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Extragalactic Astronomy

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... in mind some of the most basic facts about galaxies which we learned a few lectures ago. Jane Turner [4246] PHY 316 (2003 Spring) Quiz 2 ... – PowerPoint PPT presentation

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Title: Extragalactic Astronomy


1
Extragalactic Astronomy Cosmology
Lecture 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 Thi
ngs which will not be in any exam/quiz
-Luminosity functions of planetary nebulae/globu
lar 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 cour
se so far except items excluded
from all exams (already noted)
No math problems on GR (had no time for homework
s on this) but there will be some descriptive que
stions on GR. Will be both types for SR and the
rest of the course. Revision lecture on Tues M
arch 18 (BH-AGN and DM will wait until after th
e 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 wil
l 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 relati
vity 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 accel
erating 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 corre
ct where any velocities are even close to c
When velocities are s is an acceptable approximation to the right
answer Thats why people did not realize the ne
ed 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
Einsteins postulates Ti
me dilation Length contraction
New velocity addition law



10
THE SPEED OF LIGHT PROBLEM
  • Relativity tells us how to relate
    measurements in different frames.
  • Galilean relativity
  • Simple velocity addition law
    vtotalvrunvtrain

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

Lets start to think about the consequences of
these postulates We will perform thought expe
riments(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 t
rain 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

d2H2 (v? tsm /2) 2

?tsm ?tp /?1-(v2/c2)
The moving clock appears to run slowly
15
?tsm 2d/c d c ?tsm /2
?tp 2H/cHc ?tp /2
station master sees a longer time elapse than the
passenger
d2H2 (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/v
2 (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 ?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 cl
ock 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 occurringthis 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 v0.87c 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 t
rain What is their separation ? The station ma
ster 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 m
oving at -v So station master and passenger ag
ree the relative speed is v To the passenger, t
he 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 o
f 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 s
ees length contraction and thus a moving car
appears shorter to them The passenger sees thin
gs in the station masters frame to be contracted,
the station master sees things in the passengers
frame to be contracted Reciprocity applies to l
ength as well as time effects





24
Example

For Concorde, travelling at twice the speed of
sound ? 1.000000000002 and length contraction1
0-8 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 mass1.15 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 li
ght is constant, rays will hit the walls at the
same time.Call these events A B.
Then perform the same experiment in a moving S
pacecraft, 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 eve
nts 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 re
lative 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 tim
e 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 v (1v/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 m
eans the frequency of light in a moving frame
appears reduced Think of the frequency of ligh
t 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 ?
timemoving timerest ?1-(v2/c2)
Time Dilation
Lorentz Contraction
lengthmoving lengthrest ?1-(v2/c2)
Mass massmoving massrest/?1-(v2/c2)
Relativistic Addition of Velocities
Relativistic Doppler z 1 v
(1v/c)/(1-v/c)
35
Derivation of Emc2

Start with mass increase formula
massmoving massrest/?1-(v2/c2)
m m01-(v2/c2)-0.5 use a mathematical appro
ximation, where ? bstitute ? -v2/c2 (1 -v2/c2)-0.5 1-0.5(-v2
/c2) Our substitution means which can be simpli
fied to we are dealing (1-v2/c2)-0.5 1v2/2c2
the case v



36
Derivation of Emc2
m m0(1v2/2c2) expand m m0 m0v2/2c2 multip
ly 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 i
s not moving, so the kinetic energy term is zero,
then the total energy is not zero, as there is th
e term m0c2 i.e. even when the vel is zero, and o
bject 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 450
Light Cone
object C would have to travel at vc so impossible
light beam follows a world line ctx , 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 b
y lines equiv to vc (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 v?(c?t)2 - (?x)2?
defines a spacetime interval
51
SPACE-TIME DIAGRAMS
in general ?r2 ?x2 ?y2 ?s v?(c?t)2 - (?x)2?
a separation in Minkowski spacetime, a spacetime
interval
Light Cone
-ve sign because time cannot be treated like a
spatial dimension ?s called a spacetime interval
and is invariant - all observers agree on the
quantity
52
SPACE-TIME DIAGRAMS
?s v?(c?t)2 - (?x)2? a separation in
Minkowski spacetime
interval2 (dist traveled by light in time ?t)2
- (dist between events)2 ?s2 0 timelike (ligh
t had more than enough time to travel between
events) ?s2 0 null (or lightlike, light had exa
ctly enough time to travel between events)
?s2 travel between the events)
Light Cone
53
Summary
We have learned the special theory of relativity
relates observations made in inertial frames to
one another, because inertial frames are special,
we call it the Special Theory Special Relativit
y showed us we had to discard the concepts of
absolute space time, space time are
inextricably linked Special Relativity brings
mechanics and electromagnetics into consistency
and provides a model for situations where
velocities approach the speed of light


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