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.
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.
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.
Ion engine test firing. Photo credit: NASA/JPL
Example 4: Warp Drive