Chapter 14 Astronomy & Cosmology

Question 1

Luminosity L is defined as

Answer 1

The total power output of radiation emitted by a star

• It is measured in units of Watts (W)

Question 2

Radiant flux intensity F is defined as

Answer 2

The observed amount of intensity, or the radiant power transmitted normally through a surface per unit of area, of radiation measured on Earth

Question 3

The best way to picture this is:

Answer 3

• The luminosity is the total radiation that leaves the star
• The radiant flux intensity is the amount of radiation measured on Earth
• By the time the radiation reaches the Earth, it will have spread out a great deal, therefore, it will only be a fraction of the value of the luminosity

Question 4

Light sources which are further away appear …

Answer 4

• fainter because the light it emits is spread out over a greater area
• The moment the light leaves the surface of the star, it begins to spread out uniformly through a spherical shell
  
  • The surface area of a sphere is equal to 4πr²
• The radius r of this sphere is equal to the distance d between the star and the Earth
• By the time the radiation reaches the Earth, it has been spread over an area of 4πd²

Question 5

• The inverse square law of flux can therefore be calculated using:

Answer 5

• Where:
  
  • F = radiant flux intensity, or observed intensity on Earth (W m^-2)
  • L = luminosity of the source (W)
  • d = distance between the star and the Earth (m)
• This equation assumes:
  
  • The power from the star radiates uniformly through space
  • No radiation is absorbed between the star and the Earth

Question 6

The inverse square law of flux equation tells us

Answer 6

• For a given star, the luminosity is constant
• The radiant flux follows an inverse square law
• The greater the radiant flux (larger F) measured, the closer the star is to the Earth (smaller d)

Question 7

• A standard candle is defined as:

Answer 7

An astronomical object which has a known luminosity due to a characteristic quality possessed by that class of object

Question 8

• Examples of standard candles are:

Answer 8

• Cepheid variable stars
  
  • A type of pulsating star which increases and decreases in brightness over a set time period
  • This variation has a well defined relationship to the luminosity
• Type 1a supernovae
  
  • A supernova explosion involving a white dwarf
  • The luminosity at the time of the explosion is always the same

Question 9

Using Standard Candles as a Distance Indicator

Answer 9

• A direct distance measurement is only possible if the object is close enough to the Earth
• For more distant objects, indirect methods must be used – this is where standard candles come in useful
• If the luminosity of a source is known, then the distance can be estimated based on how bright it appears from Earth
  
  • Astronomers measure the radiant flux intensity, of the electromagnetic radiation arriving at the Earth
  • Since the luminosity is known (as the object is a standard candle), the distance can be calculated using the inverse square law of flux
• Each standard candle method can measure distances within a certain range
• Collating the data and measurements from each method allows astronomers to build up a larger picture of the scale of the universe

Question 10

Hubbles Law

Answer 10

Question 11

Wien’s displacement law relates the

Answer 11

• observed wavelength of light from a star to its surface temperature, it states:

The black body radiation curve for different temperatures peaks at a wavelength which is inversely proportional to the temperature

• This relation can be written as:
• λ_max is the maximum wavelength emitted by the star at the peak intensity

Question 12

A black-body is an object which:

Answer 12

• Absorbs all the radiation that falls on it, and is also a good emitter
• Does not reflect or transmit any radiation
• A black-body is a theoretical object, however, stars are the best approximation there is
• The radiation emitted from a black-body has a characteristic spectrum that is determined by the temperature alone

Question 13

The intensity-wavelength graph shows how thermodynamic temperature links to the peak wavelength for four different stars

Answer 13

Question 14

• The full equation for Wien’s Law is given by

Answer 14

λ_maxT = 2.9 × 10^-3 m K

• Where:
  
  • λ_max = peak wavelength of the star (m)
  • T = thermodynamic temperature at the surface of the star (K)

Question 15

Wien’s Law tells us the higher the temperature of a body

Answer 15

λ_maxT = 2.9 × 10^-3 m K

• The shorter the wavelength at the peak intensity, so hotter stars tend to be white or blue and cooler stars tend to be red or yellow
• The greater the intensity of the radiation at each wavelength

Question 16

Table to compare surface temperature and star colour

Answer 16

Question 17

A star’s luminosity depends on two factors:

Answer 17

• Its surface temperature
• Its radius
• The relationship between these is known as the Stefan-Boltzmann Law, which states:

The total energy emitted by a black body per unit area per second is proportional to the fourth power of the absolute temperature of the body

Question 18

Stefan-Boltzmann Law Equation

Answer 18

L = 4πr²σT⁴

• Where:
  
  • L = luminosity of the star (W)
  • r = radius of the star (m)
  • σ = the Stefan-Boltzmann constant
  • T = surface temperature of the star (K)

Question 19

Estimating the Radius of Stars

Answer 19

• The radius of a star can be estimated by combining Wien’s displacement law and the Stefan–Boltzmann law

Question 20

Estimating the Radius of Stars: The procedure for this is as follows:

Answer 20

• Using Wien’s displacement law to find the surface temperature of the star
• Using the inverse square law of flux equation to find the luminosity of the star (if given the radiant flux and stellar distance)
• Then, using the Stefan-Boltzmann law, the stellar radius can be obtained

Question 21

Summary of Equations

Answer 21

-Stefan-Boltzmann law= L = 4πr²σT⁴

Question 22

Emission Spectra

Answer 22

• Astronomers are very limited in how they can investigate objects in the space
• All of the techniques used involve analysing the light emitted from the star, or galaxy
• One of these techniques involves analysing the emission and absorption spectra of stars
  
  • More details on this can be found in the revision notes “Line Spectra” in the Quantisation of Energy topic
• Elements in the star, predominantly hydrogen and helium, absorb some of the emitted wavelengths

Question 23

• The top emission spectra shows spectral lines of hydrogen measured on Earth*
• The bottom emission spectra shows the shifted spectral lines of hydrogen measured from a distant galaxy*

Answer 23

Question 24

When astronomers observe light from distant galaxies, they observe differences in the

Answer 24

• spectral lines to the light from the Sun
• The lines have the same characteristic pattern, meaning the element can still be easily identified, they just appear to be shifted sightly
  
  • The lines in the spectra from distant galaxies show an increase in wavelength
  • The lines are moved, or shifted, towards the red end of the spectrum

Question 25

the Doppler effect is defined as:

Answer 25

The apparent change in wavelength or frequency of the radiation from a source due to its relative motion away from or toward the observer

• On Earth, the Doppler effect of sound can be easily observed when sound waves moves past an observer at a notable speed

Question 26

In space, the Doppler effect of light can observed when spectra of distant stars and galaxies are observed, this is known as:

Answer 26

• Redshift if the object is moving away from the Earth, or
• Blueshift if the object is moving towards the Earth

Question 27

• Redshift is defined as:

Answer 27

The fractional increase in wavelength (or decrease in frequency) due to the source and observer receding from each other

Question 28

• For non-relativistic galaxies, Doppler redshift can be calculated using:

Answer 28

• Where:
  
  • Δλ = shift in wavelength (m)
  • λ = wavelength emitted from the source (m)
  • Δf = shift in frequency (Hz)
  • f = frequency emitted from the source (Hz)
  • v = speed of recession (m s^-1)
  • c = speed of light in a vacuum (m s^-1)

Question 29

An Expanding Universe

Answer 29

• After the discovery of Doppler redshift, astronomers began to realise that almost all the galaxies in the universe are receding
• This lead to the idea that the space between the Earth and the galaxies must be expanding
• This expansion stretches out the light waves as they travel through space, shifting them towards the red end of the spectrum
• The more red-shifted the light from a galaxy is, the faster the galaxy is moving away from Earth

Question 30

The expansion of the universe can be compared to dots on an inflating balloon

Answer 30

• As the balloon is inflated, the dots all move away from each other
• In the same way as the rubber stretches when the balloon is inflated, space itself is stretching out between galaxies
• Just like the dots, the galaxies move away from each other, however, they themselves do not move

Question 31

Edwin Hubble investigated the

Answer 31

• light spectra emitted from a large number of galaxies
• He used redshift data to determine the recession velocities of these galaxies, and standard candles to determine the distances
• From these measurements, he formulated a relationship, now known as Hubble’s Law

Question 32

• Hubble’s Law states:

Answer 32

The recession speed of galaxies moving away from Earth is proportional to their distance from the Earth

Question 33

Hubble’s Law can be calculated using:

Answer 33

v = H₀d

• Where:
  
  • v = the galaxy’s recessional velocity (m s^-1)
  • d = distance between the galaxy and Earth (m)
  • H₀ = Hubble’s constant, or the rate of expansion of the universe (s^-1)
• This equation tells us:
  
  • The further away a galaxy, the faster it’s recession velocity
  • The gradient of a graph of recession velocity against distance is equal to the Hubble constant

Question 34

A key aspect of Hubble’s law is that the furthest galaxies appear to move away the fastest

Answer 34

Question 35

Age of the Universe

Answer 35

• If the galaxies are moving away from each other, then they must’ve started from the same point at some time in the past
• If this is true, the universe likely began in an extremely hot, dense singular point which subsequently began to expand very quickly
  
  • This idea is known as the Big Bang theory
• Redshift of galaxies and the expansion of the universe are now some of the most prominent pieces of evidence to suggest this theory is true
• The data from Hubble’s law can be extrapolated back to the point that the universe started expanding ie. the beginning of the universe

Question 36

the age of the universe T₀ is equal to:

Answer 36

• Current estimates of the age of the universe range from 13 – 14 billion years
• There is still some discussion about the exact age of the universe, therefore, obtaining accurate measurements for the Hubble constant is a top priority for cosmologists