A-level Physics/Cosmology/Models of the known universe

As more sophisticated tools have been developed, our understanding of the universe has improved. Some proposed models of the universe were proven wrong, and other ideas are still with us today.

Measuring distances in the universe
The distances at the scale of the universe are gigantic, and our everyday metres and even kilometres are too small to be used. We need to use units that are more appropriate for large distances. Often, other units are convenient to use because of the way they are measured.

The light-year
One light-year is defined as the distance light travels in one year. As you know, light travels at $$3 \times 10^{8}$$ $$\text{ms}^{-1}$$, and so the distance it covers in one year is enormous. One light year is approximately $$10^{16}\text{m}$$.

The astronomical unit
The astronomical unit is defined as the average distance between the Earth and the Sun. It originates from the fact that it was possible to measure the distances of the planets, but only in multiples of the distance between the Earth and Sun. It is still useful today for distances within the solar system. It is approximately equal to $$1.5 \times 10^{11}\text{m}$$.

The parsec
One parsec is simply the reciprocal of half the angle of parallax of a star, when observed from Earth at two opposite points of its orbit. Parallax is the apparent change in position of an object against a fixed background when the position of the observer changes, like how buildings seem to move faster than background hills when you're in a car. It is convenient to find from the measured angle, and is therefore used mainly for the distances of stars. This concept is covered in more detail in Stars & Galaxies. One parsec is approximately $$3 \times 10^{16}\text{m}$$, or 3.26 light-years.

Overview of the solar system
Our solar system consists of the Sun, the planets and an asteroid belt. Additionally, there are comets that have highly elliptical orbits, and return to the solar system at regular intervals.

Planets
There are eight planets orbiting the Sun (Pluto being reclassified as a dwarf planet), which is at the centre of the solar system. Most planets also have natural satellites, or moons, orbiting them. The table below outlines the main features of the planets, relative to the Earth:

Asteroid belt
There is a concentration of small, rocky asteroids between Mars and Jupiter, which is known as the asteroid belt. There are hundreds of thousands of these planetoids orbiting the Sun, and are sometimes called minor planets.

Comets
Comets are lumps of rock, frozen water, methane and ammonia that orbit the Sun, and are typically only a few kilometres in diameter. They have very eccentric (elliptical) orbits and therefore vary greatly in their distance from the Sun. When they are near the Sun, they have long tails of approximately 1AU, due to the Sun's radiation.

The progress in the understanding of the universe
The accepted model of the Solar System has been subject to great controversy over the decades. In the old geocentric model, the Earth was originally placed in the centre of the Solar System, and had the other planets and the Sun orbiting it. Now, the accepted model places the Sun in the centre, with the Earth and other planets orbiting around it.

Copernicus
Nicolaus Copernicus found the old geocentric model unnecessarily complicated. Instead of having the Earth in the centre of the universe, he decided to place the Sun in the centre, which we now call the heliocentric model. This model could very easily explain the movement of the planets and the Sun across the sky, and in particular the retrograde motion of Mars, where it would appear to move "backwards" across the sky for several weeks. This retrograde motion of Mars was previously explained by epicycles where it would "loop-the-loop" around at certain points. With Copernicus' new model, it was explained that since the Earth was closer to the Sun than Mars, there will be sections where the Earth will "overtake" Mars, and will make Mars apparently move backwards across the sky.

Opposition to Copernicus
Copernicus' heliocentric model was rejected by most people mainly because of religious beliefs at the time, and although it seemed to simplify the motion of the planets, it was less accurate than the geocentric model at fitting the observed movements of the planets.

People also argued that if the Earth was moving, the stars would have a detectable parallax. Copernicus claimed that the stars were too far away to detect any parallax, and with more sensitive equipment, he has now been proved correct. Another argument against the heliocentric model was that objects all fall towards the Earth, and so it must be the centre of the universe. This was the intuitive conclusion before Newton revolutionised our ideas about motion.

Kepler
Johannes Kepler improved upon Copernicus' original model by using elliptical orbits instead of circular ones. He devised three laws of planetary motion:

Kepler's first law
Kepler found that the planets fit the observed pattern better with the heliocentric model if they travelled in ellipses, not circles, and had the Sun at one of the foci of these ellipses. Therefore Kepler's first law states:

The planets orbit the sun in elliptical orbits with the sun at one focus.

Kepler's second law
Now that the planets had elliptical orbits, it would not make sense for them to travel at the same speed at all points of their orbit. The planets would speed up nearer the Sun, and move slower when they were further away from the Sun. Kepler observed that the imaginary triangle formed between the planet at two points in its orbit and the Sun always had the same area provided the two points of the planets orbit had the same time difference between them. From this it follows that a planets orbit is faster nearer the Sun than further away from it. Kepler's second law states that:

The line connecting a planet to the sun sweeps out equal areas in equal amounts of time.

Kepler's third law
Kepler realised that the distance of a planet from the Sun and its orbital period were related by the formula: $$T^2 \propto d^3$$, where T is the time taken for one orbit, and d is the distance from the Sun, although it is actually the length of the semi-major axis (which is half of the longest diameter of the elipse).

The square of the orbital period is proportional to the cube of the distance from the Sun.

Galileo
Galileo Galilei was the first person to use a telescope to look at the night sky. He was able to view many things that weren't visible to the naked eye, such as the imperfectness of the surface of the moon, and the fact that there were many faint stars in the sky. Both of these supported Copernicus' ideas.

Galileo and Venus
When Galileo observed Venus with his telescope. he noticed that it went through phases, like the Moon. He also noticed that when Venus was a crescent, it was much larger than when it was full. This observation was evidence that Venus was orbiting around the Sun and not Earth.

Galileo and Jupiter's Moons
Galileo also observed four objects orbiting Jupiter, which are now known as the Galilean moons They supported the view that not everything orbits the Earth.

Newton's universal law of gravitation
When Issac Newton created his universal law of gravitation, he attempted to show that Kepler's observations of planetary motion agreed with it. This was significant evidence to show that he was correct.

Newton's law of gravitation can be used to give a formula for the planets in the form $$T^2 \propto d^3$$:

The force of gravitational attraction between the Sun and a planet is equal to the centripetal force required to keep the planet in its orbit:
 * $$F=G\frac {m_1 m_2}{d^2} = \frac{m v^2}{d}$$

The period of the planets orbit can be given by:


 * $$\text{Time taken} = \frac {\text{distance}}{\text{velocity}}$$

Where the distance is the circumference of a circle, $$2 \pi \times d$$ (note that d is distance from Sun, and is therefore the radius, not the diameter). This gives us:


 * $$T = \frac{2 \pi d}{v}$$

Which we can re-arrange to make v the subject and substitute into $$v^2$$ in the centripetal force equation:


 * $$v = \frac{2 \pi d}{T}$$


 * $$v^2 = \frac{4 \pi^2 d^2}{T^2}$$


 * $$G\frac {m_1 m_2}{d^2} = \frac{4 \pi^2 m d}{T^2}$$

Eliminating m, the mass of the planet, and tidying up:


 * $$G \frac{m T^2}{4 \pi^2 d^3} = 1$$

And finally, making $$T^2$$ the subject:


 * $$T^2 = \frac{4 \pi^2}{Gm} d^3$$

We now have a formula in the form $$T^2 \propto d^3$$, with $$\frac {4 \pi^2}{Gm}$$ as the constant of proportionality, where m is the mass of the Sun.

The discovery of Neptune
In 1821, Alexis Bouvard published very accurate observations in the orbit of Uranus. However, soon after this, the orbit of Uranus was observed to deviate from the published values. In 1845 John Adams, using Newton's universal law of gravitation, calculated the orbit of another planet outside of Uranus whose gravity would account for the perturbations in Uranus' orbit. Neptune was discovered in its predicted position a year later. Pluto was discovered in a similar way, since it was causing further perturbations in the orbits of Uranus and Neptune.

Problems encountered with Newton's theory
Although Newton's theory was very successful in explaining the motion of the planets, and had even been used to discover unknown planets, there were still some problems with it:


 * The orbit of Mercury was observed to have a different orbit to the one predicted by the theory. This has now been resolved by Einstein's general theory of relativity.
 * If every object in the universe attracts each other, then the entire universe should have collapsed because of the gravitational attraction. To solve this, Newton came up with the idea that the universe was infinitely large, and that matter was uniformly spread throughout. This led to its own problems, though, namely Olber's paradox, which states that an infinitely large universe will always have a star on any given line of sight, and so the night sky should actually be bright. This has been resolved with the observations of an expanding universe by Edwin Hubble.