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