The planets 441 The wanderers

The Greeks and other ancient cultures observed certain heavenly bodies which looked like very bright stars and seemed to wander across the heavens (as shown in Figure 4.11), unlike the normal fixed stars which exactly keep their positions relative to one another. The fixed stars remain in patterns we now call constellations while these bright objects move along paths which are certainly not straight lines or curves, and to the Greeks they were quite puzzling. They called them planets, from the Greek word for 'wanderer'. The path shown in the photograph took about six months to complete. By coincidence the track of Uranus appears as the dotted line above and to the right.

4.4.2 Ptolemy's geocentric model

The Greeks knew of five planets: Mercury, Venus, Mars, Jupiter and Saturn. They made careful measurements of their paths, which appeared quite irregular and quite typical of a 'wanderer' moving through the skies. As illustrated in Figure 4.11, planetary paths contain loops where the planet changes direction and

Figure 4.11 The path of Mars through the constellations. Courtesy of Tunc Trezel NASA Astronomy Picture Of the Day 16-12-2003.

doubles back for a short period, before continuing in approximately the same direction as before.

Hipparchus of Rhodes (c. 180 BC-125 BC) produced a catalogue of about 850 stars which was later expanded by Claudius Ptolemy (c. 85-165 AD).

Both Hipparchus and Ptolemy were great astronomers and mathematicians. Considering that their measurements were made with the naked eye, looking through sighting holes aligned along wooden sticks, their accuracy was quite astounding. Some of Hipparchus' data is accurate to within 0.2° of modern values.

The two astronomers applied their mathematical skills to the data to produce a model of planetary motion. Ptolemy's work comprised 13 volumes which contained the results of many years of observation. He developed the so-called geocentric model, according to which the planets were in circular motion around the earth, each planet once around in its own 'year'. At the same time each planet moved in epicycles, or smaller circles around a point which travelled along the main orbit, as illustrated in Figure 4.12.

Ptolemy found that the simple epicycle scheme did not give predictions which were absolutely accurate, and made adjustments to his model, which made it more complicated. He had to move the centre of the planet's orbit a small distance away from the earth, and the epicycle itself was tilted from the main circle.

Figure 4.12 Ptolemy's geocentric model of planetary motion.

He finally produced a model which predicted the positions of all five planets with great accuracy and published his results in a book called the Almagest, which served astronomers and navigators for many centuries.

While the models of Hipparchus and Ptolemy predicted well the observed positions of the planets in the sky, they did not attempt to give a reason for planetary motion. We had to wait more than 1500 years for that!

4.5 The Copernican revolution

We now move rapidly through history to the 15th century. From the pseudo-religious point of view, the earth had to be at the centre of the universe. Natural philosophers, on the other hand, wanted to establish a model in which the earth, the other planets and the sun, and their relationship to one another, could be represented in a simpler and more logical way.

4.5.1 Frames of reference

Whether we realise it or not, all measurements are made with respect to a frame of reference. We generally to choose a frame with respect to which we get the simplest description.

Suppose that we want to describe the motion of a reflector mounted near the rim of a bicycle wheel. Viewed by an observer standing by the roadside and relating to his frame of reference with its origin where he is standing, the motion will appear as a series of cusps, rather like the epicycles of Ptolemy, as illustrated in Figure 4.13. However, such a description does not make it easy, either to visualize what is actually happening to the reflector, or to formulate the laws of physics which govern its motion.

A much simpler description is obtained by using a frame of reference with its origin at the centre of the bicycle wheel (which travels along with the wheel). As seen in that frame, the reflector is simply going round and round in a circle. There is

Figure 4.13 Path of a point on the frame of a bicycle wheel using the roadside as frame of reference.

essentially no fundamental reason for using one frame of reference rather than another.

Skipping ahead for a moment to more recent times, Albert Einstein put forward a formal postulate at the beginning of the 20th century, that all unaccelerated frames of reference are equivalent. This formed the first step in his theory of special relativity, to which we shall come later (Chapters 15 and 16).

4.5.2 Copernicus and the heliocentric model

Nicolas Copernicus (1473-1543), a Polish astronomer, mathematician and physician, and Canon at the Cathedral of Olsztyn, realised that the difficulties experienced by the Greeks in developing a model of planetary motion were largely due to the fact that they were using a most unsuitable frame of reference.

They were constrained by the idea that the earth is the centre of the universe, and also of the solar system. All measurements had to be made in a coordinate system in which the earth was at the origin and stationary. Everything else, including the sun, the other planets, the stars and the moon, had to revolve around the earth. Copernicus realised that it would be much simpler to adopt a heliocentric theory, in which the sun, and not the earth, is at the origin of our frame of reference. In this frame the sun is stationary and the planets orbit the sun. The earth is just one of these orbiting planets.

The idea was not completely original. It had been suggested more than a millennium earlier by Aristarchus, but dismissed by his peers on philosophical grounds. The Greek philosophers had found it impossible to accept that the earth was not at the centre of the universe and Aristarchus' suggestion seems to have been dismissed out of hand. Indeed, there was also considerable opposition to the work of Copernicus and later of Galileo on the same 'philosophical' grounds. Copernicus' book De Revolutionibus Orbium Celestium was not published until the year of his death, in 1543. Galileo's confrontation with the Roman Inquisition on the basis of his 'heretical' writings will be described in the 'historical interlude' at the end of this chapter.

Once it is accepted that the sun, and not the earth, is at the centre of the planetary system, the model becomes much easier to visualise. Copernicus assumed the orbits to be circular, with Mercury and Venus nearer to the sun than the earth, and Mars, Jupiter and Saturn in the outer orbits. Using the then-current astronomical data, he was able to calculate both the radius of each orbit, and the time taken for each planet to complete a full orbit. Most of his values agreed to within a fraction of one per cent of the modern value, an outstanding achievement.

Figure 4.14 Copernican model of the solar system.

Figure 4.14 represents a simplified version of the model. The orbits are drawn approximately to scale, but are shown as being in the plane of the ecliptic. We now know that the orbital planes of the planets are inclined with respect to the ecliptic, but the inclinations are small. The greatest inclination is that of the orbit of Mercury, at 7.0° to the ecliptic.

Copernicus found that, in order to fit astronomical observations, some modifications were necessary. He had to make some of the orbits eccentric, and also make some other minor adjustments. Nevertheless the overwhelming feature of the model is its great simplicity. There is no doubt that the frame of reference built around the sun as origin gives the clearest description of the facts and, more important, gives a good framework for the formulation of the physical laws which govern the motion.

Copernicus identified two characteristic time intervals for each planet:

Sidereal period: time taken to complete one full orbit of the sun (sidereal — measured in relation to the stars).

Table 4.2 Vital statistics for planets known at the time of Copernicus.

Radius of orbit

Average distance

Sidereal period

(Copernican

from sun (modern

Planet

('planet year')

value, AU)

value, AU)

Mercury

88 days

0.38

0.39

Venus

225 days

0.72

0.72

Earth

1 year

1.00

1.00

Mars

1.9 years

1.52

1.52

Jupiter

11.9 years

5.22

5.20

Saturn

29.5 years

9.07

9.54

1 AU (astronomical unit) = 149.6 x 106 km = average distance of the earth from the sun.

1 AU (astronomical unit) = 149.6 x 106 km = average distance of the earth from the sun.

Synodic period: time taken to reach the same configuration as seen from the earth. This is subjective, in the sense that it has meaning only for an observer on the earth, which itself is in orbit about the sun.

4.5.3 Where did the epicycles come from?

The reason why the paths of the 'wandering stars' appear to be so complicated becomes apparent when we analyse more closely the features of the Copernican model.

Copernicus chose the most suitable reference frame because that used by the people before him simply was not suitable. Take the path of Mars, illustrated in Figure 4.15 as an example. We are observing from the earth, which is travelling in an orbit 'inside' the Martian orbit, with a 'lap time' of one year. Mars, orbiting further from the sun, takes 1.9 earth years to complete a full revolution. As the earth 'overtakes' Mars on the inside, Mars appears to be moving backwards, but this is no longer true away from the

apparent path \ On

Mars orbit

Figure 4.15 The path of Mars, as seen from the earth.

Both planets travel anti-clockwise. Arrows represent directions of sight at equal time intervals. The path of Mars as seen from earth makes a characteristic loop or epicycle.

Mars orbit

Figure 4.15 The path of Mars, as seen from the earth.

'overtaking zone'. (Incidentally, we can see Mars in the middle of the night with the sun 'behind us', and therefore the orbit of Mars must be outside the orbit of the earth. Conversely, we can only see Mercury and Venus at sunrise or sunset, when they are in the direction of the sun relative to the earth.)

4.6 After Copernicus

Tycho Brahe (1546-1601), who became known for his very accurate astronomical observations, set out to test the theory of Copernicus. He reasoned that, if the earth really was in orbit, we had the advantage of observing the sky from points seper-ated by vast distances. The pattern of distant stars should appear different, when observed from diametrically opposite points on the orbit.

Brahe presumed that since some of the distant stars must be further away than others, the effect of parallax would alter their relative positions as seen from points at opposite ends of the diameter of the earth's orbit. Despite his high accuracy he could not detect any evidence of parallax, and concluded that either the theory of Copernicus was wrong or the distance to the distant stars was too large for parallax to be measurable.

Looking back with hindsight: Why Brahe did not see any parallax?

Beyond our own solar system the distance to the nearest star (Proxima Centauri) is 4.3 light years (272,000 AU) and the diameter of the earth's orbit is about 2 AU.

earth 1

earth 1

another star earth 2

Figure 4.16 Why there is no parallax.

another star earth 2

Figure 4.16 Why there is no parallax.

If Figure 4.16 were drawn to scale, the distance to Proxima Centauri would be about 1.5 km. What Brahe was attempting to do can be compared to trying to detect parallax between objects more than 1 km away by 'waggling' your eye back and forth through 1 cm.

Johannes Kepler (1571-1630), a former assistant to Tycho Brahe, inherited his astronomical records. In 1609 he published a book entitled The New Astronomy, in which he descibed his mathematical analysis of the planetary system. He showed that the reason why Copernicus had found it neccesary to make small adjustments to his model was not that there was a basic fault in the Johannes Kepler

model, but that the paths of the planets were not circles, but ellipses.

Kepler's discovery

Kepler tried to find a relationship between the size of the orbits of the planets and their orbital speeds. By trial and error, he finally found a formula which gave correct results for the six planets known at that time:

Kepler's first law: The orbit of a planet about the sun is an ellipse*, with the sun at one focus.

Kepler's second law: A line joining the planet to the sun sweeps out equal areas in equal times.

According to Kepler's second law, the areas ASB and DSC are equal if the times for the planet to go from A to B and from C to D are equal. This means that AB must be greater than CD, and therefore the orbital speed is higher when the planet is nearer to the sun.

The eccentricity* is greatly exaggerated in Figure 4.17. As an example, the eccentricity of the earth's orbit is 0.017, which means that the major axis is just 1.00014 times longer than the minor axis.

Kepler's third law: The square of the sidereal period of a planet is proportional to the cube of the semi-major axis.

* For further information see Appendix 4.1, 'Mathematics of the ellipse'.

Figure 4.17 Kepler's second law.

Figure 4.17 Kepler's second law.

The ratio T /a is constant for all planets.

The value of the constant of proportionality depends on the units.

If T is expressed in years and a in astronomical units, T 2/a3 = 1.

Kepler was fascinated by this formula and tried to relate it to some property of harmony among the heavenly bodies. He even speculated that these harmonies represented celestial music made by the planets as they travelled in their orbits. One fact was beyond contention: the formula described the relationship between the period and the radius of orbit with great accuracy. The values of T2/a3 for each planet are equal to or better than 1%, as seen in Table 4.3.

Table 4.3 Kepler's third law in action.

Planet

Sidereal period T (years)

Semi-major axis a (AU)

T 2

a3

T 2/a3

Mercury

0.2408

0.3871

0.0580

0.0580

1.0

Venus

0.6150

0.7233

0.3782

0.3784

0.999

Earth

1.0000

1.0000

1.00

1.00

1.0

Mars

1.8809

1.5237

3.54

3.60

0.983

Jupiter

11.86

5.2028

140.66

140.83

0.999

Saturn

29.46

9.588

867.9

881.4

0.985

Galileo Galilei (1564-1642)

When Galileo heard of the construction of an optical arrangement of lenses by Hans Lippershey which made distant things look large and close, he immediately set about making

his own telescopes. In a letter dated August 29, 1609, about this new invention, Galileo wrote: 'I undertook to think about its fabrication, which I finally found, and so perfectly that one which I made far surpassed the Flemish one.

He stepped up the magnifying power of his telescope from about 6 to 30, and used it to make astronomical observations, which he described in his book Message from the Stars, published in 1610.

Using his telescope, Galileo was able to see mountains and craters on the moon, and was even able to estimate the heights of the mountains from their shadows; the stars appeared brighter and were much more numerous but still appeared to be point objects. (Even the most powerful modern telescopes cannot magnify the images of most stars beyond anything more than point objects.) In contrast to the stars, the planets were seen as bright discs, and Galileo was able to distinguish the 'sunny side' of Venus, and show that the planet exhibits phases, just like the moon.

This latter observation he described thus: 'I discovered that Venus sometimes has a crescent shape just as the moon does. This discovery may seem inconsequential, but in fact was critically inconsistent with the geocentric model of the universe. In this model Venus was always between us and the supposed orbit of the sun. Accordingly, we should never be able to see its entire illuminated surface. On the other hand, the observed phases of Venus were in good agreement with the Copernican model, in which Venus orbits the sun in the same manner as the moon orbits the earth. Galileo announced his conclusion in cryptic fashion, which literally translated states:

'The Mother of Love (Venus) imitates the phases of Cynthia (the Moon).'

The moons of Jupiter

Perhaps the most dramatic of Galileo's discoveries was that four moons orbit Jupiter, just as our moon orbits the earth. He was

Table 4.4 The moons of Jupiter (modern measurements).

Sidereal

Mean dist.

period

a from

T2

Moon

T (years)

T 2

Jupiter (AU)

a3

a3

Io

4.84 x 10-3

23.43 x 10-6

2.818 x 10-3

2.238 x 10-8

1.047 x 103

Europa

9.72 x 10-3

94.48 x 10-6

4.485 x 10-3

9.028 x 10-8

1.047 x 103

Gannymede 19.59 x 10-3

38.38 x 10-5

7.152 x 10-3

3.658 x 10-7

1.049 x 103

Callisto

45.69 x 10-3

20.87 x 10-4

1.257 x 10-2

1.986 x 10-6

1.051 x 103

able to make an estimate of the relative sizes of their orbital radii and orbital times. When he communicated his results to Kepler, the latter immediately realised that his third law also applies to Jupiter's moons, although the proportionality constant is different. His mysterious law seemed to apply generally, right across the Universe! Galileo's measurements were not very accurate, but modern observations have confirmed this remarkable result, as shown in Table 4.4.

The same units of length and time have been used in Tables 4.3 and 4.4 to facilitate comparison.

Galileo was now more convinced than ever that the Copernican theory must be right. Here was another heavenly body, Jupiter, around which things were in orbit. The earth was not the only centre of a planetery system. Our moon was in orbit around us, but that was all. The sun, and the rest of the universe, did not revolve around us!

Checking numbers

Edward Barnard discovered the fifth moon of Jupiter in 1892. Its sidereal period is 0.498 days and its orbital radius is 181,300 km.

Let us check whether this moon also obeys Kepler's law.

181,300 km = 1.2120 x 10-3 AU a3 = 1.780 x 10-9 0.498 days = 1.3634 x 10-3 years T2 = 1.859 x 10-6

The value of T2/a3 is 1.044 x 103, which is in very good agreement with the values shown in Table 4.4.

A total of 13 moons are now known to orbit Jupiter. The 8 moons discovered since Galileo's time are much smaller than the Galilean moons, but every single one of them obeys Kepler's law. Io is 3600 km in diameter and orbits Jupiter every 42 hours at a mean distance of 420,000 km from the centre of the planet. A close-up picture of Jupiter's moon Io, taken from the spacecraft Galileo, shows Io casting its shadow on the surface of Jupiter as it orbits the massive planet.

4.7 The solar system in perspective

In comparison to the dimensions of the earth, the dimensions of the solar system are vast. The sun has a mass that is 300,000 times the mass of the earth, and the furthest known planet, Pluto, is about 40 times further from the sun than the earth. Still, in the perspective of our entire galaxy, this is very much a 'local neighbourhood'. Our local galaxy, the Milky Way, contains many billions of stars, some of them much bigger than the sun. Some stars have planets but, with present techniques, we have no way of seeing them and can only infer their presence indirectly (see Chapter 7). The star Proxima Centauri, our nearest stellar neighbour, is about 4 light years away, and it may also have planets too small to be detected, even indirectly, with current techniques.

The Milky Way belongs to a class of galaxies which are described as spiral, with a shape roughly that of a disc with spiral arms. The diameter of the disc is about 100,000 light years

Transit of Io across Jupiter. Courtesy of NASA IJPLI University of Arizona.

and the sun is situated ■ about 30,000 light years from the centre of the galaxy, in one of the spiral arms. The mass of the galaxy has been estimated to be 150 billion times the mass of the sun. NGC7331. Courtesy of NASA/JPL-Caltech. (Compare that to the diameter of our solar system, which is about 10 light hours, or the diameter of the earth's orbit, which is 16 light minutes.) The image of NGC7331 (often referred to as the 'twin' of the Milky Way) shows how our galaxy might look to an extra-galactic viewer millions of light years away.

On a clear night, the Milky Way may be seen as a broad band of stars across the sky.

The Milky Way as seen from Death Valley, California. Courtesy of Dan Driscoe, US National Park Service.

According to Albert Einstein, no information can travel faster than the speed of light. The news of the rise and fall of the Roman Empire has travelled barely 1/50 th of the way across our galaxy. If anyone is observing us from outside the Milky Way, they see the earth as it was before the arrival of Homo sapiens!

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