Space Flight Dynamics Chapter 1
Author: Eze-Odikwa Tochukwu Jed
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Before we begin our technical discussion of space flight dynamics, this first chapter will provide a condensed historical overview of the principle contributors and events associated with the development of what we now commonly refer to as space flight. We may define space flight as sending a human-made satellite or spacecraft to an Earth orbit or to another celestial body such as the moon, an asteroid, or a planet. Of course, our present ability to launch and operate satellites in orbit depends on knowledge of the physical laws that govern orbital motion. This brief chapter presents the major developments in astronomy, celestial mechanics, and space flight in chronological order so that we can gain some historical perspective.
1.2 Early Modern Period
The fields of astronomy and celestial mechanics (the study of the motion of planets and their moons) have attracted the attention of the great scientific and mathematical minds. We may define the early modern period by the years spanning roughly 1500–1800. This time frame begins with the late Middle Ages and includes the Renaissance and Age of Discovery. Figure 1.1 shows a timeline of the important figures in the development of celestial mechanics during the early modern period. The astute reader will, of course, recognize these illuminous figures for their contributions to mathematics (Newton, Euler, Lagrange, Laplace, Gauss), physics (Newton, Galileo), dynamics (Kepler, Newton, Euler, Lagrange), and statistics (Gauss). We will briefly describe each figure’s contribution to astronomy and celestial mechanics. The first major figure is Nicolaus Copernicus (1473–1543), a Polish astronomer and mathematician who developed a solar-system model with the sun as the central body. Galileo Galilei (1546–1642) was an Italian astronomer and mathematician who defended Copernicus’ sun-centered (or “heliocentric”) solar system. Because of his heliocentric view, Galileo was put on trial by the Roman Inquisition for heresy and spent the remainder of his life under house arrest.
Johann Kepler (1571–1630) developed the fundamental laws for planetary motion based on astronomical observations of the planet Mars compiled by the Danish nobleman Tycho Brahe (1546–1601). Kepler’s three laws are:
- The orbit of a planet is an ellipse, with the sun located at a focus.
- The radial line from the sun to the planet sweeps out equal areas during equal time intervals.
- The square of a planet’s orbital period for one revolution is proportional to the cube of the planet’s “mean distance” from the sun.
The third law notes the planet’s “mean distance” from the sun. In Chapter 2 we will define this “mean distance” as one-half of the length of the major axis of an ellipse. Kepler published his first two laws of planetary motion in 1609 and his third law in 1619. Kepler developed an expression for the time-of-flight between two points in an orbit; this expression is now known as Kepler’s equation.
Isaac Newton (1642–1727) was an English astronomer, mathematician, and physicist who developed calculus and formulated the laws of motion and universal gravitation. Newton’s three laws of motion are:
- A body remains at rest or moves with a constant velocity unless acted upon by a force.
- The vector sum of the forces acting on a body is equal to the mass of the body multiplied by its absolute acceleration vector.
- When a body exerts a force on a second body, the second body exerts an equal-and-opposite force on the first body.
The first and second laws hold relative to a fixed or inertial reference frame. Newton published the three laws of motion in Principia in 1687. Newton’s universal law of gravitation states that any two bodies attract one another with a force that is proportional to the product of their masses and inversely proportional to the square of their separation distance. Newton’s laws of motion and gravitation explain the planetary motion that Kepler described by geometrical means.
Leonhard Euler (1707–1783), a Swiss mathematician, made many mathematical and scientific contributions to the fields of calculus, mathematical analysis, analytical mechanics, fluid dynamics, and optics. Euler also developed equations that govern the motion of a rotating body; these equations serve as the foundation for analyzing the rotational motion of satellites in orbit. Johann Heinrich Lambert (1728–1777), also a Swiss mathematician, formulated and solved the problem of determining the orbit that passes through two known position vectors with a prescribed transit time. Known today as Lambert’s problem, its solution provides a method for the orbit-determination process as well as planning orbital maneuvers. Joseph-Louis Lagrange (1736–1813) was an Italian-born mathematician who made significant contributions in analytical mechanics and celestial mechanics, including the determination of equilibrium orbits for a problem with three bodies and the formulation of Lagrange’s planetary equations for orbital motion. Pierre-Simon Laplace (1749–1827) was a French mathematician who, among his many mathematical contributions, formulated the first orbit-determination method based solely on angular measurements. Carl Friedrich Gauss (1777–1855), a German mathematician of great influence, made significant contributions to the field of orbit determination. In mid-1801 he predicted the orbit of the dwarf planet Ceres using a limited amount of observational data taken before Ceres became obscured by the sun. In late 1801, astronomers rediscovered Ceres just as predicted by Gauss.
1.3 Early twentieth Century
Let us next briefly describe the important figures in the early twentieth century. It is during this period when mathematical theories are augmented by experimentation, most notably in the field of rocket propulsion. It is interesting to note that the important figures of this period were inspired by the nineteenth century science fiction literature of H.G. Wells and Jules Verne and consequently were tantalized by the prospect of inter-planetary space travel.
Konstantin Tsiolkovsky (1857–1935) was a Russian mathematician and village school teacher who worked in relative obscurity. He theorized the use of oxygen and hydrogen as the optimal combination for a liquid-propellant rocket in 1903 (the same year as the Wright brothers’ first powered airplane flight). Tsiolkovsky also developed theories regarding rocket propulsion and a vehicle’s velocity change – the so-called “rocket equation.”
Robert H. Goddard (1882–1945), a US physicist, greatly advanced rocket technology by combining theory and experimentation. On March 16, 1926, Goddard successfully launched the first liquid-propellant rocket. In 1930, Goddard moved his laboratory to New Mexico and continued to develop larger and more powerful rocket engines.
Hermann J. Oberth (1894–1989) was born in Transylvania and later became a German citizen. A physicist by training, he independently developed theories regarding human Historical Overview 3 space flight through rocket propulsion. Oberth was a key figure in the German Society for Space Travel, which was formed in 1927, and whose membership included the young student Wernher von Braun. Von Braun (1912–1977) led the Nazi rocket program at Peenemünde during World War II. Von Braun’s team developed the V-2 rocket, the first long-range rocket and the first vehicle to achieve space flight above the sensible atmosphere.
At the end of World War II, von Braun and members of his team immigrated to the US and began a rocket program at the US Army’s Redstone Arsenal at Huntsville, Alabama. It was during this time that the US and the Soviet Union were rapidly developing long-range intercontinental ballistic missiles (ICBMs) for delivering nuclear weapons.
1.4 Space Age
On October 4, 1957, the Soviet Union successfully launched the first artificial satellite (Sputnik 1) into an Earth orbit and thus ushered in the space age. Sputnik 1 was a polished 84 kg metal sphere and it completed an orbital revolution every 96 min. The US successfully launched its first satellite (Explorer 1) almost 4 months after Sputnik on January 31, 1958. Unlike Sputnik 1, Explorer 1 was a long, tube-shaped satellite, and because of its shape, it unexpectedly entered into an end-over-end tumbling spin after achieving orbit.
Our abridged historical overview of the first half of the twentieth century illustrates the very rapid progress achieved in rocket propulsion and space flight. For example, in less than 20 years after Goddard’s 184ft flight of the first liquid-propellant rocket, Nazi Germany was bombarding London with long-range V-2 missiles. Twelve years after the end of World War II, the USSR successfully launched a satellite into orbit. Another point of interest is that in this short period, rocket propulsion and space flight transitioned from the realm of the singular individual figure to large team structures funded by governments. For example, the US established the National Aeronautics and Space Administration (NASA) on July 29, 1958.
The US and USSR space programs launched and operated many successful missions after the space age began in late 1957. Table 1.1 summarizes notable robotic space missions (i.e., no human crew). A complete list of successful space missions would be quite long; Table 1.1 is not an exhaustive list and instead presents a list of mission “firsts.” It is truly astounding that 15 months after Sputnik 1, the USSR sent a space probe (Luna 1) to the vicinity of the moon. Equally impressive is the first successful interplanetary mission (Mariner 2), which NASA launched less than 5 years after Explorer 1. Table 1.1 shows that spacecraft have visited all planets in our solar system and other celestial bodies such as comets and asteroids.
On April 12, 1961, the USSR successfully sent the first human into space when Yuri Gagarin orbited the Earth in the Vostok 1 spacecraft. Less than 1 month later, the US launched its first human into space when Alan Shepard flew a suborbital mission in a Mercury spacecraft. Table 1.2 presents notable space missions with human crews (as with Table 1.1, Table 1.2 focuses on first-time achievements). Tables 1.1 and 1.2 clearly illustrate the accelerated pace of accomplishments in space flight. Table 1.2 shows
Table 1.1 Notable robotic space missions.
|Sputnik 1||October 4, 1957||First artificial satellite to achieve Earth orbit||USSR|
|Luna 1||January 2, 1959||First satellite to reach the vicinity of the moon||USSR|
|Mariner 2||December 14, 1962||First spacecraft to encounter (fly by) another planet (Venus)||US|
|Mariner 4||July 14, 1965||First spacecraft to fly by Mars||US|
|Luna 9||February 3, 1966||First spacecraft to land on another body (moon)||USSR|
|Luna 10||April 3, 1966||First spacecraft to orbit the moon||USSR|
|Venera 7||December 15, 1970||First spacecraft to land on another planet (Venus)||USSR|
|Mariner 9||November 14, 1971||First spacecraft to orbit another planet (Mars)||US|
|Pioneer 10||December 3, 1973||First spacecraft to fly by Jupiter||US|
|Mariner 10||March 29, 1974||First spacecraft to fly by Mercury||US|
|Viking 1||July 20, 1976||First spacecraft to land on Mars||US|
|Voyager 1||March 1979, November 1980||Fly by encounters with Jupiter, Saturn, and Saturn’s moon Titan||US|
|Voyager 2||January 1986, August 1989||First spacecraft to fly by Uranus and Neptune||US|
|Galileo||December 8, 1995||First spacecraft to orbit Jupiter||US|
|Mars Pathfinder||July 4, 1997||First rover on the planet Mars||US|
|NEAR Shoemaker||February 12, 2001||First spacecraft to land on an asteroid (433 Eros)||US|
|Cassini-Huygens||July 2004, January 2005||First spacecraft to orbit Saturn (Cassini) and first spacecraft to land on the moon Titan (Huygens)||US and Europe|
|Stardust||January 16, 2006||First spacecraft to return samples from a comet||US|
|MESSENGER||March 18, 2011||First spacecraft to orbit Mercury||US|
|New Horizons||July 14, 2015||First spacecraft to fly by Pluto||US|
Table 1.2 Notable space missions with human crews.
|Vostok 1||April 12, 1961||First human to reach space and orbit the Earth||USSR|
|Vostok 6||June 16, 1963||First woman in space||USSR|
|Voskhod 2||March 18, 1965||First human “spacewalk” outside of orbiting spacecraft||USSR|
|Gemini 6A||December 15, 1965||First orbital rendezvous||US|
|Apollo 8||December 24, 1968||First humans to orbit the moon||US|
|Apollo 11||July 20, 1969||First humans to land and walk on the moon||US|
|Salyut 1||April 19, 1971||First orbiting space station with crew||USSR|
|STS-1||April 12, 1981||First flight of a reusable spacecraft (Space Shuttle)||US|
|International Space Station||November 20, 1998||First multinational space station and largest satellite placed in Earth orbit||Russia, US, Europe, Japan, Canada|
Table 1.3 Significant advances in space flight dynamics in the twentieth century.
|Dirk Brouwer |
|Developed pioneering work in the field of analytical satellite theory, including the perturbing effects of a non-spherical Earth|
|Theodore Edelbaum||Obtained analytical optimal trajectory solutions for spacecraft propelled by low-thrust electric propulsion engines|
|Richard Battin||Developed guidance and navigation theories for lunar and interplanetary spacecraft|
|Rudolf Kalman||Developed an optimal recursive estimation method (the Kalman filter) that has been applied to orbit determination and satellite navigation|
|W.H. Clohessy and R.S. Wiltshire||Developed closed-form solutions for the motion of a satellite relative to an orbiting target satellite (i.e., orbital rendezvous)|
|Derek Lawden||Developed theories for optimal rocket trajectories|
|A.J. Eggers and H.J. Allen Dean Chapman||Obtained analytical solutions for the entry flight phase of a ballistic capsule or lifting spacecraft returning to Earth from space|
|Robert Farquhar||Conceived of and managed space missions that targeted orbits where the satellite is balanced by the gravitational attracting of two celestial bodies|
|Ronald Bracewell Vernon Landon||Developed theories regarding the stability of a spinning satellite in orbit|
|Paul Cefola||Developed the Draper Semianalytical Satellite Theory (DSST) for rapid orbital calculations over a long time period|
The very rapid progress in space missions with human crews in the 1960s, culminating with the first Apollo lunar landing on July 20, 1969. To date, three countries have developed human space flight programs: USSR/Russia (1961); US (1961); and China (2003). We end this chapter with a brief summary of the significant twentieth century figures in the field of space flight dynamics. Table 1.3 presents these figures and their accomplishments. This list is certainly not exhaustive; furthermore, it is difficult to identify single individuals when the tremendous achievements in space flight over the past 60 years involve a large team effort.