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Sam Davyson
PHY//12//1
19/09/05
Mr Ford – Chapter 12 – Revision Notes
Light Intensity
In space the light intensity from distant stars can be used to approximate the distances
between the stars and us. This idea can be mimicked in the laboratory with a light
box, a lux meter and a measuring rule. Taking measurements and plotting them
reveals a linear relationship between light intensity and the reciprocal of the distance
squared. It turns out that they are in fact proportional to one another.
There are some obvious problems with this technique:
 Don’t know how bright the stars really are.
 We must assume that the intensity is not affected by different mediums
(including the atmosphere).
Measuring Distance using Parallax
Parallax is a technique that uses trigonometry and the difference
star
between the apparent positions (referring to the “fixed”
2p
background stars) of a distant object from two different points
a
along a base line. The longer the base line the further you can
b
see. On earth the best we can do is looking at the object twice
SUN
with 6 months between each viewing. This gives a baseline of
2Au = 300 Million km. The angle subtended by the distant star
d
and the two viewing points in two lots of the parallax angle. So
ab
using the diagram parallax angle =
. This method is only really usable over
2
relatively short distances.
The Parsec
A parsec, or 1 pc, is a unit of distance. It is defined as the
1 au
distance between the Sun and an object when the parallax
00’1’’
1
1 pc
angle is equal to one second of arc (
of a degree).
3600
The 1 Au on the diagram is the distance between the Earth and the Sun. Using this
definition it is found that 1pc = 3.08 x 1016 m = 3.26 light yrs. If a star is further from
Earth then the parallax angle is reduced. Roughly speaking:
1
Distance (in pc) =
(when the angle is measured in seconds of arc)
Parallax Angle
Velocity of an Asteroid
By sending two pulses at an asteroid and timing the difference between the lengths of
time each pulse takes to reach the asteroid and return, the velocity (relative to you)
can be found. When doing such calculations it is very important to remember the
factor of two involved. Generally it can be shown that:
v t back  t out

where: v is relative velocity of asteroid, c = 3 x 108 ms-1, and
c t back  t out
change in tback and change in tout are time between getting two pulses back and time
between sending two pulses respectively.
Sam Davyson
PHY//12//2
19/09/05
The Doppler Shift
out and back are fairly self explanatory, and obviously
back > out if the object is moving away, and the
opposite if it is not. This differs from the method used
with the asteroid as continuous waves rather than
pulses are used. The formula that is used is not very
BACK
different to the previous one though:
v 
v back  out

and if v << c (much less) then 
. The speciality of being able
c 2
c back  out
to use continuous waves rather than pulses is important when a one-way measuring
method is used. In this the object is too far away to send a pulse to, however the
object is emitting EM radiation. The change in wavelength of this radiation as it
moves through space can be used to determine the relative velocity of the object. As
this is one-way measuring the formula is modified (no need to account for both ways)
v 

to:
. To determine  the black lines in the emission spectrum that are
c

absorbed by abundant elements in space (H and He) are tracked, and the movement of
the pattern of lines, from what we know to be the true values from experiments on
Earth, is used to determine how much the wavelength of the radiation has changed.
OUT
The Hubble Constant
Hubble measured the distances to galaxies using Cepheid Speed of
Variables (pulsating stars, slower the pulsation the brighter the recession
star) and the speed of recession by looking at the red-shift of the
black lines in the emission spectrum. He found a remarkably
strong correlation. Suggesting that v  r so: v  kr  H 0 r where
Distance to galaxy
H0 is known as the Hubble Constant. It automatically follows
v
1
that: H 0  
and this time is known as Hubble Time. This time is the
r time
1
 time elapsed since all galaxies in one place.
reciprocal of the Hubble constant
H0
This gives a way to find the age of the universe. However there are some problems.
 This assumes that the speed is constant for a galaxy. Due to gravity it isn’t
which gives an over-estimate on the age of the universe.
 H0 is not precisely known.
Using different ways of measuring distances (eg. Type II Supernovae vs. Rotation of
galaxies etc. ) different values for H0 are found. If H0 is low (which is about 50 kms1
Mpc-1) then this implies an old universe (about 20 Gyr). But this data is not what
most sources currently obtain. It is increasingly popular to think that H0 is high (say
85) giving a 14 Gyr old universe. However some parts of the universe are known to
be more than 15 Gyrs old. Which is an utter nonsense. It is thought that dark matter
could be the missing piece that would solve this conflict. Obviously its` presence
would change the rate of recession of galaxies due to gravity.
Cosmological Red-Shift
As light travels from one galaxy to another, the universe expands, which stretches the
wavelength of the light (or other radiation). This causes a red-shift of the radiation,
which should not be confused with the red-shift caused by the relative motion of two
Sam Davyson
PHY//12//3
19/09/05
objects (which is known as the Doppler effect).
The mathematics behind it works like this:
R observed λ observed λ emitted  Δλ
Δλ


 1
 1 z
 observed
R emitted
λ emitted
λ emitted
λ emitted
R observed
This result makes sense as if  = 0 then the
ratio of the distances is 1, implying that they are the same. And also if  > 0 then the
answer will always be greater than 1 suggesting a greater observed than emitted
distance. Once again the  is determined from the movement of the black lines in the
spectrum. The fact that everything measured shows a  > 0 indicates that the
distances between every pair of objects is increasing (except locally where gravity can
cause the reverse effect) and that the universe is expanding. This points to everything
(once upon a time) being at the same place. Making this evidence for the Big Bang
theory.
R emitted
 emitted
Cosmic Microwave Background Radiation
As time has passed since the Big Bang the background radiation from the Big Bang
has been cosmologically red-shifted by the expansion of the universe. Initially high
energy photons were exchanged between particles but by 300 000 yrs after the bang,
the  was stretched to 1m and the temperature had been reduced to 3000 K. This
temperature fall is observed because as  gets longer, f gets lower, so by E = hf, E
gets lower, and E  T giving a temperature fall. Today (14 Gyr later) the  is 1mm
(microwaves) and T = 2.7 K. This is also considered to be evidence for the Big Bang.
Ulber`s Paradox
This is a thought-experiment that suggests that if the Universe is infinite then
everywhere you looked in the sky your line of sight would (eventually) end with a
star, giving a white sky. Ulber found that the sky was black rather than white. He
claimed that there were not enough stars for an infinite Universe, so it must be finite.