Note: This was a blog entry used as the background for a presentation made in a Calculus I class, during spring semester of 2017. So, unless you were in the class, you may only imagine my elaborating, tying in the contents of this entry to the bigger picture, and giving examples of how our class content was relevant to the average spacetrucker. Anyway, please enjoy.
My background is in political science and the humanities, and my applications to graduate schools among the social sciences have included proposals for research in subjects such as vegan abolitionism, international sex trafficking, the prison industrial complex, linguistics, and law. I was looking at graduate schools overseas, and actually was accepted for a master of arts program in Barcelona, but I didn’t have the funding to move overseas to go back to college. So, I decided to do some leveling among the empirical sciences, in order to potentially sweeten up my applications for international scholarships which often give a “full ride” that covers cost of living, tuition, and housing expenses for international students. So here I am, at Cochise College, adding an associate of science in mathematics degree, to my bachelor of arts in journalism from the University of North Texas.
What can mathematics do for me? Or what has it already done? It can make my research among the social sciences more tractable in a quantitative sense. But I admit, it lures me away from the aforementioned graduate research, and toward research in math, physics, and computer science.
Mathematics enables higher precision thought. With that, mathematics does threaten my political career, which is a funeral I would welcome after working fifteen years as a journalist. In any case, I suspect it will enable me further in my current and more general vocation as a communicator, author, publisher, journalist, scholar, and athlete. I am a novelist, and I do not intend to quit writing prose. In fact, I understand that mathematics will improve my creative content e.g. through literature. I worry a little, though, that some theoretical physics project, some quantum software modeling project, some bottomless love affair with galactic ion drives, or some particle accelerator on Triton (or maybe at Betelgeuse) could eclipse my literary efforts. Maybe I’ll just switch to writing science/speculative fiction?
And I also do admit to a longstanding philosophical yen, the further pursuit of which, leveling in mathematics is crucial. As I remarked to Dr. Ritter last month, a thorough philosophical survey has linguistic denotations, and a proper linguistic examination has quantitative roots. Please reference the PhilPapers.org database, which contains a huge repository of academic papers about the Philosophy of Physical Science which includes subcategories in the philosophy of cosmology, philosophy of physics, quantum theory, quantum mechanical interpretation, and metaphysics, space, and time; also PhilPapers.org has a lovely Philosophy of Mathematics repository with sub-categorizations including epistemology of mathematics, ontology of mathematics, theory of mathematics, and the history of mathematics.
And it’s worth remarking that the history of mathematics is critically important to understanding the current state of the art, and specifically the historical personnel. In my opinion, a good place to dive in to the very large cast of characters could be English theoretical physicist Paul Dirac, and French mathematician, theoretical physicist, and philosopher of science Henri Poincaré. One may introduce themselves to them, and your other colleagues, on Wikipedia.org.
Meanwhile back to strictly business, the Cornell University Library’s Arxiv.org repository provides open access to more than one million e-prints of academic papers in Physics, Mathematics, Computer Science, Quantitative Biology, Quantitative Finance, and Statistics.
Whether I’m a science and technology writer, a math teacher, a post-doc researcher, still a novelist, an unlicensed bush pilot, or an un-dead radioactive astronaut, the mainstay, calculus, serves me as a fundamental curriculum of the thinking faculty generally, and it is the gateway to empirical science unequivocally.
What subjects among the sciences are beckoning most, to me as a writer and researcher? Typically, they’re fields which cannot be dabbled in properly without an immersive indoctrination into the family of modes of analytical thought. What things, for example? Propulsion systems. Quantum mechanics. Theoretical physics. Digital philosophy. Logic. Modern (and of course historical) metaphysics. Nanotechnology. Astronomy. Architecture. Engineering. Genetics and organic software. These subjects quarter endless interdisciplinary research opportunities, and all of their epistemologies are substantially mathematical.
Let’s, for example, briefly focus on spacecraft propulsion.
Example 1: Fusion Rocket
In the paper “Advanced Deuterium Fusion Rocket Propulsion For Manned Deep Space Missions,” we catch a glimpse of our friend Bernhard Riemann. The average velocity averaged over the momentum of the charged fusion products is a measure of the maximum specific impulse with respect to the maximum exhaust velocity:
Source: Winterberg, Friedwardt. “Advanced Deuterium Fusion Rocket Propulsion For Manned Deep Space Missions.” Department of Physics, University of Nevada, Reno. 3 June 2009. arXiv:0906.0740. Web. 3 April 2017.
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Example 2: Electric (Solar) Sail
The authors of “TI Tether Rig for Solving Secular Spinrate Change Problem of Electric Sail” present a control algorithm consisting of six throttling factors which are multiplied together to yield the voltage throttling of each of the E-sail’s maintethers.
The angular momentum L used by the control algorithm below is a time-averaged version of L inst which is obtained by continuously solving the differential equation
where τ L = 1200 s is timescale used in the time-averaging. The authors estimate the E-sail thrust on the tether rig as
where p is the momentum of the tether rig relative to the spacecraft, and m(rig) and m(tot) is the mass of the tether rig and the total mass, respectively. The first term is due to acceleration of the tether rig with respect to the spacecraft body, and the second term is due to acceleration of the spacecraft with respect to an inertial frame of reference. The time average of the first term is obviously zero, but its instantaneous value is usually nonzero and it carries information about tether rig oscillations that we want to damp.
The instantaneous thrust exerted on the whole system (spacecraft plus tether rig) is
From the instantaneous F(tot) is calculated a time-averaged version F(ave)(tot) by continuing to solve the time-dependent differential equation
where τ(d6) = 1200 s is another damping timescale parameter. Finally the overall throttling factor f(6) is calculated as
where ∆t d = 20 s is the timestep how often the damping algorithm is called, f(6)(old) is the previous value of f(6) and f(6)(max) = 1.01 is f(6)’s maximum allowed value. The equation (15) resembles solving a differential equation similar to (2) and (14), except that (15) also clamps the solution if it goes outside bounds (0, f(6)(max)).
The total throttling factor factor is
where the maximum is taken over the main tethers.
Source: Janhunen, Pekka; Toivanen, Petri. “TI tether rig for solving secular spinrate change problem of electric sail.” Finnish Meteorological Institute, Helsinki, Finland. Preprint submitted to Acta Astronautica. 17 March 2016. arXiv:1603.05563. Web. 3 April 2017.
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Example 3: Laser Sail
Even with low power lasers, mitigation of flux (conduction of energy from laser to sail) is important in laser sail design. The matter is discussed in “A Roadmap to Interstellar Flight.” Since the laser line is very narrow, a sail’s reflectivity can be made extremely high and absorption rate very low with multiple layer dielectric coatings. Coatings on glass can achieve 99.999% reflectivity, which is fine in most cases except the extreme flux of true interstellar probes which use small reflectors of about one meter. Relativistic aspects of the highest speed missions present another challenge because the laser wavelength is shifted at the reflector.
The flux is proportional to the thickness and density on a smaller sail, and inversely proportional to the mass on a larger sail, which means lower mass payloads have high flux requirements on the sail.
On this subject, the author gives two sails scenarios: 1) where light is either reflected or absorbed but none is transmitted through the sail (for both dielectric and metal coatings) and 2), where some light is transmitted through the reflector (for dielectric only coatings).
Source: Lubin, Philip. “A Roadmap to Interstellar Fligh.” Physics Department, UCSB. Journal of the British Interplanetary Society Vol. 69, 40- 72 Feb. 2016. arXiv:1604.01356. Web. April 3, 2017.
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Ion engine test firing. Photo credit: NASA/JPL
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Example 4: Warp Drive
“The geodesic incompleteness of the Alcubierre-Hiscock space at the event horizon for superluminal warp bubbles has been eliminated by first extending the metric beyond the horizon according to the Kruskal procedure, and then by an embedding in a three-dimensional Minkowski space. It has been also shown that, if the Minkowski space is provided with the topological identifications that correspond to the universal covering of the three-dimensional Misner space, then one can convert the interior of the warp spaceship into a multiply connected space with closed timelike curves which is able to behave like a time machine.”
“The most consistent quantum treatment for dealing with vacuum fluctuations in the multiply connected case leads also to the result that the size of the spaceship bubble and its closed timelike curves must necessarily be placed at scales close to the Planck length.”
Source: Catalán, Miguel. “On the Warp Drive Space Time.” Centro de Fı́sica, Instituto de Matemáticas y Fı́sica Fundamental, Consejo Superior de Investigaciones Cientı́ficas, Madrid. Physical Review D, 62 (2000). arXiv:gr-qc/9907026. Web. 10 April, 2017.
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Example 5: Relativistic Gravity for Propulsion
The paper “Test of relativistic gravity for propulsion at the Large Hadron Collider” presents a lab experiment to test for relativistic gravity for spacecraft propulsion at relativistic speeds.
Within the weak-field approximation of general relativity, exact solutions have been derived for the gravitational field of a mass moving with arbitrary velocity and acceleration (Felber, 2005a). The solutions indicated that a mass having a constant velocity greater than 3 – 1/ 2 times the speed of light c gravitationally repels other masses at rest within a narrow cone.” At high Lorentz factors ( λ >> 1 ), the force of repulsion in the forward direction is about -8λ5 times the Newtonian force. A large mass moving faster than 3 – 1/ 2 c could serve as a driver to accelerate a much smaller payload from rest to a good fraction of the speed of light. That a particle with a radial speed exceeding 3 – 1/ 2 c is repelled in a static Schwarzschild field was first correctly noted by (Hilbert, 1917). The same critical speed of repulsion was found for radial motion along the rotation axis of a spinning stationary source in (Mashhoon, 2005).
The radial component of the electric ‘velocity field’ never changes sign. The radial component of the gravitational ‘velocity field’, on the other hand, can change sign at sufficiently high source velocity and repel masses within a narrow cone:
If a mass m approaching or receding from a stationary test particle at a speed greater than 3 – 1/ 2 c repels the test particle, then a stationary mass M should also repel a test particle that approaches or recedes from it at a speed greater than 3 – 1/ 2 c . In the Schwarzschild field of a stationary mass M, the exact equation of motion of a test particle having purely radial motion (zero impact parameter) is
At extreme relativistic velocities, such as are attained in particle accelerators and storage rings, the repulsive gravitational field of particle bunches in their forward direction can be many orders of magnitude greater than the Newtonian field.
“In the weak-field approximation of general relativity, the relativistically exact gravitational field of a particle having any velocity and acceleration is given in equation (2) by (Felber, 2005a). This exact weak-field solution is the first to show that a distant inertial observer sees masses with velocity greater than 3 – 1/ 2 c repel stationary particles within an ‘antigravity beam’ in the forward and backward directions. This result should perhaps not be surprising, since (Hilbert, 1917) showed over 90 years ago that a distant inertial observer sees a stationary mass repel particles moving radially towards it or away from it with a velocity greater than 3 – 1/ 2 c . That is, it should not be surprising to anyone who believes that if A repels B, then, by conservation of momentum and mass dipole moment, B should repel A.”
Source: Felber, Franklin (Starmark, Inc.). “Test of relativistic gravity for propulsion at the Large Hadron Collider.” Appeared in the American Institute of Physics’ Interplanetary Physics proceedings of the Space, Propulsion & Energy Sciences International Forum (SPESIF-2010). arXiv:0910.1084. Web. 3 April 2017.
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Image source: Belbruno, E.A. “Lunar Capture Orbits, a Method of Constructing Earth Moon Trajectories and the Lunar Gas Mission” JPL. AIAA-87-1054. 19th AIAA/DGLR/JSASS International Electric Propulsion Conference. (1987). edbelbruno.com. Web. 10 April 2017.
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Shuttle Main Engine Test Fire, NASA/JPL public domain photo
Quoting Stephen Wolfram from his book “A New Kind of Science”: “Mind is part of the universe being studied, not apart or outside of it.”
Source: Wolfram, Stephen. A New Kind of Science|Online. Wolfram Media. Champaign, Ill. 2002. Web.
Source: Contributors various, Spacecraft Propulsion, Wikipedia. 20 April 2017. Web.
See also: List of Fictional Spacecraft, Wikipedia. 22 April 2017. Web.
