The Expanding Universe (HSC SSCE Physics): Revision Notes

The Expanding Universe

Introduction

In the early 20th century, astronomers made a remarkable discovery about the enormous size of galaxies and the vast distances between them. This raised an important question: why hasn't the Universe collapsed in on itself under the force of gravity? Understanding how the Universe expands has been crucial to answering this question.

Distances in the Universe

The light year

A light year is the distance that light travels through a vacuum in one Earth year. This equals $9.46 \times 10^{15}$ m.

Light travels at a speed of $c = 3.0 \times 10^8$ $m \cdot s^{-1}$ in a vacuum. At this incredible speed:

• Light takes just over 1 second to reach the Moon
• Light from the Sun takes 8 minutes and 20 seconds to reach Earth
• Light from Proxima Centauri takes 4.3 years to reach us

Using the Hubble Space Telescope, astronomers have observed light that has travelled for nearly 14 billion years across the Universe.

Other distance units

Astronomers also use other units such as the parsec (approximately 3.26 light years) and the astronomical unit (AU), which is the distance between Earth and the Sun.

Early theoretical predictions

Einstein's static Universe

In 1916, Albert Einstein published his General Theory of Relativity. At that time, he firmly believed the Universe was static - neither expanding nor contracting. This seemed like common sense to many scientists.

Friedmann's mathematical solutions

In 1922, Russian mathematician Alexander Friedmann discovered something surprising. When he worked through the mathematical equations of Einstein's General Theory of Relativity, he found solutions suggesting the Universe could either be expanding or contracting.

Lemaître's expanding Universe

Independently, in 1927, Georges Lemaître, a Belgian priest, mathematician and astronomer, reached similar conclusions to Friedmann. However, Lemaître went further. He proposed that the early Universe was like a "primeval atom" containing all the mass in the Universe. This primeval atom then expanded, spreading mass throughout the Universe.

Importantly, Lemaître used actual observational data from cosmology to support his ideas. He correctly worked out that the Universe must be expanding at a speed $v$ that is proportional to the distance $D$ from Earth:

$\frac{v}{D} = \text{constant}$

The value Lemaître calculated for this expansion rate was remarkably similar to the rate later measured by Edwin Hubble in 1929.

Einstein's changing view

Einstein initially didn't accept these ideas. He thought Friedmann's work was mathematically interesting but didn't represent physical reality. He also rejected Lemaître's expanding Universe theory at first, even after Lemaître showed him the observational evidence. However, by the early 1930s, Einstein accepted that the expanding Universe solution was correct and acknowledged that Friedmann had been right. Sadly, Friedmann had already died from typhoid by this time.

Measuring cosmic distances

To confirm whether the Universe really was expanding, astronomers needed reliable ways to measure the enormous distances to faraway galaxies. They also needed to measure how fast these galaxies were moving.

Luminosity and stellar brightness

Stars radiate energy outward in all directions. The radiation we observe on Earth comes from the surface of stars. By carefully measuring this radiation, we can work out the distance to stars.

Luminosity (symbol $L$, measured in watts, W) is the total rate of energy output from a star's surface. This is also called absolute brightness - it's how bright the star actually is.

Apparent brightness (symbol $m$, measured in watts, W) is how bright a star appears to us when we observe it from Earth. The difference between apparent and absolute brightness tells us how far away the star is.

The Stefan-Boltzmann Law

Luminosity is related to a star's surface temperature $T$ (measured in kelvin, K) and its surface area $A$ (measured in square metres, $m^2$). This relationship is called the Stefan-Boltzmann Law:

$L = \sigma A T^4$

where $\sigma = 5.7 \times 10^{-8}$ $W \cdot m^{-2} \cdot T^{-4}$ is the Stefan-Boltzmann constant.

The intensity of a star's output is the luminosity per unit area of the star's surface:

$I = \frac{L}{A} = \sigma T^4 \text{ W} \cdot \text{m}^{-2}$

Calculating distance from brightness

Energy from a star radiates outward in all directions (spherically). At a distance $d$ (metres) from the star, this energy is spread over a spherical surface with area $4\pi d^2$. Therefore, the apparent brightness $m$ (W) of a star with luminosity $L$ (W) is:

$m = \frac{L}{4\pi d^2}$

Cepheid variables - cosmic distance indicators

In 1912, Henrietta Leavitt made a crucial discovery whilst analysing a group of stars called Cepheid variables. These are stars whose apparent brightness changes periodically (they pulse).

Leavitt was one of many women employed to analyse astronomical photographs. At the time, they received low wages whilst men operated the telescopes. Despite these conditions, Leavitt made a groundbreaking observation: the period of a Cepheid variable star is directly related to its luminosity.

Hubble's observations and law

Edwin Hubble knew about Leavitt's discovery when he began observing stars in the nearby Andromeda galaxy (known as M31).

Measuring the distance to M31

In 1923, Hubble identified a Cepheid variable in M31, which he later used to measure the galaxy's distance. In 1929, Hubble calculated that M31 was around 900,000 light years away. This was later refined - in 1953, Walter Baade showed that M31 is actually 2.5 million light years away.

The Doppler effect and spectral shifts

Hubble's calculation of this enormous distance prompted him to observe the spectra of other similar galaxies. By carefully comparing the wavelengths of spectral lines (particularly hydrogen) from distant galaxies to the same spectral lines produced in laboratories on Earth, he could determine how fast these galaxies were moving relative to Earth.

This technique uses the Doppler effect:

• If wavelengths are shortened (shifted towards the blue end of the spectrum), the galaxy is approaching us - this is called blueshift
• If wavelengths are lengthened (shifted towards the red end of the spectrum), the galaxy is receding from us - this is called redshift

Hubble's crucial discovery

Hubble measured the speeds of several galaxies. He quickly realised they were moving much faster than any object within our own galaxy. He concluded they must be separate galaxies in their own right, located far from our own.

He made a remarkable observation: in general, the further away a galaxy is, the faster it is moving away from us. By plotting recession speed against distance, he discovered a linear relationship.

Hubble's Law

This relationship is now known as Hubble's Law:

$v = H_0 D$

where:

• $v$ = recession speed (how fast the galaxy is moving away)
• $D$ = distance to the galaxy
• $H_0$ = Hubble's constant

Many refinements have been made to the value of $H_0$ over the years, and with each refinement, our estimate of the Universe's age changes. Currently, astronomers estimate the Universe is approximately 13.7 billion years old.

An important exception

Understanding the expansion

If the Universe is expanding, where did it originate from? The remarkable thing about this expansion is that from wherever you observe the Universe, it appears to be expanding away from that point. There is no identifiable "centre" of the Universe that we can point to.

The balloon analogy

Connection to the Big Bang Theory

If we imagine running time backwards, the expanding Universe means that everything must have been closer together in the past. Eventually, we reach a point where everything came together at the very beginning. This gives further support to the Big Bang Theory, which explains the origin of the Universe.

Remember!