12 OUR PLACE IN THE UNIVERSEThe parallax method
• Review knowledge and understanding of cosmology
• Learn how to use the parallax method to determine distances to stars
• Appreciate its limitations
The distance-brightness method• Explain why observed stellar
intensity follows an inverse-square law
• Explain how Cepheid variable stars can be used to determine absolute/intrinsic brightness (luminosity)
• Use the relation between apparent brightness and luminosity to determine distances to stars
The Doppler effect• Explain the pulse-echo method of
radar ranging of near-Earth objects, appreciating its limitations
• Explain the origin of the Doppler effect, and how it may be used when analysing spectra of astronomical objects to determine velocities
Special relativity
• Describe experimental evidence for the constancy of the speed of light in all reference frames
• State the Einstein postulates
• Use space-time diagrams to derive the relativistic Doppler relations
Special relativity• Describe experimental
evidence for the constancy of the speed of light in all reference frames
• State the Einstein postulates
• Use space-time diagrams to derive the relativistic Doppler relations
Postulate 1: Physical behaviour cannot
depend on any ‘absolute velocity’. Physical
laws must take the same form for all observers, no matter what their state of
uniform motion in a straight line.Postulate 2: The speed of light c is a universalconstant. It has the same value, regardless of the motion of the platform from which it is observed. In effect, the translation between distance and time units is the same for everybody.
Einstein’s postulates
• If the Earth was travelling in the direction of the beam the travel time for the light would change
• By slowly rotating the apparatus they ensured that the beams at some point would point along/across the Earth’s direction of travel.
• The fringes were expected to change as the apparatus turned
• They stayed put – either the Earth wasn’t moving in space or light wasn’t affected by the movement
Michelson-Morley experiment
Radar ranging and velocity of asteroid
relativevelocity v
asteroid
asteroid speed of light= 3 108 m s–1
first radarpulse returns0.2 s later
second radarpulse out
first radarpulse out
second radarpulse returns0.22 s later
first pulse second pulse
time between first andsecond pulses 100 s
relativevelocity v
first pulseout
first pulsereturns
second pulsereturns
second pulseout
0.2 s 0.22 s
100 s
distance out and back= 0.2 light-seconds
distance of asteroid= 0.1 light-seconds= 30000 km
distance out and back= 0.22 light-seconds
distance of asteroid= 0.11 light-seconds= 33000 km
increase in distance3000 km = 0.01 light-seconds
relative velocity v = 3000 km100 s
v = 30 km s–1
v/c = 0.01 s/100 s = 10–4
time taken= 100 s
What are the limitations of this method?
Reminder
Space time diagrams
Two-way radar speed measurement
time t/s
10
8
6
4
2
03 ls 6 ls
distance x/c
worldline ofmovingobject
5
pulse 1 back tback
pulse 1 out tout
pulse 1 out and back instantly
c (tback + tback – tout – tout)12
(tback + tout)12
12pulse 2 back tback + tback
tback
tout
pulse 2 out tout + tout
(tback – tout)12
v (tback + tback + tout + tout)12
c (tback – tout)12 v (tback + tout)
12
c (tback – tout)12 v (tback + tout)
12
(tback – tout)
(tback + tout)
tbacktout
Two-way radar speed measurement 2
Speed is measured by comparing the interval between returning pulseswith the interval at which they were sent
extra timebetweenreflections
extra distancebetweenreflectionsc
distance whenpulse 2 reflected
from pulse travel time from object travel time
distance whenpulse 1 reflected
subtractextra distancebetween reflections
compare
v=c
1+v/c= 1–v/c
time t 10
8
6
4
2
02 6
distance x/c
Let the one-way Doppler shift = k
Pulses sent from A at intervals twill arrive at B at intervals ktout
k2 =1 + v/c1 – v/c
4
pulses return toobserver at intervalstback = k ktout
pulses sent outat intervals tout
The same shift k must apply to pulsessent from B to A
Therefore:pulses arrive back at A at intervalstback = k ktout
Two-way Doppler shift
One-way Doppler shift
k =1 + v/c1 – v/c
The two-way Doppler shift = k2pulses arrive atmoving object atlarger intervalsktout
A B
Doppler shift – two-way and one-way
The Doppler shift k is the observed quantity that measures the speed of a remote object
Space time diagrams
Construct a space-time diagram for the following situation.
A spacecraft, initially 4 light-seconds distant from the Earth, travels towards the Earth at a speed 0.5c.After 1 second, a radar pulse from the Earth is sent out towards the spacecraft.
Use the space-time diagram to determine:
(a) At what time and where the radar pulse hits the spacecraft,(b) At what time the reflected radar pulse is received back on Earth,(c) Where the spacecraft is located when the reflected pulse is
received back on Earth,(d) At what time the spacecraft reaches Earth.
Time dilation
Time dilation
• Use the concept of the light clock to explain time dilation
• Resolve the Twins Paradox
• Use experimental data on muon lifetimes to illustrate time dilation effects
Twins Paradox
Time dilation
The half life of a sub-atomic particle in the laboratory rest frame is 1 microsecond.
Q1. What fraction of these particles would be expected to survive to a detector located 6 km away from the experiment? Assume the particles are travelling effectively at 3 x 108 ms-1.
In fact, 25 % of the particles produced in the experiment survive out to the detector located at 6 km from the experiment.
Q2. Use this information to calculate the relativistic factor γ (gamma) for the sub-atomic particle.
Q3. At what speed are they actually travelling?
The expanding Universe• Review evidence for
expansion
• Use Hubble’s law to estimate an age for the Universe
• Explain the origin of red shifts > 1 in terms of relativistic Doppler effect and the expansion of space itself
The expanding Universe• Review evidence for
expansion
• Use Hubble’s law to estimate an age for the Universe
• Explain the origin of red shifts > 1 in terms of relativistic Doppler effect and the expansion of space itself
Ladder of astronomical distancesRed-shiftAssume that the speed of recessionas measured by wavelength shift isproportional to distance
Brightest galaxyAssume that brightest galaxiesin clusters are all equally bright
SupernovaType 1a supernovae all havethe same absoute brightnessComa cluster
Virgo cluster of galaxies
M31 Andromeda
Magellanic clouds
Tully-FisherFaster rotating galaxies havegreater mass and are brighter
Blue supergiantsAssume that the brighteststar in a galaxy is as brightas the brightest in another
Cepheid variablesThese very bright pulsing starscan be seen at great distances.The bigger thay are the brighterthey shine and the slower theypulsate.
Colour-luminosityThe hotter a star the brighter its light, andthe brighter it shines. If the type of star canbe identified there is a known relationshipbetween colour and brightness. Distancethen found comparing actual with apparentbrightness
ParallaxShift in apparent position as Earth movesin orbit round Sun. Recently improvedby using satelite Hipparcos: nowoverlaps Cepheid scale.
Baselineall distances based on measurement of solar system, previously using parallax, today usingradar
1010
109
108
107
106
105
104
103
102
10
1
Ladder of astronomical distancesRed-shiftAssume that the speed of recessionas measured by wavelength shift isproportional to distance
Brightest galaxyAssume that brightest galaxiesin clusters are all equally bright
SupernovaType 1a supernovae all havethe same absoute brightnessComa cluster
Virgo cluster of galaxies
M31 Andromeda
Magellanic clouds
Tully-FisherFaster rotating galaxies havegreater mass and are brighter
Blue supergiantsAssume that the brighteststar in a galaxy is as brightas the brightest in another
Cepheid variablesThese very bright pulsing starscan be seen at great distances.The bigger thay are the brighterthey shine and the slower theypulsate.
Colour-luminosityThe hotter a star the brighter its light, andthe brighter it shines. If the type of star canbe identified there is a known relationshipbetween colour and brightness. Distancethen found comparing actual with apparentbrightness
ParallaxShift in apparent position as Earth movesin orbit round Sun. Recently improvedby using satelite Hipparcos: nowoverlaps Cepheid scale.
Baselineall distances based on measurement of solar system, previously using parallax, today usingradar
1010
109
108
107
106
105
104
103
102
10
1
Hubble’s LawRecession velocity = constant x distance
v = H0r
The bigger the Hubble constant the faster the universe expands, and the younger it must be to have got to its present size.
Hubble constant – time scale for the universe
This time is its reciprocal – the Hubble time:
t = 1/H0
Optical telescopes can see out to about 1000 million light-years, What red shift does this correspond to? (by Hubble’s Law)
0.078
Radio and infrared telescopes can detect red-shifts (z = /) up to 3 or 4
At large distances , the red shift is best thought of not as a velocity of recession, but simply as the waves stretching as the space stretches
z = 3-4, (with z =v/c) would imply a recession velocity of more than c.
List the difficulties associated with measuring the distance to the furthest and faintest galaxies
The history of the Universe• Explain the origin of the
cosmic microwave background
• Review evidence for the generally accepted model of the history of the Universe
• Appreciate that there are many unsolved problems in cosmology
Robserved/Remitted = observed/emitted
Robserved/Remitted = (observed+)/emitted
Robserved/Remitted = 1+z