FEATURE: Kepler’s laws

Kepler’s laws

 

I measured the skies, now the shadows I measure, sky bound was the mind, earth-bound the body rests

Kepler’s epitaph

Johannes Kepler’s three laws of planetary motion are a cornerstone of modern physics they describe the elliptical paths taken by planets around the sun, the time it takes to complete one orbit and how distant planets more slowly than nearby ones. These laws are applied today in the detection of dark matter and can also be applied to planets orbiting distant stars. Kepler 1571-1630 grew up in Germany with his mother living at his grandfather’s inn.  He became interested in astronomy as a child and by the time he was ten he had recorded a comet and a lunar eclipse in his diary.  Kepler studied at the university of Tubingen and went on to teach mathematics at Graz.  Kepler thought that god had created the universe according to a mathematical plan.  His theory of cosmology was published in ‘the sacred mystery of the cosmos’ he later assisted Tycho Brahe at his observatory outside Prague inheriting his position as imperial mathematician in 1601.  There Kepler prepared horoscopes and analysed Tycho’s astronomical tables publishing his theories of non-circular orbits and the first and second laws of planetary motion in ‘New Astronomy’ the third law of planetary motion was published in ‘harmony of the worlds’

                Modern astronomy began in 1609 with the publication of his masterwork ‘Astronomia nova’ Kepler had derived equations to describe the orbits of the planets based on careful records of the motions of mars taken by Tycho Brahe a Danish astronomer and aristocrat for whom Kepler worked as an instrument builder.  Kepler’s measurements of mars where much more accurate than had been achieved before.

Kepler’s first law

Kepler’s first law states that planets trace out an elliptical path with the sun at one focus of the ellipse. Until this radical theory was put forward everyone believed the orbits of planets to be perfect circles, it was thought that nature loved perfection and abhorred deviation from it Kepler inherited this belief at first imagining that planets were arranged about the sun in a nested series of crystal spheres spaced according to mathematical ratios derived from polygons.  But Tycho’s data caused him to change his mind.  When viewed from earth Mars's speed seems to seems to vary considerably it also seems to take backward steps drawing out loops in the sky.  Before Kepler many had tried to explain away the retrograde movements by adding small extra circles known as epicycles to large circular orbits.  In fact these days the phrase adding epicycles is a byword for bad science.  Kepler spotted that an ellipse did a much better job of explain Mars’s motion and to realize that it is because we are viewing the solar system from a moving platform that the other planets seem to back track.

Kepler’s second law

In his second law Kepler details how quickly a planet moves around its orbit; as it progresses along its elliptical path, it sweeps out a segment of equal area in an equal time. The segment like a slice of pie is measured by drawing a line from the planet to the sun and again at a given period.  When the planet is close to the sun it moves quickly and it draws out a broad pie slice; when it is further from the sun it travels more slowly subtending a smaller angle in the same amount of time. But states Kepler’s second law, the area of this long thin pie slice is the same as that of the short fat one.  Kepler figured this out by noting how fast mars moved around its orbit

Nature uses as little as possible of anything

Johannes Kepler

Kepler’s third law

Kepler’s third law goes one step further and tells us how the orbital periods scale up for different sized ellipses at a range of distances from the sun.  It states that the squares of the orbital periods are proportional to the cube power of the longest axis of the elliptical orbit.  The larger the elliptical orbit, the slower the period of time taken to complete an orbit, So planets further from the sun orbit more slowly than nearby planets.  Mars takes nearly two Earth years to go around the Sun, Saturn 29 years and Neptune 165 years mercury circles the sun in just 80 earth days.  If Jupiter travelled at the same speed it would take 3.5 Earth years to complete an orbit when in fact it takes 12.

 

Modern man

Kepler’s laws have stood the test of time. They apply equally to anybody in orbit around another from comets asteroids and moons in our solar system to planets around other stars and even artificial satellites whizzing around the Earth.  Kepler succeeded in unifying the principles into geometric patterns of nature.  It took Newton to unify these laws into a universal theory of gravity

580 BC	Pythagoras suggests that planets orbit on the surface of perfect spheres
150	Ptolemy explains retrograde motions with epicycles
1543	Copernicus proposes that planets orbit the Sun
1576	Tycho Brahe maps planets positions
1609	Kepler publishes first and second laws
1619	Kepler publishes his third law
1687	Newton proposes theory of gravity
2009	Kepler satellite launched by NASA to find planets around distant stars

FEATURE: Heliocentrism

 Heliocentrism
 

Finally we shall place the sun himself at the centre of the universe

Nicolaus Copernicus

 

It is surely harmful to souls to make it a heresy to believe what is proved

Galileo Galilei

Since as far back as 270bc it has been posited that the sun is in the centre of the solar system.  The idea didn’t catch on until the 17^th century when a complete heliocentric theory emerged. At the time the church had a lot more influence and since the bible seems to point to a geocentric model that is what the prevailing theory was.  Heliocentrism shattered our world view mankind was no longer literally the centre of the universe.

                In antiquity models of the cosmos placed earth at the centre with everything radiating outward from there, the orbits executed perfect circles. Philosophers were attracted to the idea that nature favoured perfect geometries as more accurate measurements where made there clockwork mathematical descriptions increasingly failed to explain them.  It wasn’t until Kepler that these perfect circles where replaced by the more correct ellipses. The ancient Greeks believed that all the heavenly bodies were affixed to crystal spheres that spun about the Earth causing the stars pinned upon them or revealed through tiny holes within them to circle the north and south celestial poles each night.

                The idea that the heavens revolve about the sun rather than the Earth a heliocentric model, after the Greek word helios for sun, was suggested by ancient Greek philosophers as far back as Aristarchus in 270bc who conveyed such ideas in his writings.  After calculating the relative sizes of the earth and the sun Aristarchus realized the sun was much larger it made more sense to him that the smaller earth would move rather than the larger Sun.

Ptolemy

                In the second century Ptolemy used mathematics to predict the motions of the stars and Planets.  He did so reasonably well but; there where patens that his equations could not match.  Most puzzling of all is that the planets seemed to change direction. Ptolemy believed , like many of his contemporises, that the planets literally turned on giant wheels in the sky.  He invented an explanation for the retrograde motion by adding extra cogs to their orbits. Theses superimposed epicycles gave the appliance of the planets occasionally looping backward.

Copernicus

                Born in Torun Poland Copernicus trained as a cannon taking classes in law, medicine, astronomy and astrology he was fascinated by but critical of Ptolemy’s ideas about the order of the universe.  Copernicus worked out his own system where the earth and the other planets rotate around the sun he published this theory in his work ‘On The Revolutions of Heavenly Spheres’ published just two months before he died. Ideas like this were occasionally mooted over the centuries but not taken seriously.  So it was not until the 16^th century that the consequences of the sun central model were fully developed.

The established church and society in general favoured Ptolemy’s geocentric view.

Galileo

Galileo Galilei notoriously challenged the roman catholic church by championing heliocentrism.  He backed up his audacity with careful observations made through the newly developed telescope.  Galileo found evidence that the earth was not central at all, that Jupiter has moons orbiting it and that Venus has phases like the moon.  He published these discoveries in his book starry messenger having made the audacious clame that the suns apparent migration across the sky was simply down to the earth’s rotation he found himself summoned to Rome. Despite the fact that Galileo’s observations where backed up by Jesuit astronomers the church refused to accept galilos theory.  Stating that although it was appealing in its simplicity that the theory could not be taken literally. Although restricted Galileo remained certain that his Sun cantered explanation was true, asked by Pope Urban VIII to write a balanced account of both sides in a ’dialogue of the two world systems’ Galileo upset the pontiff by expressing a bias for his own view over that of the church.  He was summoned back to Rome and put on trial for breaking his ban and placed on house arrest for the rest of his life.

Kepler

Meanwhile German astronomer Johannes Kepler was also working on the mathematics of planetary motions. Kepler published his analysis of the path of Mars in his 1609 book Astronomia nova.  Kepler found that an ellipse rather than a circle gave a better description of the red planets orbit around the sun.  now considered to be a basic law of physics Kepler’s vision was advanced for its time and took a long time to be accepted, Galileo for one took no notice.

Gradual acceptance

Evidence of the fact of heliocentric theory accumulated over the Century’s.  Kepler’s mechanics of orbits influenced Newton’s theory of gravity, as further planets were discovered, the fact that they orbited the sun was obvious. Man’s place at the centre of things is no longer tenable

270bc	Ancient Greeks propose sun centred model
200	Ptolemy adds epicycles to explained retrograde motions
1543	Copernicus publishes his heliocentric model
1609	Galileo discovers Jupiter’s moons.  Kepler models orbits as ellipses
1633	Galileo put on trial for teaching heliocentrism