Extragalactic Astronomy & Cosmology Lecture SR1

1. Extragalactic Astronomy & CosmologyLecture SR1 [4246] Physics 316 Jane Turner Joint Center for Astrophysics UMBC & NASA/GSFC 2003 Spring

2. Quiz 2 Revision Guide: • You should be able to:- • describe Hubbles key breakthroughs and use the Hubble law • -describe the general approach of the Cosmic Distance ladder ( an overview ) plus describe at least some of the steps in detail - and note the problems limiting use of some key ‘standard candles’ • You should be familiar with the use of Cepheid variables • and SNe type 1a and what results from those have told us • In addition, try to keep in mind some of the most basic facts about galaxies which we learned a few lectures ago

3. Quiz 2 Things which are not included in Quiz 2 (but will be in the Mid-term exam) -Lives of stars Things which will not be in any exam/quiz -Luminosity functions of planetary nebulae/globular clusters -Anything from the telescope session -Fusion processes in stars

4. Mid-Term Exam March 20 (Thursday), usual lecture room and time (25% of final grade) Will cover the entire course so far except items excluded from all exams (already noted) No math problems on GR (had no time for homeworks on this) but there will be some descriptive questions on GR. Will be both types for SR and the rest of the course. Revision lecture on Tues March 18 (BH-AGN and DM will wait until after the spring break)

5. Mid-Term Exam Revision lecture on Tues March 18 Also will give out project choices for the next half of the semester List of options, you will be able to select and mail in your choice the first week after the break Also, student presentations will be Tues April 22

6. Special Relativity-what is it? Einstein tried to fit the idea of an absolute speed for light into Newtonian mechanics. He found the transformation from one reference frame to another had to affect time-this led to the theory of special relativity. In special relativity the velocity of light is special, inertial frames are special. Anything moving at the speed of light in one reference frame will move at the speed of light in other inertial frames. Other velocities are not preserved. Have to worry about applicability of SR to accelerating frames

7. Special Relativity-what is it? So, special relativity is a theory which takes into account the absoluteness of the speed of light It is necessary to get calculation correct where any velocities are even close to c When velocities are << c then Newtonian mechanics is an acceptable approximation to the right answer That’s why people did not realize the need for SR for a long time, Newtonian mechanics fit everyday life.

8. Special Relativity-what is it? Special Relativity was constructed to satisfy Maxwells Equations, which replaced the inverse square law electrostatic force by a set of equations describing the electromagnetic field.

9. Special Relativity Einstein’s postulates Time dilation Length contraction New velocity addition law

10. “Relativity” tells us how to relate measurements in different frames. Galilean relativity Simple velocity addition law : vtotal=vrun+vtrain THE SPEED OF LIGHT PROBLEM

11. Einsteins Postulates Einstein threw away Galilean Relativity Came up with two “Postulates of Relativity”

12. Einsteins Postulates Postulate 1 – The laws of nature are the same in all inertial frames of reference Postulate 2 – The speed of light in a vacuum is the same in all inertial frames of reference Let’s start to think about the consequences of these postulates We will perform “thought experiments”(Gedankenexperiment)… For now, we will ignore effect of gravity – suppose we are performing these experiments in the middle of deep space

13. Time Dilation Imagine a pulse of light from a bulb on a train travelling at velocity v. A passenger on the train sees the light hit a mirror and bounce back. A person outside the train, at rest, sees the light path to be longer….lets call them the station master E(see Hawley & Holcomb page 175 - read chapters 6 & 7)

14. Frame of passenger Frame of Station Master mirror mirror tsm =2d/c tp =2H/c H d vtr d2=H2+ (v tsm /2) 2 tsm = tp /[1-(v2/c2)] The moving clock appears to run slowly

15. tp =2H/cH=c tp /2 tsm =2d/c d = c tsm /2 station master sees a longer time elapse than the passenger d2=H2+ (v tsm /2) 2 4(d2-H2)/v2=  tsm2 sub for d, H 4/v2 (c2 tsm2 /4) - (c2tp2 /4) =  tsm2 c2/v2(tsm2 - tp2) = tsm2 c2/v2(1 - tp2 / tsm2) = 1 1 - tp2 /tsm2 =v2/c2 tp2 /tsm2 =1 -v2/c2 tp2 =1 -v2/c2 (tsm2) tsm = tp /[1-(v2/c2)] if v< c then tsm > tp The moving clock appears to run slowly

16. Time Dilation... A moving clock appears to run more slowly ! Now, invert this, the station master has a bulb & mirror, the passenger sees this person as moving at speed -v relative to them. The station masters clock is a moving clock from the passengers view. The passenger sees the station masters clock running slowly. This is the “Principal of Reciprocity” No frame is preferred. Any clock at rest w.r.t an inertial observer will show “proper time”-the time between two events in the rest-frame in which those events occurred.

17. Time Dilation Hawley & Holcomb page 177 This effect is called Time Dilation The moving clock slows by a factor The Lorentz factor  v/c The shortest time for an event is that measured by an observer in the same inertial frame as the event is occurring…this is the “proper time”

18. Time Dilation -Example The moving clock slows by a factor The Lorentz factor If we have a spacecraft traveling at v=0.87c then =2. An event taking 30s to an astronaut on the spacecraft, appears to take 60s to an outside observer in their own inertial frame

19. Length: Lorentz Contraction Measure length by comparison of an object to a fixed standard ruler, where the two ends of an object are measured at the same specific time Consider two telephone poles beside our moving train What is their separation ? The station master measures the time the front of the train passes each pole, and then calculates the distance between them to be xsm = v tsm

20. Length: Lorentz Contraction xsm = v tsm The passenger sees the poles moving at -v So station master and passenger agree the relative speed is v To the passenger, the dist between each pole passing the window is xp = v tp We already have tsm = tp /[1-(v2/c2)] Which gives us...

21. Length: Lorentz Contraction xp=tp = [1-(v2/c2)] xsm tsm xp =xsm [1-(v2/c2)] or …if v =0.5c xp =0.87 xxsm the passenger measures a shorter distance than the station master The poles are in the frame of the station master, who sees them separated by the maximum length anyone ever will, the “proper length”

22. Length: Lorentz Contraction xp =xsm [1-(v2/c2)] The passenger is taking a measurement of their separation from a frame which has a relative velocity, so sees a contraction in length Note: the contraction appears only in the direction of relative motion, the heights of the telephone poles would be seen as the same by both observers!

23. Reciprocity: Lorentz Contraction Now consider the length of a car on the moving train The passenger, moving with the car, sees it at its “proper length” The station master sees length contraction and thus a moving car appears shorter to them The passenger sees things in the station masters frame to be contracted, the station master sees things in the passengers frame to be contracted Reciprocity applies to length as well as time effects

24. Example For Concorde, travelling at twice the speed of sound  =1.000000000002 and length contraction=10-8 cm (out of 60m proper length) !! Note: the contraction appears only in the direction of relative motion

25. Mass Increase Similar arguments as for length contraction can be used to relate moving mass to rest mass such that moving mass =rest mass/[1-(v2/c2)] v=0.5c -> moving mass=1.15 x rest mass

26. Simultaneity Consider an observer in a room. Suppose there is a flash bulb exactly in the middle of the room. Suppose sensors on the walls record when the light rays hit the walls. Since the speed of light is constant, rays will hit the walls at the same time.Call these events A & B. Then perform the same experiment in a moving Spacecraft, observed by somebody at rest

27. Simultaneity Flash hits both front/back of train simult in the train frame, seen by passenger In station masters frame, light hits the back of the train before the front The concept of events being simultaneous is different for observers in different reference frames

28. The order of events • Consider same experiment seen by three observers • Moving astronaut thinks events A and B are simultaneous • Observer at rest thinks A occurs before B

29. What about a 3rd observer who is moving faster than astronauts spacecraft? • 3rd observer sees event B before event A • So, order in which events happen depends on frame of reference.

30. Addition of Relativistic Velocities Need a new formula for adding relativistic velocities Suppose you see an astronaut moving at vel V1 and she sees a second object moving relative to her at V2 -the Newtonian approx. says the outside observer sees the 2nd object move at (V1 + V2) But once we take account of the way time and distance depend on v, we find No matter how close to c V1 and V2 are, Vadd cannot exceed c because the speed of light is absolute

31. Relativistic Doppler Formula Classic Doppler effect seen when there is relative motion, as the crests of the waves bunch or stretch out Relativity adds the effect that the frequency of the light (which is ~ 1/time) is smaller at the source than the receiver, due to time dilation z + 1 = √[ (1+v/c)/(1-v/c) ] (Hawley & Holcomb page 183)

32. Transverse Doppler Effect A relativistic Doppler effect also occurs in the direction perpendicular to the relative motion The observation of a moving clock running slow means the frequency of light in a moving frame appears reduced Think of the frequency of light as a clock with a number of cycles per second, if that clock in the moving frame runs slow, we see fewer cycles completed per second Freq reduction is like a redshift

33. Summary of Formulae Lorentz Factor  or  Velocity v/c Gamma value 0 1 0.1 1.005 0.87 2 0.9 2.29 0.99 7.1 0.999 22.4

34. Summary of Formulae Lorentz Factor  or  Time Dilation timemoving = timerest[1-(v2/c2)] Lorentz Contraction lengthmoving =lengthrest [1-(v2/c2)] Mass massmoving =massrest/[1-(v2/c2)] Relativistic Addition of Velocities Relativistic Doppler z + 1 = √[ (1+v/c)/(1-v/c) ]

35. Derivation of E=mc2 Start with mass increase formula massmoving =massrest/[1-(v2/c2)] m =m0[1-(v2/c2)]-0.5 use a mathematical approximation, where  << 1 (1+)-0.5 ≈ 1-0.5 and substitute = -v2/c2 (1+ -v2/c2)-0.5 ≈ 1-0.5(-v2/c2) Our substitution means which can be simplified to we are dealing (1-v2/c2)-0.5 ≈ 1+v2/2c2 the case v << c m =m0(1+v2/2c2)

36. Derivation of E=mc2 m =m0(1+v2/2c2) expand m =m0 + m0v2/2c2 multiply both sides by c2 mc2 =m0c2+ m0v2/2 m0v2/2 is the kinetic energy mc2 is the total energy of the object (E) what about an object which is not moving, so the kinetic energy term is zero, then the total energy is not zero, as there is the term m0c2 i.e. even when the vel is zero, and object has energy due to its rest mass, E= m0c2 more often written E= mc2

37. II : EXAMMASS TO ENERGY • Nuclear fission (e.g., of Uranium) • Nuclear Fission – the splitting up of atomic nuclei • E.g., Uranium-235 nuclei split into fragments when smashed by a moving neutron. One possible nuclear reaction is • Mass of fragments slightly less than mass of initial nucleus + neutron • That mass has been converted into energy (gamma-rays and kinetic energy of fragments)

38. From web site of Georgia State University

39. Nuclear fusion (e.g. hydrogen) • Fusion – the sticking together of atomic nuclei • Much more important for Astronomy than fission • e.g. power source for stars such as the Sun. • Explosive mechanism for particular kind of supernova • Important example – hydrogen fusion. • Ram together 4 hydrogen nuclei to form helium nucleus • Spits out couple of “positrons” and “neutrinos” in process

40. Mass of final helium nucleus plus positrons and neutrinos is less than original 4 hydrogen nuclei • Mass has been converted into energy (gamma-rays and kinetic energy of final particles) • This (and other very similar) nuclear reaction is the energy source for… • Hydrogen Bombs (about 1kg of mass converted into energy gives 20 Megaton bomb) • The Sun (about 4109 kg converted into energy per second)

41. EXAMPLES OF CONVERTING ENERGY TO MASS • Particle/anti-particle production • Energy (e.g., gamma-rays) can produce particle/anti-particle pairs • Very fundamental process in Nature… shall see later that this process, operating in early universe, is responsible for all of the mass that we see today!

42. Particle production in a particle accelerator • Can reproduce conditions similar to early universe in modern particle accelerators…

43. Spacetime From SR we found time intervals, space separations and simultaneity are not absolute space and time have to be considered together to understand events - so we need to consider 4-dimensional spacetime Difficult to think in 4D, but we can make nice spacetime diagrams! First developed in 1908 by Hermann Minkowski

44. SPACE-TIME DIAGRAMS Only plot one dimension of space for simplicity “Light Cone” Any point is an event, a line/curve connecting points is a worldline

45. SPACE-TIME DIAGRAMS “Light Cone” time axis often renormalized and plotted as ct, so it has same dimensions as space axis

46. SPACE-TIME DIAGRAMS object B traveling at v<c has worldline > 450 “Light Cone” object C would have to travel at v>c so impossible light beam follows a world line ct=x , using x versus ct - this is a line at 450

47. SPACE-TIME DIAGRAMS Inertial Observers Accelerated Observer “Light Cone”

48. SPACE-TIME DIAGRAMS “Light Cone”

49. SPACE-TIME DIAGRAMS Event A constrained to lie within cones defined by lines equiv to v=c (45o) future “Light Cone” elsewhere elsewhere past

50. SPACE-TIME DIAGRAMS How do we define a ‘separation’ between two events on a worldline? “Light Cone” in general r2= x2+ y2 s= √(ct)2 - (x)2 defines a spacetime interval