant values for the positions of distant celestial objects. Based on latitude-longitude coordinate system for the Earth. Declination - coordinate on celestial sphere analogous to latitude and is measured in degrees north or south of the celestial equator Right Ascension - coordinate on celestial sphere analogous to longitude and is measured eastward along the celestial equator from the vernal equinox to its intersection with the objects hour circle Hour circle Positions on the Celestial Sphere The Equatorial Coordinate System • • • • Hour Angle - The angle between a celestial object ’ s hour circle and the observer ’ s meridian, measured in the direction of the object ’ s motion around the celestial sphere. Local Sidereal Time(LST) - the amount of time that has elapsed since the vernal equinox has last traversed the meridian. Right Ascension is typically measured in units of hours, minutes and seconds. 24 hours of RA would be equivalent to 360. Can tell your LST by using the known RA of an object on observer’s meridian Hour circle What is a day? The period (sidereal) of earth’s revolution about the sun is 365.26 solar days. The earth moves about 1 around its orbit in 24 hours. • Solar day – – • Is defined as an average interval of 24 hours between meridian crossings of the Sun. The earth actually rotates about its axis by nearly 361 in one solar day. Sidereal day – Time between consecutive meridian crossings of a given star. The earth rotates exactly 360 w.r.t the background stars in one sidereal day = 23h 56m 4s Precession of the Equinoxes • Precession is a slow wobble of the Earth’s rotation axis due to our planet’s nonspherical shape and its gravitational interaction with the Sun, Moon, etc… • Precession period is 25,770 years, currently NCP is within 1 of Polaris. In 13,000 years it will be about 47 away from Polaris near Vega!!! • A westward motion of the Vernal equinox of about 50” per year. Elliptical Orbits 3 Kepler’s Laws of Planetary Motion Kepler’s First Law: A planet orbits the Sun in an ellipse, with the Sun at one focus of the ellipse. Kepler’s Second Law: A line connecting a planet to the Sun sweeps out equal areas in equal time intervals Kepler’s Third Law: The Harmonic Law P2=a3 Where P is the orbital period of the planet measured in years, and a is the average distance of the planet from the Sun, in astronomical units (1AU = average distance from Earth to Sun) Kepler’s First Law Kepler’s First Law: A planet orbits the Sun in an ellipse, with the Sun at one focus of the ellipse. • a=semi-major axis • e=eccentricity • r+r’=2a - points on ellipse satisfy this relation between sum of distance from foci and semimajor axis b 2 = a 2 (1- e 2 ) a(1- e 2 ) r= 1+ e cosq A = pab Kepler’s Second Law Kepler’s Second Law: A line connecting a planet to the Sun sweeps out equal areas in equal time intervals Kepler’s Third Law Kepler’s Third Law: The Harmonic Law P2=a3 • Semimajor axis vs Orbital Period on a loglog plot shows harmonic law relationship Newton’s Laws of Motion • Newton’s First Law: The Law of Inertia. An object at rest will remain at rest and an object in motion will remain in motion in a straight line at a constant speed unless acted upon by an external force. • Newton’s First Law: The net force (thesum of all forces) acting on an object is proportional to the object’s mass and its resultant acceleration. n Fnet = å Fi = ma i=1 • Newton’s Third Law: For every action there is an equal and opposite reaction Newton’s Law of Universal Gravitation • Using his three laws of motion along with Kepler’s third law, Newton obtained an expression describing the force that holds planets in their orbits…(derivation in book, done on blackboard) Mm F =G 2 r Work and Energy • • • • • • Energetics of systems Potential Energy Kinetic Energy K = 1 mv 2 2 Total Mechanical Energy Conservation of Energy Gravitational Potential Mm energy U = -G r • Escape velocity v esc = 2GM /r Derivation of Kepler’s Third Law • Integration of the expression of the 2nd law over one full period dA 1 L = dt 2 m • • Results in A= 1L P 2m 2 4 p P2 = a3 G(m1 + m2 ) • Derived on pp 48-49 Stellar Parallax • • Trigonometric Parallax: Determine distance from “triangulation” tan q = B /d d = B /tanq Parallax Angle: One-half the maximum angular displacement due to the motion of Earth about the Sun (excluding proper motion) d= 1AU 1 » AU tan p p With p measured in radians PARSEC/Light Year • • • 1 radian = 57.2957795 = 206264.806” Using p” in units of arcsec we have: 206,265 d» AU p" Astronomical Unit of distance: PARSEC = Parallax Second = pc 1pc = 2.06264806 x 105 AU • The distance to a star whose parallax angle p=1” is 1pc. 1pc is the distance at which 1 AU subtends an angle of 1” d» • • 1 pc p" Light year : 1 ly = 9.460730472 x 1015 m 1 pc = 3.2615638 ly •Nearest star proxima centauri has a parallax angle of 0.77” •Not measured until 1838 by Friedrich Wilhelm Bessel •Hipparcos satellite measurement accuracy approaches 0.001” for over 118,000 stars. This corresponds to a a distance of only 1000 pc (only 1/8 of way to centerof our galaxy) •The planned Space Interferometry Mission will be able to determine parallax angles as small as 4 microarcsec = 0.000004”) leading to distance measurements of objects up to 250 kpc. The Magnitude Scale • Apparent Magnitude: How bright an object appears. Hipparchus invented a scale to describe how bright a star appeared in the sky. He gave the dimmest stars a magnitude 6 and the brightest magnitude 1. Wonderful … smaller number means “bigger” brightness!!! • The human eye responds to brightness logarithmically. Turns out that a difference of 5 magnitudes on Hipparchus’ scale corresponds to a factor of 100 in brightness. Therefore a 1 magnitude difference corresponds to a brightness ratio of 1001/5=2.512. • Nowadays can measure apparent brightness to an accuracy of 0.01 magnitudes and differences to 0.002 magnitudes • Hipparchus’ scale extended to m=-26.83 for the Sun to approximately m=30 for the faintest object detectable Flux, Luminosity and the Inverse Square Law • Radiant flux F is the total amount of light energy of all wavelengths that crosses a unit area oriented perpendicular to the direction of the light’s travel per unit time…Joules/s=Watt • Depends on the Intrinsic Luminosity (energy emitted per second) as well as the distance to the object • Inverse Square Law: F= L 4pr 2 Absolute Magnitude and Distance Modulus • • Absolute Magnitude, M: Defined to be the apparent magnitude a star would have if it were located at a distance of 10pc. Ratio of fluxes for objects of apparent magnitudes m1 and m2 . F2 = 100(m1 -m 2 )/ 5 F1 • Taking logarithm of each side æ F1 ö m1 - m2 = -2.5log10ç ÷ è F2 ø •Distance Modulus: The connection between a star’s apparent magnitude, m , and absolute magnitude, M, and its distance, d, may be found by using the inverse square law and the equation that relates two magnitudes. F10 æ d ö (m-M )/ 5 100 = =ç ÷ F è10 pc ø 2 Where F10 is the flux that would be received if the star were at a distance of 10 pc and d is the star’s distance measured in pc. Solving for d gives: d =10(m-M +5)/ 5 pc The quantity m-M is a measure of the distance to a star and is called the star’s distance modulus æ d ö m - M = 5log10 (d) - 5 = 5log10 ç ÷ è10 pc ø Einstein’s Postulates of Special Relativity • • The Principle of Relativity. The laws of physics are the same in all inertial reference frames. The Constancy of the Speed of Light. Light moves through vacuum at a constant speed c that is independent of the motion of the light source. A reference frame in which a mass point thrown from the same point in three different (non coplanar) directions follows rectilinear paths each time it is thrown, is called an inertial frame. – L. Lange (1885) as quoted by Max von Laue in his book (1921) Die Relativitätstheorie, p. 34, and translated by Iro). Proper Time And Time Dilation t2 - t1 = (t2¢ - t1¢ ) + (x 2¢ - x1¢ )u /c 2 1- u 2 /c 2 (x 2¢ - x1¢ ) = 0 Proper Length and Length Contraction • Measure positions at endpoints at same time in frame S’ and in frame S, L’=x2’-x1’ (x 2 - x1 ) - u(t2 - t1 ) ¢ ¢ x 2 - x1 = 1- u 2 /c 2 L ¢ L= 1- u2 /c 2 Lmoving = Lrest 1- u2 /c 2 = Lrest / g Redshift zº lobs - lrest lrest n obs = n rest lobs = lrest 1- v r /c 1+ v r /c 1+ v r /c 1- v r /c (radial motion) • Can determine radial velocity of object by measuring shift in spectral lines…. • See example 4.3.2 p99 • For v<<c we have z= 1+ v r /c -1 1- v r /c z +1= 1+ v r /c 1- v r /c v r (z + 1) 2 -1 = c (z + 1) 2 + 1 vr l - lrest = z = obs c lrest Relativistic Momentum and Energy Momentum p= mv 1- u /c 2 2 = gmv Kinetic Energy K = mc 2 (g -1) Total Energy E = gmc 2 Rest Energy E = mc 2 Momentum Energy Relation E 2 = p2c 2 + m 2c 4 Speed of Light • Ole Roemer(1644-1710) measured the speed of light by observing that the observed time of the eclipses of Jupiter’s moons depended on how distant the Earth was from Jupiter. He estimated that the speed of light was 2.2 x 108 m/s from these observations. The defined value is now c=2.99792458 x 108 m/s (in vacuum). The meter is derived from this value. • Measurement of speed of light is the same for all inertial reference frames!!! Special Relativity (will come back to this topic..soon) Takes an additional 16.5 minutes for light to travel 2AU The Wave Nature of Light • Light impinging on double slit • Exhibits Inerference pattern Interference condition ì nl ï d sinq = í 1 ïî(n - )l 2 (n=0,1,2,…for bright fringes) (n=1,2,…for dark fringes) INTERFERENCE http://vsg.quasihome.com/interfer.htm WAVE Electromagnetic Waves Electromagnetic Wave speed Light is indeed an Electromagnetic Wave Waves are Transverse Electromagnetic Spectrum Region Gamma Ray X-Ray Wavelength nm 1 nm<10 nm Ultraviolet 10 nm<400 nm Visible 400 nm<700 nm Infrared 700 nm<1 mm Microwave Radio 1mm<10 cm 10 cm< Particle-like nature of light Photons • Photon = “Particle of Electromagnetic “stuff”” Light is absorbed and emitted in tiny discrete bursts • Blackbody Radiation Failure of Classical Theory Radiation is “quantized” • Photo-electric effect (applet) Blackbody Radiation • • • Any object with temperature above absolute zero 0K emits light of all wavelengths with varying degrees of efficiency. An Ideal Emitter is an object that absorbs all of the light energy incident upon it and re-radiates this energy with a characteristic spectrum.Because an Ideal Emitter reflects no light it is known as a blackbody. Wien’s Law: Relationship between wavelength of Peak Emission max and temperature T. Blackbody lmax T = 0.002897755mK • Stefan-Boltzmann equation: (Sun example) 4 L = AsT L:Luminosity A:area T:Temperature Blackbody Radiation Spectrum Planck’s Law for Blackbody Radiation •Planck used a mathematical “sleight of hand” to solve the ultraviolet catastrophe. •The energy of a charged oscillator of frequency f is limited to discrete values of Energy nhf. •During emission or absorption of light the change in energy of an oscillator is hf. •The mean energy at high frequencies tends to zero because the first allowed oscillator energy is so large compared to the average thermal energy available kBT that there is almost zero probability that this state is occupied. u( l,T) = •Planck seemed to be an “Unwilling revolutionary”.He viewed this “quantization” merely as a calculational trick…Einstein viewed it differently…light itself was quantized. 8phc l5 (e hc / lk T -1) B Stefan-Boltzman derived etotal c = 4 òl ¥ =0 u(l,T)dl = ò 0 2phc 2 dl l5 (e hc / lkB T -1) 2p 5 kB 4 = T = sT 4 2 3 15c h 4 etotal ¥ Monochromatic Luminosity and Flux Monochromatic Luminosity Ll dl = 4p 2 R2 Bl dl 8p 2 R 2 hc 2 / l5 Ll dl = hc / lkB T dl e -1 Monochromatic Flux received at a distance r from the model star is: 2 Ll 2phc 2 / l5 æ R ö Fl dl = dl = hc / lkB T ç ÷ dl 4p × r 2 e -1 è r ø mbol = -2.5log10 (ò ¥ 0 ) Fl dl + Cbol S Fd is the number of Joules of starlight energy with wavelengths between and dthat arrive per second per one square meter of detector aimed at the model star, assuming that no light has been absorbed or scattered during its journey from the star to the detector. Earth’s atmosphere absorbs some starlight, but this can be corrected. The values of these quantities usually quoted for stars have been corrected and would correspond to what would be measured above Earth’s atmosphere. Why do we keep the wavelength dependence? Filters!!! Sf( The Color Index UVB Wavelength Filters • • • Bolometric Magnitude: measured over all wavelengths. UBV wavelength filters: The color of a star may be precisely determined by using filters that transmit light only through certain narrow wavelength bands: – U, the star’s ultraviolet magnitude. Measured through filter centered at 365nm and effective bandwidth of 68nm. – B,the star’s blue magnitude. Measured through filter centered at 440nm and effective bandwidth of 98nm. – V,the star’s visual magnitude. Measured through filter centered at 550nm and effective bandwidth of 89nm U,B,and V are apparent magnitudes Sensitivity Function S() Filter Response U = -2.5log10 (ò ¥ 0 ) Fl SU dl + CU æ U - B = -2.5log10 ç ç è B = -2.5log10 ò ò Fl SU dl ö 0 ÷+C U -B ¥ Fl SB dl ÷ø ¥ 0 (ò ¥ 0 ) Fl SB dl + CB æ U - B = -2.5log10 ç ç è V = -2.5log10 ò ò ¥ 0 ¥ 0 Fl SU dl ö ÷+C U -B Fl SB dl ÷ø http://astro.unl.edu/naap/blackbody/animations/blackbody.html (ò ¥ 0 ) Fl SV dl + CV Spectral Type, Color and Effective Temperature for Main-Sequence Stars Spectral Type B-V Te(K) O5 -0.45 35,000 B0 -0.31 21,000 B5 -0.17 13,500 A0 0.00 9,700 A5 0.16 8,100 F0 0.30 7,200 F5 0.45 6,500 G0 0.57 6,000 G5 0.70 5,400 K0 0.84 4,700 K5 1.11 4,000 M0 1.24 3,300 M5 1.61 2,600 40,000 35,000 30,000 Temperature 25,000 20,000 15,000 10,000 5,000 0 -1 -0.5 0 0.5 1 1.5 B-V From Frank Shu, An Introduction to Astronomy(1982), Adapted from C.W. Allen, Astrophysical Quantities Note that this table does not quite agree with our text!!!! 2 Spectral Type, Color and Effective Temperature for Main-Sequence Stars Spectral Type, Color and Effective Temperature for Main-Sequence Stars (continued) Interstellar Reddening One also needs to correct color indices for interstellar reddening. As the light propagates through interstellar dust, the blue light is scattered preferentially making objects appear to be redder than they actually are… Fraunhofer Spectral Lines • Josef von Fraunhofer(17871826) had catalogued 475 dark spectral lines in the solar spectrum. • Fraunhofer showed that we can learn the chemical composition of the stars. Identified a spectral line of sodium in the spectrum of the Sun Spectral Lines Kirchoff’s Laws of Spectra • Robert Bunsen (1811-1899) created a burner that produced a “colorless” flame ideally suited for studying the spectra of heated substances. • Gustav Kirchoff (1824-1887) and Bunsen designed a spectroscope that could analyze the emitted light. • Kirchoff determined that 70 of the dark lines in the solar spectrum corresponded to the 70 bright lines emitted by iron vapor. • Chemical Analysis by Spectral Observations “Spectral Fingerprint”. •A hot dense gas or hot solid object produces a continuous spectrum with no dark spectal lines •A hot, diffuse gas produces bright spectral lines (emission lines) •A cool, diffuse gas in fron of a sources of a continuous spectrum produces dark spectral lines (absorption lines) in the continuous spectrum. Spectral Lines Application of Spectral Measurements • Stellar Doppler Shift • Galactic Doppler Shifts • Quasar Doppler Shifts Spectral Lines Spectrographs • Spectroscopy • Diffraction grating equation dsinq = nl (n=0,1,2,…) • Resolving Power Photons Photoelectric Effect • Photoelectric Effect • Kinetic energy of ejected electrons does not depend on intensity of light! • Increasing intensity will produce more ejected electrons. • Maximum kinetic energy of ejected electrons depends on frequency of light. • Frequency must exceed cutoff frequency before any electrons are ejected Einstein took Planck ’ s assumption of quantized energy of EM waves seriously. Light consisted of massless photons whose energy was: E photon = hn = hc l K max = E photon - F = hn - F = hc l -F Einstein was awarded the Nobel Prize in 1921 for his work on the photo-electric effect Photons Compton Scattering • • • • • Compton Scattering Wikipedia entry Arthur Holly Compton (18921962) provided convincing evidence that light manifests particle-like properties in its interaction with matter by considering how (x-ray) photons can “collide” with a free electron h at rest. lC = = 0.00243nm me c Compton Wavelength: Conservation of energy and momentum leads to the following: is the characteristic change in wavelength in the scattered photon. Showed that photons are massless yet carry momentum!!! E photon = hn = hc l = pc Rutherford-Bohr Model of the Atom • Ernest Rutherford (1871-1937) • Rutherford Scattering Observation consistent with scattering from a very small (10,000 times smaller radius than the atom) dense object… The nucleus Wavelengths of Hydrogen Bohr’s Semi-classical Model of the Atom • Bohr assumed for the electron proton system – to be subject to Coulomb’s Law for electric charges – Quantization of angular momentum for the electron orbit. L=nh/2 • Quantization condition prevents electron from continuosly radiating away energy • Discrete energy levels for electron orbit Bohr’s Semi-classical Model of the Atom Bohr’s Semi-classical Model of the Atom Bohr’s Semi-classical Model of the Atom Bohr’s Semi-classical Model of the Atom Bohr Hydrogen atom and spectral lines DeBroglie Matter Wave Quantum Mechanics and Wave-Particle Duality • Fourier • Wave Packets Uncertainty Principle Additional Quantum Numbers and splitting of spectral lines • Normal Zeeman Effect Can measure magnetic fields by examining spectra!!!! Spin and the Pauli Exclusion Principle No two electrons can share the same set of four quantum numbers Chemistry….The periodic table…. Electron Degeneracy Pressure and White Dwarfs • Exclusion Principle for Fermions (spin-1/2 particles) and uncertainty principle provide electron degeneracy pressure that is the mechanism that prevents the further collapse of white dwarves…. Reflection and Refraction Reflection q1 = q2 Refraction n1 sinq1 = n2 sinq2 Reflection and Refraction Lenses • Focal length determined by radii of curvature , R1and R2, of the lens surfaces and the index of refraction of the lens material • Convention for sign of radii of curvature is concave --> negative convex --> positive • Wavelength dependent 1 1 1 = (nl -1)( - ) f R1 R2 Lensmaker formula for thin lenses Mirrors • For spherical mirrors • f=R/2 • Wavelength independent Focal Plane • Image size proportional to focal length y = f tanq y » fq Plate Scale dq 1 = dy f Resolution and Rayleigh Criteria Diffraction • Single Slit diffraction • Consider a point in the slit D/2 away from another point in slit. The pair of rays from these two points will exhibit destructive interference if they arrive at a point on the screen being 1/2 wavelength out of phase D l sinq = 2 2 sin q = l D • Now divide the slit into fourths D l sinq = 4 2 sin q = 2 l D sinq = m l D Condition for minimum in light intensity on screen Resolution Single Slit Diffraction Diffraction from a Circular Aperture Airy Disk Intensity as a function of angle from center of the image is given by a Bessel function Diffraction limited spatial resolution for a telescope with an aperture of diameter D is given by: Rayleigh Criterion for Resolution • Somewhat arbitrary definition of barely resolvable is that the maximum of one of the Airy disks lies at the first minimum of the other Airy disk http://astronomy.swin.edu.au/cosmos/R/Resolution Atmospheric Seeing • Why do stars twinkle? • Stars are so distant as to effectively be point sources. • Planets are resolvable. • Plane wavefronts of light from a distant point source become distorted as the wavefronts passes through a turbulent layer in the atmosphere. • The index of refraction in the turbulent layer is not uniform Brightness of Image Refracting Telescope • Angular Magnification Reflecting Telescopes Telescope Mounts • Altitude Azimuth Equatorial Radio Telescopes • Can observe the universe in radio wavelengths from the surface of the earth Infrared, Ultraviolet, X-ray and Gamma-ray Astronomy • Can view universe at other wavelenghts as well….