Black as night

Olbers popularised this question as a paradox: if the Universe is infinitely large and old, why is it dark at night?

Possible (overlapping) answers:

• The Universe is not infinite.
• Not all the light has reached us yet.
• Could something be absorbing light?
• Expansion of the Universe.

Constructing Hubble's law

Constructing a Hubble's law plot involves measuring the following for many galaxies

• the redshift, which tells us the speed at which the galaxy is moving away from us
• the distance, originally deduced from Cepheids, but now done using a variety of methods - see previous lecture

Cepheids (Type I)

• Cepheids are a class of variable stars named after the star δ Cephei.
• δ Cephei varies between apparent magnitudes 4.4 and 3.5 with a period of 5.4 days.
• They are variable stars that are between 4 and 20 solar masses.
• They can be up to 100,000 more luminous than the Sun.
• They vary because they pulsate in size, by up to a quarter of their radius.
• Type I Cepheids are also known as classical Cepheids.
• Type II Cepheids are much less massive and less luminous.

Delta Cephei - light curve

Source: ThomasK Vbg CC-BY SA 3.0

Cepheid - Period-luminosity graph

Source: Marcia Rieke Copyright not clear.

Method to determine distance to a galaxy

• Find a Cepheid in a galaxy.
• Measure times between maximum magnitudes to measure the period.
• Use the period and the period-luminosity relationship to estimate the luminosity.
• The luminosity plus the inverse square will yield a distance.

Cepheids in practice

• The Cepheid method can be used to calculate distances of up to about 100 million light years.
• The limitation on distance arises from being able to accurately observe Cepheid variables in the galaxy.
• The largest Cepheid determined distances come from the Hubble Space Telescope.
• Uncertainties arise because of: absorption of light; location within the galaxy; variations from the standard period-luminosity relationship.

Redshift and the Doppler effect

• The Doppler effect applies when a source of waves moves relative to an observer
• If the source is moving towards the observer there is a shift to shorter wavelengths - a blue shift
• If the source is moving away from the observer there is a shift to longer wavelengths - a redshift
• The size of the shift is proportional to the speed of the source.
• BUT although galaxy redshifts seem to be due to the Doppler effect, it was later understood that there was another explanation, of which, more later.

Redshift example

Source: Georg Wiora CC-BY SA 2.5

Redshift formula

If v is the speed of the source, c is the speed of light then the redshift z is:

z = v / c

If λ₀ is the stationary wavelength and λ is the observed (shifted) wavelength, then the redshift is given by

z = (λ - λ₀) / λ₀

Notes:

• Doubling the shift λ - λ₀, doubles the speed v
• This formula holds only when v is much less than c. There is a slightly more complicated relativisitc formula in which z can be larger than 1.

Redshift calculations

• Consider the Hα line which has λ₀=656nm
• A galaxy is observed where the Hα line is at λ₀=722nm
• So λ - λ₀=66 nm
• Then z = (λ - λ₀) / λ₀ = 0.1
• The speed of light is c = 300,000 km^-1
• So v = z × c = 30,000 km^-1
• To what speed does λ - λ₀=22 nm correspond?

Hubble's law - 1929

Source: Hubble (1929) Public Domain (probably)

Hubble's law - 2005

Source: Reindl et al (2005) Published in ApJ

The Hubble constant

• The speed v is usually measured in kilometres per second - km/s or km s^-1
• The distance d is usually measured in mega parsecs - Mpc = 1 000 000 pc
• So the Hubble constant H₀ is quoted in units of speed per unit distance.
• The current accepted value is about H₀ = 70 km s^-1 Mpc^-1
• The uncertainty of this value is no more than 10%.
• The speed of light is 300 000 km s^-1
• 1 parsec = 3.26 light years

Estimates of the Hubble constant

Source: Alexander.stohr CC-BY SA 3.0

The age of the Universe

An estimate of the age of the Universe can be worked out from the Hubble constant.

The age of the Universe T is given by this equation:

T = 70/H₀ × 14 billion years

Taking the current accepted value we can calculate the age of the Universe as 14 billion years.

More refined models of the universe are used to adjust this estimate, but this is still a plausible estimate of the age of the Universe.

Note: Look at the units of the Hubble constant and notice that if you convert Mpc to km then the units become just s^-1.

Other distance techniques

At galactic scales, the following distance measuring methods can be used:

• Cepheid period-luminosity relation
• Type Ia supernovae.
• Tully-Fisher relation

In astronomy, the term 'standard candle' refers to an object of known luminosity that can be used to calculate a distance.

For a long list of methods see http://www.astro.ucla.edu/~wright/distance.htm

Type 1a supernova

• Recall from before that a type Ia supernova occurs when a white dwarf accretes mass from another star and approaches the 1.44 solar mass Chandrasekhar limit.
• This means that all type Ia supernovae have a similar luminosity.
• This, together with the fact their luminosity is so large, makes them good standard candles.
• They can be used to measure distances of up to almost 1000 million light years.

Tully-Fisher

• A galaxy's luminosity depends on its mass.
• The orbital speed of stars in a galaxy also depends on the galaxy's mass.
• An indication of these orbital speeds can be found by looking at the width of spectral lines in a galaxy's spectrum.
• In 1977 Tully and Fisher found that there was a good correlation between a spiral galaxy's luminosity and the width of spectral lines.
• So, by measuring the width of spectral lines and then using the Tully-Fisher relation, it is possible to estimate the luminosity of galaxy and thus estimate the distance to it.