Fibonacci numbers in the sky
In addition to his fame as an astronomer Johannes Kepler was a talented mathematician. He contributed many insights into the sequence of fibonacci numbers that have intrigued mathematicians, artists, musicians, architects, naturalists and others for centuries. An intensely religious man, he was devoted to understanding and worshipping God’s creation. He noticed early on that the Fibonacci numbers represented what he called the seminal faculty, fundamental to the biological process of self-replication. He examined the structure of flowers and found in them an emanation of sense of form and feeling for beauty from the soul of the plant. His work encouraged biologists to revive the art of phyllotaxis, the study of leaf arrangements in plants. Many other examples of the Golden ratio’s role in the natural world have been discovered: too many to list here.
Kepler went on to devise his three laws of planetary motion and his third law is a direct application of the Golden Ratio. Fibonacci numbers are not about rabbits but about the measurement of time, and that was Kepler’s specialty. In his book The harmonies of the World he illustrated the orbits of the planets on a music staff. Mercury sings Soprano, Earth Alto, Mars Tenor, Jupiter and Saturn Bass. He does not suggest this music will be heard in Space but the mystery of numbers is part of God’s great plan. Mathematics is silent music and Fibonacci numbers are the orchestra.
The mystery of numbers is profound. The word ‘harmony‘ indicates the number 2. The Pythagoreans explored harmony through music and found the most pleasing sounds were created by two tones one octave apart, meaning one tone was twice the pitch of the other. Experimenting with stringed instruments they found a relationship between the length of a taut string and the pitch of the sound it created. The number 2 is the common property possessed of an octave of music and a pair of shoes, which is a mystery in itself. How would you define the connection between an octave of music and a pair of shoes? Hence, how would you define any number?
What is a Number? Does anyone know?
Claudius Ptolemy, the famous Alexandrian astronomer, also took an interest in the Golden Ratio, which he illustrated in Book 1 of the Almagest. It features the defining property of the Golden ratio, namely a right-angled triangle with hypotenuse square root of 5. Phi The circumference of the triangle is 3+√5 and its area is ½. The triangle is the path of the knight on a chess board: 2 steps forward and 1 to the side. The angles of the triangle are 63.435 degrees (arctan 2) and 26.56505 degrees (arctan ½)The numbers can be expressed not only in degrees but also in hyperbolic measure and this is a feature of all fibonacci numbers. Tan 26.565° = sinh lnφ. =½ sec 6349° =√5=cosh3lnφ.
The Molniya Orbit.
Kepler’s laws of planetary motion are working overtime these days. Our sky is cluttered with various forms of space vehicles, so numerous that collisions have been known to occur. The junk is piling up like the plastic junk in the oceans. Out of sight out of mind in both cases. The shape of any particular orbit depends upon the obliquity of the orbital plane relative to the equator and on the eccentricity of the ellipse. Artificial satellites provide many different services. Most scientific satellites and many weather satellites follow a near-circular low Earth orbit in a narrow band over the equator. They gather information on such matters as tropical rain forests and deforestation.
The semi-synchronous orbit is nearly circular with high obliquity at an altitude about 4 times the Earth’s radius. It is the orbit used by the GPS and takes 12 hours. It crosses over the same two spots on the equator every 24 hours.
At an altitude of about 6 times the Earth’s radius the orbit matches the Earth’s rotation and the satellite hovers over the same latitude but may drift north and south. This is called a geosynchronous orbit. A geostationary satellite has both eccentricity and obliquity zero. It remains stationary over a fixed point on the equator. Geostationary satellites are used for communications by phone, television or radio. They also act as search and rescue beacons for sailors in distress.
Geostationary satellites are difficult to access in high latitudes. Northern parts of Russia experienced interruptions until they devised the Molniya (meaning lightning) orbit. A satellite in a highly eccentric orbit spends most of its time in the neighbourhood of apogee, which is the point where the Earth is farthest from the Sun. In the northern hemisphere the sub-satellite point is in latitude 63.435 degrees. That sounds familiar. That is one of the angles in the triangle two steps forward and one to the side described by Ptolemy – arctan2. The satellite over apogee is at an altitude of about 40,000 kilometres giving it a wide view over large swathes of the northern hemisphere, thus improving communications. To gain continuous high elevation of the northern hemisphere, at least three Molniya spacecraft are needed.
While the northern hemisphere is at apogee the southern hemisphere is at perigee =closest approach to the Sun, and the numbers are reversed. Australia experiences catastrophic bush fires in summer while the northern hemisphere experiences catastrophic melt-down of the northern ice cap: the two extremes of global warming.
The obliquity of a satellite’s orbit is a function of phi. The obliquity of the Molniya orbit is sec 63.435 degrees = sq rt of 5, which is the same as cosh 3 ln phi. For every numeric value of the obliquity there is a hyperbolic function in terms of the Golden Ratio. Square root of five is the hypotenuse of the Fibonacci Time triangle, which is not the same as the so-called Golden Triangle. It is a fundamental player in the Golden Spiral known to navigators as a loxodrome. Furthermore, any Fibonacci number, φ^n, is given by 103682 ^n/24. 103682 is Phi raised to the power 24. It can be regarded as the limit of time measured as 24 hours in a day. It is the twelfth of never. Phi is an exponential clock. We can measure time more accurately and conveniently with logarithms to base phi. The ancient Babylonians were clever to invent 24 hours in a day but it’s time to move on. Buried in there also is the sailors’ rule of twelfths. We have not heard the last of that yet. Pi/12 = 15 degrees = 1 hour of longitude = 1 time zone. Fibonacci numbers are deeply involved with climate change.