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Moving worlds

Wandering stars

The word 'planet' derives from the Greek for 'wanderer' and the five bright planets were originally recognised as 'wandering stars' as they moved through the zodiac. The way they appear to move depends on both their distance from the Sun and whether they are inside or outside of the orbit of the Earth.

Consequently Mercury moves very quickly across the sky as it races around the Sun, whilst Saturn crawls round, taking almost 30 years to complete a single circuit.


Inferior planets

Mercury and Venus are closer to the Sun than the Earth and are described as inferior planets.

The changing phases of an inferior planet as it moves around the Sun

The changing phases of an inferior planet as it moves around the Sun


As they move around the Sun, they exhibit a full range of phases like our Moon. When an inferior planet is directly between the Earth and the Sun it is at inferior conjunction. At this time they have a large angular diameter, but a 'new' phase and are invisible except in the rare circumstances of a transit. When inferior planets are on the far side of the Sun from the Earth they are at superior conjunction and have a full phase but a small angular diameter.

The key positions of an inferior planet with respect to the Earth and Sun



The top illustration shows the key positions of an inferior planet with respect to the Earth and Sun:

1. Inferior conjunction
2. Superior conjunction
3. Eastern elongation
4. Western elongation







This illustration shows the appearance of the inferior planets at these points.

Appearance of inferior planets

The changing positions of Venus and Mercury
At greatest eastern elongation Mercury and Venus are at their maximum angular distance from the Sun and have a 50% phase, appearing half-illuminated. At this time they are to the east of the Sun, so are visible to the unaided eye just after sunset.

Similarly, at greatest western elongation inferior planets are also at maximum distance from the Sun and at 50% phase, but are best seen just before sunrise.

The changing positions of Venus and Mercury. The lines show the apparent path of each planet in the evening sky as they move around the Sun


Transits of Mercury and Venus

If an inferior planet is directly between the Earth and the Sun and exactly in line with both bodies then for a few hours it will appear to cross or transit the solar surface, appearing as a silhouetted black disk.

Transits of Mercury occur 13 or 14 times each century. The last one took place on 7 May 2003 (see left) and the next will be in November 2006.

Transit of Mercury, 7 May 2003
Ultraviolet image of the transit of Mercury from the TRACE satellite (left) and an image of the transit of Venus taken from the Royal Observatory, Greenwich on 8 June 2004
An image of the transit of Venus taken from the Royal Observatory's 28-inch refractor on 8 June 2004
Images: Brian Handy, Montana State University, TRACE project (left) and © NMM London

Transits of Venus are much rarer. Two successive transits separated by 8 years occur at intervals of 121.5 and 105.5 years. A pair were observed in the 19th century. The last pair, the first was seen on 8 June 2004 and the next can be seen in 2012.


Superior planets

The superior planets are those further from the Sun than the Earth and include Mars, the gas giants and Pluto.

The key positions of a superior planet with respect to the Earth and Sun
The key positions of a superior planet with respect to the Earth and Sun
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Opposition
Superior conjunction
Eastern quadrature
Western quadrature 

Superior planets are best viewed at opposition, when the Sun and planet are on opposite sides of the Earth. The planet is then closest to the Earth, culminates at midnight and is visible for all or most of the night. At conjunction a superior planet is on the other side of the Sun from the Earth and so is invisible.

A sketch of Mars showing a gibbous phase
At quadrature a planet makes a right angle with the Earth and Sun. This is also the time when a superior planet has a phase furthest away from full, but only Mars really exhibits this to a significant extent.


A sketch of Mars showing a gibbous phase
Image: Sally Russell

Just 85% of the visible hemisphere of Mars is illuminated at this time, giving it an obvious gibbous appearance but all the more distant planets show minimum phases of 99% or above.


Retrograde motion

With the exception of Pluto, both inferior and superior planets move within the zodiac constellations and remain close to the ecliptic.

Inferior planets move east to west and then west to east as they move around the Sun. The long-term movement of superior planets is from west to east. This is direct motion. But around opposition their motion appears to loop or zigzag and they temporarily move from east to west. This is retrograde motion. The stationary points are where the planet apparently changes direction.

The retrograde motion of a superior planet in the sky
The retrograde motion of a superior planet in the sky. The markers indicate where the planet is at opposition and its two stationary points.

Retrograde motion can be explained by considering the relative orbital speeds and inclinations of the superior planets and the Earth in the heliocentric model of the solar system. Planets further than the Earth from the Sun move more slowly in their orbits, so the Earth will catch up with them as the two planets move towards opposition.

Retrograde motion explained
Retrograde motion explained. Looking at positions 1-3, the direct motion of Mars appears to slow as Earth approaches. In positions 4 and 5, Mars appears to move backwards as Earth overtakes. Mars returns to its eastward drift at 6 and 7.


Models of the solar system: from Ptolemy to Kepler

Explaining the movements of the planets and predicting their positions was complex. The models of the solar system evolved from geocentric (Earth-centred) to heliocentric (Sun-centred).

Ptolemy
The Greek astronomer Ptolemy devised a widely accepted theory to explain the apparent motion of the Sun and planets across the sky. He placed the Earth at the centre of the universe - this was a geocentric model. The Sun and planets then moved around the Earth in circles.

A diagram explaining the Ptolomaeic heliocentrical view of the universe
A diagram explaining the Ptolomaeic heliocentrical view of the universe

This basic model could not explain why planets underwent retrograde motion - they sometimes appeared to reverse direction and move in loops against the background stars. To account for this a more complicated version placed the Earth very slightly away from the centre of the planet's orbits - the deferent. Two further complications were added. Firstly planets were thought to travel in loops - epicycles - superimposed on their circular orbits. Secondly, the motion of the centre of a planetary epicycle was thought to be uniform (they kept the same speed) with respect to the equant - a point on the other side of the centre of the circle from the Earth.

Further variations in such a complicated system allowed astronomers to explain the observed motions of objects in the solar system and the geocentric model survived for the next 1500 years.


Copernicus

The Polish canon Nicholas Copernicus published a new sun-centred (heliocentric) model of the solar system in 1543, the year in which he died.

In his book De Revolutionibus he put forward the idea that the planets moved in circles around a stationary central Sun. This model explained the retrograde motion of the planets but could still not predict their exact locations.


Johannes Kepler
In the early-17th century, Johannes Kepler finally produced a simple and precise description of planetary motion.

Diagram of a planet in an elliptical orbit
Diagram of a planet in an elliptical orbit
He used the observational records of Danish astronomer Tycho Brahe to produce a complete mathematical description of the motion of a planet as it moves around the Sun.

The description also applies to a satellite as it moves around a planet.

Diagram explaining the key principles of Kepler's laws

He created three laws that govern the motion of each object.

These are explained below and relate to the diagram on the left.


1. Each planet moves around the Sun in an elliptical orbit with the Sun at one focus (the other focus is empty).

2. Each planet moves so that an imaginary line joining the centre of the planet to the centre of the Sun sweeps out equal areas in equal times. In the diagram above the planet moves from A to B and from C to D in the same amount of time - the two shaded wedges have equal areas. This means that planets move fastest when they are closest to the Sun (perihelion) and slowest when they are furthest away (aphelion).

3. The square of the time taken for a planet to complete one orbit is proportional to the cube of its mean distance from the Sun.

This is expressed mathematically as:

Equation of Kepler's third law (the orbital period of a planet squared is equal to constant times the mean distance of the planet to the Sun cubed)

where T is the orbital period of the planet, r is the mean distance of the planet from the Sun and k is a constant.

If two bodies are involved, this can be written as an equation:

Equation of Kepler's third law where two bodies are involved

where 1 and 2 are the two orbiting bodies.

Kepler's laws also apply to any system of satellites, for example the moons of Jupiter as well as to comets and asteroids in orbit around the Sun.

The new planets discovered in orbit around other stars also move in the same way.

For the special case of planets in orbit around our own Sun, a simpler version of Kepler's third law can be used. All the planetary orbits can be scaled with respect to the Earth, which takes 1 year to travel around the Sun in an orbit at a mean distance of 1 astronomical unit (AU).

Using Kepler's third law:

Equation of Kepler's third law (the orbital period of a planet squared is equal to constant times the mean distance of the planet to the Sun cubed)

12 = k x 13

so in this case k = 1.

If all the other planetary orbits are set out in years and astronomical units, then their orbits can be scaled with respect to the Earth, just by measuring how long it takes them to travel around the Sun.


Worked examples

1. Saturn takes 29.5 years to orbit the Sun. Calculate Saturn's mean distance from the Sun.

In this case the simple version of Kepler's third law applies where k=1, T is in years and r is in astronomical units (AU).

Equation of Kepler's third law (the orbital period of a planet squared is equal to constant times the mean distance of the planet to the Sun cubed)

29.52 = 1 x r3

r3 = 29.52

So r = 9.5 AU


2. The two small satellites, Phobos and Deimos orbit at mean distances from the centre of Mars of 9400 km and 23,500 km respectively. If Phobos takes 0.32 days to complete each orbit, calculate the orbital period of Deimos.

Here the full version of Kepler's third law is used where TD and TP are the orbital periods of Deimos and Phobos, rD and rP are the mean distances of Deimos and Phobos from Mars.

Equation of Kepler's third law where two bodies are involved

(TD/0.32)2 = (23500/9400)3

TD2 = 15.625 x 0.1024

So TD = 1.3 days


Questions to think about

1. Why do the bright planets move in the Zodiac constellations, instead of wandering about all over the sky?


2.

a) Why does Mercury show phases, whilst Jupiter does not?
b) When is it possible to see Mercury and Venus at Inferior Conjunction?


3.

a) Draw a diagram to show the orbits of Earth and Mars. Indicate the positions of the two planets when Mars is at opposition.
b) Use your diagram to explain why are there times when Mars seems to have apparent retrograde motion against the stars.


4. What are inferior and superior planets?


5. Jupiter orbits the Sun at an average distance of 5.2 AU. Calculate its period in years.


6. Uranus takes 84 years to complete one solar orbit. Calculate its mean distance from the Sun.


7. Two satellites, Io and Europa, orbit Jupiter at mean distances of 420,000 km and 607,000 km respectively. Io completes one orbit in 1.8 days. Calculate the orbital period of Europa.


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