Abstract
The phrase “It’s not rocket science!” has become one of the great clichés of our age and yet, before the mid-twentieth century, its meaning would have been obscure to most people. In more contemporary times this phrase, delivered with suitable scorn, will be familiar to all of us who have fouled up from time to time. Always delivered with dripping sarcasm, it is applied to tasks that are regarded as simple, and so by inference rocket science is regarded as hard—intellectually or mathematically difficult. In financial circles, the theorists who earn Ph.D.s in economics and write learned papers that fail to predict the next crash are often known as “rocket scientists”, just because their work is esoteric and, perhaps, a little removed from the day-to-day world of financial management and personal investments.
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- 1.
We’re not entirely sure when the phrase was first used in this way. Some claim it dates from newspaper reports of American Football matches in the early 1980s. Others say it was coined by ex-NASA engineers at least a decade earlier. Previously, the demanding skill used in sarcastic comparisons was brain surgery.
- 2.
The question of whether math is the fundamental language of the Universe or just a tool constructed by humans to aid understanding of that Universe is one we leave for philosophers.
- 3.
We use the word propellant rather than the more everyday term fuel ; there is a difference, which we introduced in Chap. 2: propellant is partly fuel, and partly oxidizer. The two components may be physically distinct materials, or they may be mixed together intimately. Thus, a common liquid rocket propellant consists of liquid hydrogen fuel plus liquid oxygen oxidizer, kept well separated until ignition. Gunpowder is a solid rocket propellant with the fuel and oxidizer components thoroughly mixed together. Later we will have much more to say about propellants.
- 4.
But not completely because even interstellar space is not entirely empty. All drag is due to friction and gets worse as you go faster. That’s true even in the vast not-quite emptiness between the stars. You just have to be going very, very fast for it to have any effect.
- 5.
The early 1980’s arcade game Asteroids did a pretty decent job of simulating two-dimensional free space movement. Indeed, modern computer games frequently have much more realistic physics than modern movies do.
- 6.
Rocket propellant is converted into hot and high-pressure gas that is ejected from the nozzle at very high speed. We will investigate the nature of rocket propellants (and nozzles) in later chapters. In this one, we are more interested in the physics of rockets, and so all we need to know about the propellant is that it results in gas of known mass being ejected at known speed. In principle, other fluids may be used. Backyard rockets can be made by ejecting water via compressed air. (For more on water rockets see Sobey (2006), and the website http://www.sciencebits.com/RocketEqs where videos and calculations are both presented.) Different scale (both physical and financial), same principle.
- 7.
Again, an area where Hollywood lies to you: air-to-air missiles in movies are frequently shown as ‘turning and burning’ like fighter jets, with a permanent rocket plume providing continual thrust throughout their flight.
- 8.
See for example the section on energy in the useful Wikipedia article Rocket.
- 9.
Physicists use the Greek letter delta (Δ) to indicate a (small, usually) change in something, so Δv means a change in velocity v. We say velocity when referring to both the speed and direction of a moving body, and speed when referring only to how fast it moves (e.g. my car’s speed is 50 kph; my car’s velocity is 50 kph northward). We will use ‘Δ’ in mathematical expositions and ‘delta’ in text throughout the book.
- 10.
Of course, Newton’s theory of gravity is only approximately true. It breaks down near very large masses such as huge stars, where Einstein’s theory works better. Einstein’s theory is also an approximation—a better one—that breaks down near the center of black holes. We will not be traveling anywhere near a large star or black hole in this book, and so we do not need to deal with the general theory of relativity. Newton’s theory is a very good approximation to the way nature works at mass scales of planets and spacecraft.
- 11.
For a report on the original experiments see Stoll (1956). Interesting physiological effects precede blackout, with the experimental subjects reporting loss of color vision (gray-out) and then tunnel vision before losing consciousness. The centrifuge experiments were aimed at providing information on the maximum G-force that a fighter pilot could tolerate in a tight turn, while still maintaining control of his jet.
- 12.
We glide over a slight complication here. This value for G ignores the contribution due to Earth’s gravity, which can add up to 1 more point (i.e. increase G to 8.87). The amount added varies with altitude, however, because the rocket changes direction: it begins its ascent vertically but then pitches over (the maneuver is known as a gravity turn) so that by the time it gets to LEO altitude it is moving tangentially to the Earth’s surface. This change is necessary because the rocket needs to maintain a certain speed to stay in a stable orbit—for example, at 350 km altitude the orbital speed is 7.7 km s−1. The change in direction changes the value of G in a time-dependent way—to simplify, we ignore it. Another point that may have occurred to you: we have not mentioned the effect of aerodynamic drag on the delta-v required to reach orbit. In fact this has been included (Δv would be lower by a couple of kilometers per second if the Earth had no atmosphere).
- 13.
The exponential cost of propellant combined with a practical limit, due to propellant chemistry, of gas exhaust speed has led some experts to deplore the ‘tyranny of the rocket equation’. The ‘tyranny’ is the low payloads that are forced upon those who seek to go up into space because of the exponentiation. In fact, they say, we are only just able to do so. If planet Earth were just a little more massive, or if chemical propellants were just a little less energetic, then it would not be possible for humans to reach space—the necessary fraction of propellant would be 100%. See the interesting essay by NASA flight engineer Don Pettit, at https://www.nasa.gov/mission_pages/station/expeditions/expedition30/tryanny.html. He also presents a TED talk on the same subject https://www.youtube.com/watch?v=uWjdnvYok4I.
- 14.
Technically, a satellite is something that is in orbit around something else. Not all spacecraft are satellites, at least not all the time, and not all satellites are spacecraft. Natural satellites can include moons, asteroids and other space rocks but if it turns out to have an engine, or an airlock, it’s not a natural one. We will generally use ‘spacecraft’ for artificial objects unless they are actual Earth-orbiting satellites when, although the term would not be wrong, it may seem incongruous. A space probe orbiting Mars is also, correctly, a satellite, but we will probably just call it a spacecraft. Nomenclature is not always straightforward.
- 15.
For example, the Earth’s orbit about the sun is nearly circular, so that our planet orbits with a speed that is almost constant. Our minimum speed is 29.28 km s−1 and our maximum speed is 30.27 km s−1. Some terminology concerning elliptical orbits: the smallest distance between an orbiting spacecraft and a celestial body is generally called the periapsis . If the celestial body is the sun, then the shortest distance is known as perihelion; if it is the Earth, it is known as the perigee.
- 16.
Technically, a LEO is a circular orbit about planet Earth at an altitude of between 160 km and 2000 km. A GSO is at an altitude of 35,786 km and orbits in the same plane as the equator. It has a period of 1 day, so that a satellite in such an orbit stays in the same place, as observed from the surface.
- 17.
The fact that planetary orbits are not exactly circular and that they are not in exactly the same plane complicates the Hohmann ellipse transfer calculations, but the results are very similar to those we get by assuming circular coplanar orbits. Another factor which is important is the alignment of the inner and outer planet for a given transfer time. The Hohmann ellipse transfer time is fixed for a given pair of orbits, and cannot be altered, so the planet positions must be just so. This fact limits the possibility of Hohmann ellipse transfer orbits to specific launch windows, which generally are no more than a couple of hours extent. For a detailed discussion of (Hohmann) transfer orbits between Earth and the Moon, see Biesbroek and Janin (2000).
- 18.
- 19.
The photos you take of Jupiter’s surface must be with a high-speed camera, in more senses than one.
- 20.
According to NASA, Voyager’s fly-by caused Jupiter to slow down by one foot every trillion years. See Johnson (2006).
- 21.
We saw in Chap. 2 that many of the early theoretical pioneers of space exploration are Russian. Yuri Kondratyuk (one of our A-list pioneers, you may recall) was born in modern Ukraine, and died at age 44 fighting for the Soviets against Nazi Germany. He led an interesting and turbulent life; see e.g. the Smithsonian’s Pioneers of Flight short article at http://pioneersofflight.si.edu/content/yuri-vasilievich-kondratyuk-0 or the New Mexico Museum of Space History biographical sketch at http://www.nmspacemuseum.org/halloffame/detail.php?id=2015. Friedrich Zander was born to Baltic Germans in the then Russian empire and died of typhus in 1933, aged 45. He made significant contributions to rocket flight theory and collaborated with other early pioneers such as Tsiolkovsky and Kondratyuk. Biographies of Zander are more readily available in English: see Freeman (2003) or the Wikipedia article Friedrich Zander.
- 22.
Cassini-Huygens was named after an eminent Italian/French astronomer of the eighteenth century, and a brilliant Dutch physicist of the seventeenth (a rival to Newton). The NASA spacecraft was called Cassini and it carried a European Space Agency probe called Huygens, which was sent down to land on the surface of Titan, a moon of Saturn. Cassini-Huygens was launched on October 15 1997; after a near-seven year flight to Saturn, its mission was due to last four more years but was extended until September 2017. Space probes that don’t fail prematurely tend to last longer than planned, generally on account of their being so well built.
- 23.
See Johnson (2006). Another example of the saving in propellant due to gravitational fly-bys, quoted from NASA’s Cassini website https://solarsyatem.nasa.gov/missions/cassini/mission/spacecraft/navigation/: “A single fly-by of Titan at an altitude of 620 miles (about 1000 kilometers) gave Cassini a change in velocity of about 1800 miles an hour (800 meters per second)—this is equivalent to one-third of Cassini’s total propellant at launch.”
- 24.
Other objects have attained higher speeds, usually through gravity assist, although few have maintained them over such a long period. For a slightly out-of-date summary see: https://www.jpl.nasa.gov/infographics/infographic.view.php?id=11489. For up-to-the-minute Voyager information see https://voyager.jpl.nasa.gov.
- 25.
We note incidentally that the sun cannot be used to gain energy via fly-bys, except for missions to distant stars. Within the solar system it has, by definition, no speed for a spacecraft to steal. The analysis of slingshot maneuvers is in e.g. Curtis (2010) and Roy (1988), and is fairly straightforward to derive, at least for the two-dimensional case in which we consider the spacecraft motion to be in the orbital plane of the planet. See da Silva and Lemos (2008), or the helpful online article by Johnson, at http://maths.dur.ac.uk/~dma0rcj/Psling/sling.pdf.
- 26.
Cosmologists would disagree, but would admit that everything is influenced by the gravitational field of everything else within billions of light years. For us puny Earthlings confined to our insignificantly tiny little solar system, however, it is almost the same thing.
- 27.
The three unstable Lagrange points L1, L2 and L3, were in fact discovered by the Swiss mathematical genius Leonard Euler around 1767. The two stable Lagrange points L4 and L5 were discovered by the French theoretical physics genius Joseph-Louis Lagrange in 1772. See e.g. Koon et al (2007). The stability of L4,5 is not obvious; in fact a spacecraft at L4 or L5 will move away from these points due to gravity, but as they pick up speed the Coriolis effect causes their trajectories to bend, forming stable closed orbits (stable if the mass ratio of the two large masses exceeds 25). For a mathematical proof, see Richard Fitzpatrick’s website at http://farside.ph.utexas.edu/teaching/336k/Newton/node126.html
- 28.
There are a number of online articles about the superhighway, including NASA’s Interplanetary Superhighway Makes Space Travel Simpler, at https://www.nasa.gov/mission_pages/genesis/media/jpl-release-071702.html, and Wikipedia’s Interplanetary Transport Network.
- 29.
In fact, as we saw in Chap. 2, rockets have a long history that predates WW2. However, they began to make a serious impact, as it were, only from this time.
- 30.
The bazooka was a novelty brass wind instrument popular in the 1930s, but rarely seen or heard today. See e.g. http://www.oxforddictionaries.com/definition/bazooka?view=uk. Continuing the musical theme, the Americans had a barrage rocket weapon analogous to the Katyusha which they called a xylophone, again due to a physical resemblance. Another version of this weapon was the Calliope.
- 31.
The panzerschreck was a shoulder-carried rocket launcher inspired by captured American bazookas. It seems to have been more effective than the original.
- 32.
- 33.
Presumably drag coefficient Cd is high because the missile is long and thin. We used the cross-sectional area of the Stinger in the equation for drag coefficient (see A12 in the Appendix), but this does not include the contribution to drag from the missile body or fins.
- 34.
This is not to say that modern artillery rounds contain no electronics—some components are mechanically robust enough to survive firing, as we have seen, but their capabilities are very limited. Thus a proximity fuze is a very crude radar system which detonates a warhead when it gets close enough to another object, without much caring what that object might be.
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Johnson, S.B. The Secret of Apollo: Systems Management in the American and European Space Programs. (Baltimore: Johns Hopkins University Press, 2006).
Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross. Dynamical Systems, the Three-Body Problem and Space Mission Design. (New York: Springer-Verlag, 2007).
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Denny, M., McFadzean, A. (2019). Theory: The Rocket Equation and Beyond. In: Rocket Science. Springer, Cham. https://doi.org/10.1007/978-3-030-28080-2_3
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