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| Fig. 1: The Apollo 11 Saturn V rocket climbing towards space. (Source: Wikimedia Commons) |
The storied relations between the United States and the now-defunct Soviet Union towards the latter half of the 20th century were strongly characterized by the famous Space Race, towards which both countries committed heavy economic and scientific investment. Both superpowers announced their intentions to further the possibility of spaceflight in 1955, signalling the start of a fierce competition in human determination and technological prowess that led to great advances in the field of rocket technology. The Soviet Union's initial successes included Sputnik 1, the first artificial satellite to orbit the Earth, and Vostok 1, the first manned mission in space. On the other hand, barely a decade and a half after the beginning of the Space Race, the United States succeeded in placing two men on the Moon, and, just as incredibly, returning them home safely afterwards.
It is without question that the legacy of the space race has been a newfound and continuing advocacy for the exploration of space, resulting in major advances in the fields of astronautical engineering and astronomy. The casual observer, who perhaps eagerly pressed against the security cordon as Yuri Gagarin, the first man in space, was paraded through the streets of Moscow, or watched with bated breath as Neil Armstrong set foot on the grey desolation of the Moon, would find it astounding to think that, barely half a century later, humanity has active and serious research into the colonization of other planets.
Gravity is a familiar force. When we jump, we fall back down, and no matter how much effort we put into it, we only rise a few feet before the gargantuan mass of the Earth beckons us back. How then, does a rocket manage to escape this powerful attraction?
A rocket gains its propulsive force by means of burning fuel and expelling exhaust. Fig. 1 shows the rocket of Apollo 11, which placed Armstrong and Aldrin on the Moon. The pillar of flame and smoke that trails it is not merely fire - it is a glowing cloud of hot gas, being blown out of the tail end (the nozzle) of the rocket in the opposite direction to which the rocket travels. The oft-quoted version of Netwon's Third Law states that to every action there is an equal and opposite reaction: consequently, as engine of the rocket combusts fuel and accelerates it out of the nozzle, the engine (and the rocket which it is affixed to) gains a thrust force that pushes it in the other direction.
The specifics of this interaction are convoluted by the fact that since the rocket is continually throwing out combusted propellant as gas, its mass is decreasing as it accelerates. The velocity v of the rocket is given by the Tsiolkovsky rocket equation, derived in 1903 [1]
| v | = | v0 × ln (M /M0) |
where v0 is the effective exhaust velocity, M is the total mass of the rocket with fuel, and M0 is the total mass of the rocket without fuel. While this basic version of the rocket equation exempts the complications of gravity and drag, among other forces, it demonstrates that to maximize the total change in velocity of the rocket, the percentage of fuel as a fraction of the total mass of the entire rocket must be high (which makes perfect sense: something with more fuel should be able to go faster and further than something with less!). Throughout the history of human spaceflight, this ratio, known as the propellant mass fraction, has hovered between 0.85 and 0.95 for rockets. [2] That is to say, between 85% and 95% of the mass of a pre-launch rocket consists of fuel. In comparison, the 2016 model of the Ford Fiesta weighs approximately 1200 kg and has a 12.4 gallon petrol tank, which corresponds to 35 kg of petrol and hence a propellant mass fraction of only 0.03. Such a disparity is primarily blamed on the strength of the Earth's gravitational well - a huge amount of energy is required to move an object from surface to orbit, and that amount of energy scales with the mass of the object.
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| Fig. 2: A basic diagram of a space elevator. (Source: Wikimedia Commons) |
The unfortunate downside to this reality is such: because the majority of volume and mass of a rocket is dedicated to fuel, and a good portion of the remainder is the actual structure of the rocket, the remaining mass - consisting of astronauts and their equipment - comes at a premium. Today, NASA estimates that the cost to deliver a pound of mass to orbit is approximately $10,000.
As materials get cheaper to devlop and technologies get more efficient, one might expect that it become more and more economical to develop rockets. Elon Musk, the founder of the private space transport services company SpaceX, reported that in the development of the Falcon series of launch systems, the dollar cost of sending a pound of mass to orbit decreased from $4,000 for Falcon I to $1,300 for Falcon V. [3]
"But why limit ourselves to rockets?" some might ask, and one will find that in a bid to avoid the rocket equation entirely, many technologies have emerged as budding alternatives to chemical rockets. Konstantin Tsiolkovsky, the rocket scientist who formulated the rocket equation, also conceptualized the space elevator: a long cable, or tether, that attaches a station in orbit around the Earth to a fixed point on the Earth itself, with cargo being able to be shuttled up and down its length. A basic visualization is shown in Fig. 2. While the image of a seemingly never-ending tower stretching up into the sky may seem like an impossible thing of magic, the mechanics involved are no different to how satellites orbit the Earth.
This particular dream of a direct physical pathway to space has been the topic of serious research for decades, but many problems have quickly presented themselves. The foremost problem is that of materials. A cable used as a space elevator must be geostationary - meaning that it rotates in synchrony with the Earth exactly, appearing as a very tall but very still tower to someone at its base. For such a cable, the point of greatest tensile stress is found at geostationary height Rg, and the equation that calculates said stress, accounting for both gravity and the inertial centrifugal force, is given by [4]
| σ(Rg) | = | GMρ × [ 1/R - 3/(2Rg) + R2/(2Rg3) ] |
where σ(Rg) is the tensile stress at Rg, G is Newton's gravitational constant, M and R are the mass and radius of the Earth, and ρ is the density of the material of the cable. We can see that to satisfy this equation, the material that the cable is made of must have sufficient tensile strength to match its density - a good way to think about this is that the heavier the material of the cable, the more that gravity tugs at it, and thus the stronger the material must be to support its own weight. Denoting the rotational angular frequency of the Earth by ω and the acceleration due to gravity at the earth's surface by g = GM/R2, we have
| Rg / R | = | [ | g/ω2R | ]1/3 | = | [ | 9.8 m sec-2 6.37 × 106 m |
× ( | 3600 sec h-1 ×
24 h 2 π |
)2 | ]1/3 | = | 6.63 |
Thus for a cable made of steel we have
| σ(Rg) | = | 0.785 × ρgR | = | 0.785 × 7900 kg m-3 × 9.8 m sec-2 × 6.37 × 106 m |
| = | 3.87 × 1011 Pa | |||
This is over 700 times steel's yield strength of 5.0 × 108 Pa. Hence, a cable made of steel, were it even possible to build, would quickly neck and snap under its own weight. This same argument applies to all known materials, and forms one of the most serious theoretical weaknesses of the space elevator movement, amongst others such as unpredictable weather effects and the danger of impacts from space debris.
Despite this fundamental deficiency, research into space elevator technologies continues, in hope of finding able solutions to its problems. A notable momentum-based variant of the space elevator is the Hypersonic Airplane Space Tether Orbital Launch (HASTOL), currently in development by Boeing, which involves a shorter cable acting as a pivoting tether that swings cargo into orbit, and targets a pound to orbit cost of $200. [5] Yet another technology being developed is to power space-bound vehicles by way of laser propulsion instead of burning chemical fuel. Such vehicles are equipped with light sails - just as the cloth sails of a medieval ship are powered by wind, these light sails are powered by means of a ground-based system that directs a laser beam at the craft throughout its flight.
To the uninitiated, it may seem like the reduction of launch costs is certainly a desirable goal, but not necessarily a relevant one. However, making it cheap to transport things into space is a necessary step to realizing future space projects. The lofty possibilities for the human race - space tourism, habitation and even the colonization and terraformation of Mars - must not only be scientifically possible but also economically viable.
© Alastair Wee. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] M. Turner, Rocket and Spacecraft Propulsion: Principles, Practice And New Developments, 2nd Ed. (Springer, 2005), p. 15.
[2] J. Holt and T. Monk, "Propellant Mass Fraction Calculation Methodology for Launch Vehicles and Application to Ares Vehicles", American Institute of Aeronautics and Astronautics, AIAA 2009-6655, 14 Sep 09.
[3] E. Seedhouse, SpaceX: Making Commercial Spaceflight a Reality (Springer, 2013), p. 171.
[4] P. K. Aravind, "The physics of the space elevator", American Journal of Physics 75, 126 (2007).
[5] "Hypersonic Airplane Space Tether Orbital Launch (HASTOL) Architecture Study, Phase II: Final Report", Boeing, October 2001.
