Dimensions beyond Our Imagination
Lim Ju Boo
Synopsis:
Most scientists as well as ordinary people-in-the-street are only familiar with small numbers, common scales and measurements which they encountered in their daily lives. Most of us are only used to small measurements like a few meters, kilometers, kilograms, watts, joules and calories, cubic meters, and square meters.
In this talk, we shall deal with distances, mass, energy, speed and other dimensions well outside the tethers of our imaginations. This talk centres on the applications of simple mathematics in astronomy to give us some beautiful conclusions that is amazingly outside our understanding. We will take a ride outside this infinitesimal world of ours into the realms of the unimaginable.
The talk shall be well illustrated by lots of coloured slides. However, because of the large amount of information collected, and the limited amount of time available, it would not be possible to include all the information in the talk. Hence these short notes below shall complement the rest of the slides in the talk.
The Horrendous Scales
Speed of light = 299 792 458 ms -1
1 light year is approximately = 9.46 x 10 12 km = 9.46 x 10 15 meters = 9.46 x 10 18 mm (based on average year). This is the distance light takes to travel in a year
1 Astronomical Unit (AU) = 149,597,870.691 kilometers = 1.495978706 x 10 11 meters = 1.495978706 x 10 14 mm. This is 7.59 times the distance from Earth to the Sun
It takes light only 499 seconds (8.3167 minutes) to cross this distance
One light year = 63,239.7 astronomical units
The nearest known quasar is 2S 0241 + 622 discovered in 1978 is 800 million light years away
The number of sheets of papers needed to represent the distance to the nearest quasar = 800 million sheets, each sheet is 1 mm thick. Each 1 mm is equivalent to 1 light year
This means the nearest quasar 800 million light years away has to be represented by a column of papers stretching 800 km high, each sheet 1 mm thick representing a light year of 9.46 x 10 12 km.
The most distant quasar is OQ 172, discovered in 1973 by Lick Observatory. It lies 18000 million light years (1.7 x 10 23 km or 1.136 x 10 15 AU) away. If a light year of nearly 10 million, million km is drastically scaled down to just a single sheet of paper 1 mm thick, then the columns of papers has to pile up till it stretches out to 18,000 km into outer space. This is 7.59 times the distance from Earth to the Sun. Remember each 1 mm sheet is 1 light year across.
But if a single sheet represents 1 AU, then the column to quasar OQ 172 shall be more than 1136, 000,000 km long. If a plane could cross all that distance to OQ 172 at a constant speed of 1,000 kph, it will take the plane 1.94 x 10 16 years (nearly 20,000 million, million years) to arrive to the furthest quasar
The average Sun-Pluto distance is 39.4 AU = 5,894,156,102 km
But we scale down the Sun-Pluto distance as 1, then Quasar OQ 172 will stretch to 2.89 x 10 13 (28.9 million, million) times the Sun-Pluto distance
Total quantity of matter in the Universe
The amount of matter in the Universe is equal to the mass of Sun x total no. of suns in the Universe
= (2 x 10 33) x 10 x 22 = 2 x10 55 gm = 20 septendecillion gm
Estimated no. of particles (nucleons & electrons) in the Universe = 2.2 x 10 79 = 22 quinvigintillion
Size of atomic particles
Effective radius of an atom 10 -10 metres (10 -7 mm)
Effective radius of nucleus = 10 -14 metres (1 x 10 -11 mm)
Effective radius of an electron = 10 -15 metres (10 -12 mm)
Thus if an atomic nucleus is enlarged to the size of a Malaysian 10 cents coin 2 cm in diameter, the effective radius of the electron will be
2 cm ÷ (10 -14) x (10 -15) = 0.2 cm
The infinitesimal to the infinite:
Since the speed of light = 299 792 458 ms -1
Light will take only 6.67128 x 10 -18 of a second to cross the diameter of an atom
‘Resonance particles’ have a life span of 10 -23 of a second (ten trillionth of a trillionth of a second)
Resonance particle exists only sufficiently long enough for light to cross 2.99 x 10 -15 metres
This means it can cross only 1.495 x 10 -3 = 0.0015 % the diameter of an average atom in the life span of a resonance particle
But for a hydrogen atom, the time light takes to cross its diameter is 4.503 x 10 -19 second = 0.45 of an attosecond = 0.45 quintillionth of a second (1 attosecond = 10 -18 second)
Light takes 8.34 x 10 -24 seconds to cross the diameter of a proton or neutron, each roughly 2.5 x 10 -15 meters in diameter, or 2.5 quadrillionth of a meter.
The largest naturally occurring nucleus is that of uranium 238 which has 92 protons and 146 neutrons. Its diameter is 1.55 x 10 -14 metres so that it is 6.2 times as wide as a proton.
In any atom, the nucleus has about 1 / 100,000 the diameter of the atom of which it forms a part
If the atom were a hollow sphere, one could fill it with about 10 15 (1 quadrillion) nuclei.
The diameter of a hydrogen atom is 1.35 x 10 -10 meters or a little over a ten-billionth of a metre. More massive atoms are a little bit larger, but even the most massive atoms known is probably not wider than 8 x 10 -10 meters in diameter
This means that 405 hydrogen atoms can be squeezed into a cubic nanometer, and that the spindle tuber disease virus, the smallest known fragment of life can contain perhaps 75,000 atoms The spindle tuber disease in potatoes virus has a mass of 8 x10 -22 kg (8 x 10 -19 gram)
Energy Values:
The energy value of petrol = 35 048 096.18 Joules (8 374 853.25 calories) per litre.
1 Imperial gallon = 4.54609188 litres. One calorie is approx. equal to 4.18 Joules
Human daily food requirements is about 2800 kilocalories (1 kilocalorie = 1,000 calorie)
An average car consumes about 1 Imperial gallon of petrol every 35 miles (12.39 km / litre). This is 38,072,852 calories (159,144,523 Joules)
This is the energy equivalent to 13.59 times the daily food requirements of an adult
But the Sun pours out 3.83 x 10 26 watts of energy through hydrogen fusion into helium. This is equivalent to 3.83 x 10 26 J sec -1 or 3.83 x 10 33 ergs / sec (nearly 4000 nonillion ergs sec -1). 1 Joule = 107 ergs
This is the energy value of 1.092 x 10 19 litres of petrol. Hence if we could put in all the energy the Sun releases in just one second into our petrol tank instead of using petrol, then it could push a car round and round, and round this Earth up to (1.092 x 10 19) ÷ 12.39 = 8.8 x 17 (880 quardrillion) km. This is 2.2 million, million times round the Earth since the equatorial circumference round the Earth is 2πr = 2 π x 6378.137 km = 40,075 km
Mass & Speed:
If light could bend round the curvature of Earth, it could whip round the equatorial circumference 7.48 times in one second
The mass of the Sun is 1.9891 x 10 30 kg (nearly 2 nonillion kg)
Mass of Earth is 5.974 x 10 24 kg (nearly 6000 sextillion kg)
Hence the mass of the Sun is 332 960 times as massive as Earth
The diameter of a human hair ranges from 1.27 x 10 -4 m to 2.54 x 10 -5 m (127 -25.4 microns)
A car speeding at 100 kph (27.78 m sec-1) crosses the diameter of a human hair 75 microns (0.075 mm) in just 2.7 x 10 -6 second.
This is the time sufficient for even light to travel only 809.439 metres. Light is 10 792 528.49 times faster than a 100 kph car.
But light takes only 2.5 x 10 -13 second (0.25 of a picosecond) to cross the breath of a human hair 75 micron in diameter
Hydrogen atoms permeate the entire diameter of the Universe. The diameter across the width of the hydrogen atom is taken as the smallest unit reference of length in the entire universe since the hydrogen atom is the symbol of the commonest element in the entire Universe.
Size of atomic particles:
Effective radius of a hydrogen atom is 10 -10 metres (10 -7 mm)
Effective radius of nucleus = 10 -14 metres (1 x 10 -11 mm)
Effective radius of an electron = 2.81794 x 10 -15 metres (2.81794 x 10 -12 mm)
Effective volume of an electron = 9.373 x 10 – 44 m-3
Effective volume of an atomic nucleus = 4.189 x 10 -42 cubic metres
Hence the volume of an average atomic nucleus is 44.7 times that of the electron
Radius of Sun: 695 500 000 m
Volume of Sun = 4/3 π r 3 = 1.409 x 10 27 cubic metre
If the volume of an average atomic nucleus were to be enlarged to the size of the Sun, then the volume of its electron shall be 3.152675 x 10 25 cubic metres. (3.15 x 10 16 cubic km).
The radius of Jupiter, the largest planet in the Solar System, is 71,492,000 meters (71, 492 kilometers). Its volume is 1.53 x 10 24 cubic metres (1.53 x 10 15 cubic km). It is placed over a mean distance of 778,330,000 kilometers away. The electron shall be 20.6 times bigger than Jupiter
Earth mean distance from the Sun is 149,600,000 km. This means an electron is 5.2 as far away its nucleus as Earth is from the Sun. The volume of Sun is 921 times that of Jupiter.
But in the case of an average atom, if the nucleus were to be magnified to the volume of the Sun, its nucleus would be about 45 times larger than its electron, and it would be placed 6.96 x 10 12 metres (6,960,000,000 km) away from its nucleus at ground state. Pluto’s average distance from the Sun is 5, 913,520, 000,000 metres. This is 7.59 times farther away than Jupiter is. But our magnified electron would slightly further than the Sun-Pluto’s distance by 1.18 times.
Thus if an atomic nucleus is enlarged to the size of a Malaysian 10 cents (2 cm) coin, the distance of an orbiting electron at ground state will be = 2 /100 metres ÷ (10 -14) x (10 -15) = 200 cm (2 metres) away
The infinitesimal to the infinite
Since the speed of light = 299 792 458 ms -1
Light takes only 6.67128 x 10 -18 of a second to cross the diameter of an atom
‘Resonance particles’ have a life span of 10 -23 of a second (ten trillionth of a trillionth of a second)
Resonance particle exists only sufficiently long enough for light to cross 2.99 x 10 -15 metres
This means it can cross only 1.495 x 10 -3 = 0.0015 % the diameter of an average atom in the life span of a resonance particle
But for a hydrogen atom, the time light takes to cross its diameter is 4.503 x 10 -19 second = 0.45 of an attosecond = 0.45 quintillionth of a second (1 attosecond = 10 -18 second)
Light takes 8.34 x 10 -24 seconds to cross the diameter of a proton or neutron, each roughly 2.5 x 10 -15 meters in diameter, or 2.5 quadrillionth of a meter.
The largest naturally occurring nucleus is that of uranium 238 which has 92 protons and 146 neutrons. Its diameter is 1.55 x 10 -14 metres so that it is 6.2 times as wide as a proton.
In any atom, the nucleus has about 1 / 100,000 the diameter of the atom of which it forms a part
If the atom were a hollow sphere, one could fill it with about 10 15 (1 quadrillion) nuclei.
The diameter of a hydrogen atom is 1.35 x 10 -10 meters or a little over a ten-billionth of a metre. More massive atoms are a little bit larger, but even the most massive atoms known is probably not wider than 8 x 10 -10 meters in diameter
This means that 405 hydrogen atoms can be squeezed into a cubic nanometer, and that the spindle tuber disease virus, the smallest known fragment of life can contain perhaps 75,000 atoms The spindle tuber disease in potatoes virus has a mass of 8 x10 -22 kg (8 x 10 -19 gram)
The Atom and the Solar System
But in most cases, the nucleus of an average atom has a diameter of about 10 -15 meter, whereas the atomic diameter is about 10-11 meter. This can vary according the quantum of energy supplied to exit the atom. The electron can jump obit to a higher or lower level unlike the planets round the Sun according to how excited it is by the energy (heat or light) pumped into the atoms. Generally, the nucleus has a diameter 10,000 times smaller than the atom. The great amount of empty space in an atom can be illustrated by the following analogy.
An Analogy
Imagine the nucleus to be the size of a golf ball. Then on this scale the first electron shell would be about one kilometer from the golf ball, the second shell about four kilometers, the third nine kilometers and so on. If you find that hard to visualize let’s try another method. The full stop at the end of this sentence probably is about 1/2 a millimeter in diameter. If that dot represents the nucleus, then the electrons in the first shell would be orbiting with a diameter anything from a meter to 50 meters around the nucleus.
This distance varies from shell to shell, meaning the radius of the electronic orbit varies according to the amount of energy it has. Their orbits are similar to all the 9 planets of our Solar System except that the planets of our Solar System do not jump from orbit to orbit. The planets’ distances from the Sun change at different times of the year only because of their eccentricity as they execute an elliptical path round the Sun. Their orbital pathways are always the same.
In fact, the actual diameter of an atom is very small and it would require some two hundred million of them side by side to form a line a centimeter long.
Let’s look again at some data:
• The diameter of a single proton is 10-15 metres
• The diameter of a hydrogen atom is 10-10 metres
• The diameter of the universe by the latest estimates is at least 30 or 40 billion light years in diameter.
This means that light traveling at a velocity of 299 792 458 metres per sec that will take only:
• 3.34 x 10-18 seconds (3.34 quintillionth of a second) to cross the diameter of a hydrogen atom at ground state.
• 3.34 x 10-23 seconds (33.4 septillionth of one second) to cross the diameter of a single proton which is one of the smallest atomic particles.
• But will take 4 x 1010 (40 billion) years to cross from one end of the universe to the other
• Thus the Universe is 3.78 x 1023 km x 1000 ÷ 10-10 metres = 3.78 x 10 36
(3.78 undecillion or 3780 decillion) times the width of the hydrogen atom
The Ultimate of Time:
Probably the shortest time possible, is to use the diameter across the width of a hydrogen atom as the smallest distance, and the speed of light in a vacuum as the fastest of all speeds as reference yardsticks to measure space & time. From the above light will take 3.34 x 10-18 seconds (3.34 quintillionth of a second) to cross the diameter of a single hydrogen atom.
The Ultimate of Temperature:
In similar light, the ultimate of all theoretical possible temperatures can only be:
V = 0.158 √ (T ÷ m).
By using a series of equations, it can be shown that the above equation may also be expressed as:
T = 40 mv2
The factor 40 only holds if we use units of temperature in (degrees) Kelvin, and velocity in km / sec.
where, V = velocity of any atomic particles in km / sec (molecular / atomic velocity)
T = temperature in degrees Kelvin
M = mass of particles at rest
Let us set the value of ‘v’ (velocity of the molecules as a gas gets heated up) as the maximum possible speed of 299, 792 km / sec. – the speed of light. Applying this value into the equations, we will get what seems the maximum possible temperature.
When the temperature reaches higher and higher, the atomic particles move faster and faster until it reaches the velocity of light. Since nothing can travel faster than light, then the highest possible theoretical temperature can only be:
T max = 3.59 x 1012 = 3,600,000,000,000 (degrees) Kelvin (3.6 million, million K)
At last! It is not as simple as that! At searing temperatures of millions of degrees all molecules and atoms break down into mere particles. Fusion reactions between simple nuclei are possible so that complicated nuclei can be created. At even still higher temperatures (if this is theoretical possible), then the reverse is true, and all nuclei must break down into even simpler particles. This gives rise to further complications between creation / destruction of matter, temperature, pressure, dimension and time.
Mass Increase with Velocity
But by then, as the velocity increase to near that of light, the mass of the particles will also increase according to Fitzgerald-Lorentz contraction derived from Einstein Theory
M = m 0 / m ÷ [√ (1- v2 / c2)]
Where M= final mass, m 0 is the initial rest mass, v2 = velocity of the particle at rest, c2 = the velocity of light.
As the particles go faster and faster their mass get more and more massive, until their mass reach infinity at the speed of light. So when v = c, the equation tells us the final mass (M) becomes infinite - exceeds that of the mass of the Universe itself. That is not possible. This means that even the temperature of 3.6 million, million Kelvin can never be reached.
A Journey to the Edge of the Universe:
Let us imagine we have a plane than can drift through the immense intergalactic valley, end to end, an immense abyss of space spanning 40,000 million light years, or 3.78 x 1023 km (378 sextillion km) across.
For accomplish that the plane theoretically has to fly non-stop at a constant speed of 1000 kph almost 48000 million, million Earth years. Fancy that! I salute the super-pilot who can live and endure that kind of journey. To solve that, he may have to marry abroad, bear children over 1.6 x 1015 (1600 million, million) generations to take over the piloting once each child attains the age of 30 years. The span of 30 years is taken as one generation.
Better Idea:
This is not possible. So, I have a better idea. It is possible for the pilot’s sperm and his wife’s eggs be frozen in liquid nitrogen as they are left to drift into the frigid coldness and darkness of space where the temperature is almost 0 Kelvin. The pilot may remain back on Earth, or he may come along, but kept in suspended animation to follow his genes aboard. A robot is programmed to take over which will only be activated towards the end of the journey.
The awaken robot will then take out the sperm-ovum from deep freeze, fertilize them in oxygen-rich nutrient broth. The robot will then nurse the foetus-baby and bring him (them) up. The robot-nurse will teach the child the world and galaxy from where he came. It will then t each the human child his language, culture and civilization, and shown images of the world of his root.
It will tell and teach him or her, the purpose of his mission, their destination, his fate and destiny. The child’s tear-filled eyes will be shown pictures and images of his biological parents, their world, humans civilizations on Earth, its plants, trees, forest, animals and all other living things from the world they came. All the images of his world will be beamed towards the plane from the day he plane left Earth and the Solar System into the frigid reaches of extra-galactic dimensions
All scenes of Earth will be continuous in time frame such that the entire length of 40,000 million years of history from the beginning to the edge of this Universe shall be continuously shown.
The TV transmissions will not be broadcast the usual way. The present technology will ‘dilute’ the energy of the transmission over a wider and wider volume into the emptiness of space. The entire energy of the signals will have to be concentrated into just a very narrow beam towards the direction of the plane in order to focus the pictures clearly on arrival without being spread, diluted and weaken out
The transmission should be continuous so that there is no gap in time in receiving the images. Even then, towards the end of the journey, all the images would have been at least 40,000 millions years old for the child. He will perhaps never be able to learn the ‘current’ scenario of this Earth, except his origin, in a distant galactic world that has long faded into oblivion over the immensity of time and space.
Even for light this is a horrendously long journey. It will take even light traveling at 299792458 metres per second ÷ 1000 (to change into km / sec) x 60 sec (to change to km per min) x 60 min (to change to km per hour) x 24 hr in a day x 365.25 days in a year x 40,000, 000,000 solar years = 3.78 x 1023 km (378 sextillion km) to span this chasm.
In order to cross this chasm, light will have to take at least 4 x 1010 (40 billion) years to transverse from one end of the Universe to the other end. This is a horrendously bizarre distance.
We can only wish our very distant generation the very best of luck for their survival.
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