Computers and Mathematics

Today physicists and engineers have at their disposal two great tools: the computer and mathematics. By using the computer, a person who knows the physical laws governing the behavior of a particular device or a system can calculate the behavior of that device or system in particular cases even if he knows only a very little mathematics. Today the novice can obtain numerical results that lay beyond the reach of the most skilled mathematician in the days before the compu­ter. What are we to say of the value of mathematics in today's world? What of the person with a practical interest, the person who wants to use mathematics?

Today the user of mathematics, the physicist or the engineer, need know very little mathematics in order to get particular numerical answers. Perhaps, he can even dispense with the complicated sort of functions that have been used in connection with configurations of matter. But a very little mathematics can give the physicist or engineer that is harder to come by through the use of the computer. That thing is insight. The laws of conservation of mechanical energy and momen­tum can be simply derived from Newton's laws of motion. The laws are simple, their application is universal. There is no need for compu­ters, which can be reserved for more particular problems.

Electronic Components for Computers

The electronic digital computer is built primarily of electronic components, which are those devices whose operation is based on the phenomena of electronic and atomic action and the physical laws of electron movement. An electronic circuit is an interconnection of electronic components arranged to achieve a desired purpose or fun­ction.

During the past two decades, the computer has grown from a fledg­ling curiosity¹ to an important tool in our society. At the same time, electronic circuit developments have advanced rapidly; they have had a profound² effect on the computer. The computer has been signi­ficantly increased in reliability and speed of operation, and also redu­ced in size and cost. These four profound changes have been primarily the result of vastly improved electronic circuit technology.

Electronic vacuum tubes were used in the earliest computers. They were replaced by solid-state electronic devices³ toward the end of the 1950's. A solid-state component is a physical device whose ope­ration depends on the control of electric or magnetic phenomena in solids; for example, a transistor, crystal diode, or ferrite core⁴. Solid-state circuits brought about the reliability and flexibility required by the more demanding applications of computers in industry. Pro­bably the most important solid-state device used in computers is the semi-conductor, which is a solid-state element which contains proper­ties between those of metal or good conductor, and those of a poor conductor, such as an insulator. Perhaps the best-known semi-conduc­tor is the transistor.

The advances in electronic circuit technologies have resulted in changes of "orders of magnitude" where an order of magnitude is equal to a factor of ten.

The number of installed computers grew from 5000 in 1960 to appro­ximately 80,000 in 1970. Also, the number of circuits employed per computer installation has significantly increased. The first computers using solid-state devices employed 20,000 circuits. Today computers using transistors may contain more than 100,000circuits. The trend is likely to continue; it has been made possible by the continued decrease in size, power dissipation, cost, and improved reliability of solid-state circuits. Note that what was used in a "high performance" computer in 1965 became commonly used in 1968. The speed of the logic circuits is given in nanoseconds, 10~⁹ seconds.

Existing Satellite Communications Systems

In a satellite communications system, satellites in orbit provide links between terrestrial stations sending and receiving radio signals. An earth station transmits the signal to the satellite, which receives it, amplifies it and relays it to a receiving earth station. At the fre­quencies involved, radio waves are propagated in straight lines, so that in order to perform its linking and relay functions, the satellite, must be 'visible'— that is, above the horizon — at both the sending and receiving earth stations during the transmission of the message.

There are at present two different types of systems by which satelli­tes are so positioned: 'synchronous' and 'random orbit'. A satellite placed in orbit above the equator at an altitude of 22,300 miles (35,000 km) will orbit the Earth once every 24 hours. Since its speed is equal to that of the Earth's rotation, it will appear to hang¹ motion­less over a single spot on the Earth's surface. Such a satellite is called a synchronous satellite, and the orbit at 22,300 miles above the equa­tor is known as 'the geostationary orbit'. A synchronous satellite is continuously visible over about one-third of the Earth (excluding extreme northern and southern latitudes). Thus a system of three such satellites, properly positioned and linked, can provide coverage of the entire surface of the Earth, except for the arctic and antarctic regions.

A satellite in any orbit other than a synchronous orbit will be si­multaneously visible to any given pair of earth stations for only a por­tion of each day. In order to provide continuous communication bet-ween such stations, more than one satellite would be required, orbiting in such a way that when the first satellite disappeared over the horizon from one station, another had appeared and was visible to both sending and receiving earth stations. The number of such satellites required to provide continuous communication depends on the angle and alti-tude of the orbit. The number could be minimized if the spacing bet­ween the satellites were precisely controlled (controlled-orbit system), but a somewhat larger number with random spacings can achieve the same result (random-orbit system).

Since the synchronous satellite remains stationary with respect to any earth station, it is relatively simple to keep the antennas at the

sending and receiving stations properly pointed at the satellite. Only small corrections for orbital errors are required. In a random-orbit system, the earth-station antenna must track the satellite across the sky. Moreover, if continuous communication is to be maintained, a second antenna must be in readiness at each earth station to pick up the following satellite as the first one disappears over the horizon.

At present, there are operating systems of both main types. Intel­sat operates a synchronous system providing global coverage, with satellites positioned above the Atlantic, Pacific, and Indian Oceans. The Soviet Orbita system, used for space network domestic communica­tions (including television distribution) within the USSR, is a random-orbit system, using eight satellites and providing continuous 24-hour communications. The satellites are spaced so that two of them are al­ways over the Northern Hemisphere⁵; and the orbits are such that during the time when it is in operation, a satellite is at the apogee of its orbit. Its apparent motion with respect to the Earth's surface is slowest at this time and the tracking problem is minimized.

Hypersonic Transport

The fastest flights within the atmosphere have been made by a roc­ket craft that carried fuel for only a few short minutes of powered flight. For short experimental flights, that is fine; for longer trips and higher payloads within the atmosphere, other forms of propulsion, such as the ramjet, are necessary.

The ramjet lies somewhere between the jet and the rocket. The next step, supersonic ramjets may even play a role in sending payloads into space.

If fossil fuels must be phased out, hydrogen may become the fuel of the future. The light weight of hydrogen in combination with its high energy value gives it the highest energy density per pound of all fuels. A hypersonic, hydrogen-fueled aircraft has already been proposed. The supercold liquid hydrogen would be not only a good fuel, it would also provide an answer to the continuing problem in high-speed flight— frictional heating due to air resistance. Temperatures of leading edges can go up into thousands of degrees, possibly causing weakening of structural members. The liquid hydrogen, at —423° F, could be used to help cool these areas.

Someday in the future our supersonic planes may seem slow and old-fashioned. Plans are already on the drawing board for aircraft capable of a speed of 11,000 mph.

And why not? Materials technology is advancing rapidly and may provide substances able to stand up against the multithousand degree temperatures that will be encountered. Programs are already under way to develop a hypersonic test engine and to develop and test light­weight structures capable of operating at high speeds.

There is no theoretical reason why these or even higher speeds cannot be attained. Some aircraft experts predict we will see such craft in service by 1985 or 1990.

At 18,000 mph, however, the craft has reached orbital speed. And at 25,000 mph, the problem is no longer how to keep it up, but how to keep it down— that is, how to keep it from flying off into space.

Even at 11,000 mph — which can probably be attained without any revolutionary development, no two cities on earth would be more than three-quarters of an hour apart. How small would our world be then?