Review Chapters 1-8

1. Review Chapters 1-8 • 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

2. Review Chapters 1-8 • 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 Wave-Particle Duality • Telescopes • Basic Optics • Optical Telescopes • Radio Telescopes • Infrared,UV,X-ray and Gamma-Ray astronomy

3. Review Chapters 1-8 • Binary Systems and Stellar Parameters • Classification of Binary Stars • Mass Determination Using Visual Binaries • Eclipsing, Spectroscopic Binaries • The Search for Extrasolar Planets • The Classification of Stellar Spectra • The Formation of Spectral Lines • The Hertzsprung-Russell Diagram

4. Midterm Exam 1 • The exam • Part I - in-class • Conceptual questions • “Easy” calculations (bring a calculator) • Three sheets (back and front) of notes (no xerographic reduction) • Part II - Take-home • Application of the material to Astrophysical Problems • More of a pedagogical leaning exercise • Preparation • Material from textbook ch 1-8 • Concepts • Equations • Examples from textbook • Homework problems!!! See solutions on web…

5. Possible Problem Topics • Model of the Atom • Wavelengths of emitted photons • Quantum Mechanics • Wave-Particle Duality • Uncertainty Principle • 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 • Binary Star Systems • Classification • Mass Determination • Classification of Stellar Spectra • H-R Diagram • …. • 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 • Wien’s and Stefan/Boltzmann • Planck Function (usage…not derivation) • Monochromatic Flux/Luminosity • Color Index, Bolometric Correction • Redshift • Spectral Lines • Kirchoff’s Laws • Spectrographs • Photons • Photoelectric, Compton effects • Energy in terms of eV-nm

6. The Celestial Sphere History The Greek tradition Copernican Revolution Positions on the celestial sphere Altitude-Azimuth coordinate system Equatorial coordinate system Daily,Seasonal changes, Precession Measurements of time

7. Positions on the Celestial SphereThe 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

8. 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

9. Positions on the Celestial SphereThe 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

10. 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

11. 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.

12. Celestial MechanicsFun with Kepler and Newton Elliptical Orbits Newtonian Mechanics Kepler’s Laws Derived Virial Theorem

13. Elliptical Orbits 3Kepler’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)

14. The Geometry of Elliptical Motion Description in book (pp25-27. ). Example 2.1.1

15. 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

16. Kepler’s Second Law Kepler’s Second Law: A line connecting a planet to the Sun sweeps out equal areas in equal time intervals

17. Kepler’s Third Law Kepler’s Third Law: The Harmonic Law P2=a3 • Semimajor axis vs Orbital Period on a log-log plot shows harmonic law relationship

18. 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. • Newton’s Third Law: For every action there is an equal and opposite reaction

19. 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)

20. Work and Energy • Energetics of systems • Potential Energy • Kinetic Energy • Total Mechanical Energy • Conservation of Energy • Gravitational Potential energy • Escape velocity Derivations on pp37-39

21. Derivation of Kepler’s First Law • Consider Effect of Gravitation on the Orbital Angular Momentum • Central Force Angular Momentum Conserved • Consideration of quantity leads to equation of ellipse describing orbit!!! • Derivation on pp43-45

22. Derivation of Kepler’s Second Law • Consider area element swept out by line from principal focus to planet. • Express in terms of angular momentum • Since Angular Momentum is conserved we obtain the second law • Derived on pp 45-48

23. Derivation of Kepler’s Third Law • Integration of the expression of the 2nd law over one full period • Results in • Derived on pp 48-49

24. Virial Theorem • Virial Theorem: For gravitationally bound systems in equilibrium the Total energy is always one half the time averaged potential energy • The Virial Theorem can be proven by considering the quantity and its time derivative along with Newton’s laws and vector identities • Many applications in Astrophysics…stellar equilibrium, galaxy clusters,…. • Derivation on pp 50-53

25. Distance and Brightness • Stellar Parallax • The Magnitude Scale

26. Stellar Parallax • Trigonometric Parallax: Determine distance from “triangulation” • Parallax Angle: One-half the maximum angular displacement due to the motion of Earth about the Sun (excluding proper motion) With p measured in radians

27. PARSEC/Light Year • 1 radian = 57.2957795 = 206264.806” • Using p” in units of arcsec we have: • 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” • 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.

28. 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

29. 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:

30. Absolute Magnitude and Distance Modulus • 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. • 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: • The quantity m-M is a measure of the distance to a star and is called the star’s 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 . • Taking logarithm of each side

31. Einstein’s Postulates of Special Relativity A reference frame in which a mass point thrown from the same point in three different (non co-planar) 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). • 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.

32. Proper Time And Time Dilation

33. Proper Length and Length Contraction • Measure positions at endpoints at same time in frame S’ and in frame S, L’=x2’-x1’

34. Redshift (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

35. Relativistic Momentum and Energy Momentum Kinetic Energy Total Energy Rest Energy Momentum Energy Relation

36. The Continuous Spectrum of Light • The Nature of Light • Blackbody Radiation • The Color Index

37. 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

38. The Wave Nature of Light Interference condition • Light impinging on double slit • Exhibits Inerference pattern (n=0,1,2,…for bright fringes) (n=1,2,…for dark fringes) INTERFERENCE WAVE http://vsg.quasihome.com/interfer.htm

39. Electromagnetic Waves Electromagnetic Wave speed Light is indeed an Electromagnetic Wave Waves are Transverse

40. Electromagnetic Spectrum

41. Radiation Pressureand the Poynting Vector Poynting Vector Radiation Pressure Radiation Pressure is significant in • extremely luminous objects such as: • early main-sequence stars • red supergiants • Accreting compact stars • Interstellar medium dust particles • The rate at which energy is carried by a light wave is described by the Poynting vector. • Instantaneous flow of energy per unit area per unit time (W/m2) for all wavelengths. • Points in the direction of the electromagnetic wave’s propagation. • Radiant Flux: Time average (over one period) of the Poynting vector • Because an electromagnetic wave carries momentum it can exert a force on a surface hit by light… absorption) reflection)

42. Particle-like nature of lightPhotons • Photon = “Particle of Electromagnetic “stuff”” • Blackbody Radiation Failure of Classical Theory Radiation is “quantized” • Photo-electric effect (applet) Light is absorbed and emitted in tiny discrete bursts

43. 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. • Stefan-Boltzmann equation: (Sun example) Blackbody L:Luminosity A:area T:Temperature Blackbody Radiation Spectrum

44. Blackbody Radiation http://www.mhhe.com/physsci/astronomy/applets/Blackbody/frame.html

45. 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. • Planck seemed to be an “Unwilling revolutionary”.He viewed this “quantization” merely as a calculational trick…Einstein viewed it differently…light itself was quantized.

46. Derivation of Stefan-Boltzmann Law The Stefan-Boltzmann constant is a derived constant depending on kB, h and c !!!!! • Stefan-Boltzmann’s Law can be obtained from Planck’s Law by simply integrating the spectral density function over all wavelengths • Subsituting and evaluating • We obtain:

47. Color/Temperature Relation • Planck’s law and Astrophysics • Monochromatic Luminosity and Flux • Bolometric Magnitude • Filters, measured flux • The Color Index

48. Planck’s Law and Astrophysics Power radiated per unit wavelength per unit area per unit time per steradian: (units are W m -2 m-1 sr -1 ) Note that: Power radiated per unit wavelength per unit area per unit time per steradian: (units are W m -2 ssr -1 ) Consider a model star consisting of a spherical blackbody of radius R and temperature T. Assuming that each patch dA emits isotropically over the outward hemisphere, the energy per second having wavelengths between  and demitted by the star is: In spherical coordinates the amount of radiant energy per unit time having wavelengths between  and demitted by a blackbody radiator of temperature T and surface area dA into a solid angleis given by: Angular integration yields a factor of  , the integration of dA over the surface of the sphere yield 4R2. In terms of B:

49. Monochromatic Luminosity and Flux Monochromatic Luminosity Fd is the number of Joules of starlight energy with wavelengths between  and dthat 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. Monochromatic Flux received at a distance r from the model star is: Why do we keep the wavelength dependence? Filters!!! Sf( S

50. The Color IndexUVB 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()