ASPECTS OF GEOMETRY FOR ROCKET PROPULSION ERNEST YEUNG Abstract. I point out aspects of differential geometry related to rocket propulsion. The dynamics of rocket flight is viewed from the point of view of smooth manifolds. General expressions that hold true for the dynamics of rockets in curved spacetime (relativistic or Newtonian) are formulated. Thermo- dynamics is also formulated from the point of view of a manifold of thermodynamics states. This note serves as an appetizer for further work, to motivate applications of topology, symplectic geometry, and numerical relativity to rocket propulsion and related aspects of rocket flight in space. Part 1. Executive Summary 2 1. Forces 2 2. Energy 3 Part 2. Perspectives and Prospects 3 3. Why? 3 4. Applications 4 Part 3. Some Fluid mechanics 5 5. Mass transport (in Continuum)5 6. Momentum (in Continuum)7 7. Cauchy Stress Tensor9 8. Some Carath´eodory's Thermodynamics 10 9. Energy Transport 12 References 13 Contents \Why does TARS have to detach?" \We have to shed the weight to escape the gravity." Date: 27 mars 2015. 1991 Mathematics Subject Classification. Fluid Mechanics. Key words and phrases. Aerodynamics, Combustion Theory, Differential Geometry, Folations, Propulsion, Rocket Propulsion, Smooth Manifolds, Symplectic Geometry, Thermodynamics, Topology. Ernest Yeung had been supported by Mr. and Mrs. C.W. Yeung, Prof. Robert A. Rosenstone, Michael Drown, Arvid Kingl, Mr. and Mrs. Valerie Cheng, and the Foundation for Polish Sciences, Warsaw University, during his Masters studies. I am on linkedin: ernestyalumni. I am crowdfunding on Tilt/Open and at Patreon to support basic sciences research: ernestyalumni.tilt.com and ernestyalumni at Patreon. Tilt/Open is an open-source crowdfunding platform that is unique in that it offers open-source tools for building a crowd- funding campaign. Tilt/Open has been used by Microsoft and Dicks Sporting Goods to crowdfund their respective charity causes. Patreon is a subscription crowdfunding service that allows you to directly support the works of artists (and scientists and educators! See the Science and Education section of Patreon), allowing you to be a patron of the arts (and the sciences!). Patreon is run by creators and artists and allows you to be flexible in your support. 1 \Newton's third law: the only way humans have figured out of getting somewhere is to leave something behind." -Interstellar (2014) Part 1. Executive Summary 1. Forces Let spacetime be a quadruple (M; O; A; g) consisting of a (1 + n)-dimensional smooth manifold M, a choice of topology O, a choice of a smooth atlas A, and a choice of metric g, so that the Newtonian and relativistic cases come into play with the choice of g. Usually n = 3 for 3 spatial directions. Consider a total system consisting of a (rigid) rocket with combustion chamber and nozzle, and fuel that will be treated as a fluid. As the rocket is a rigid body, it can be shown that the total momentum of the rocket is such that the rocket acts as if the momentum is at the center of mass and concentrated there like a point particle [1]. Let the mass of the rocket be M0. Let the 4-acceleration of the rocket be a. I claim that the full expression for the external forces and dynamics accounted for in rocket propulsion, written in a covariant manner, of a total system consisting of the rocket, with its (rigid) combustion chamber and nozzle, and fuel, on a spacetime that is a 4-manifold M equipped with a metric g, is the following: (1.1) Z @ui Z @ui Z @e n i _ j n i n i M0a + ρ vol ⊗ ei + u dm ⊗ ei + ρu j vol ⊗ ei + ρu vol ⊗ + [u; ei] = B @t B @x B @t X Z = Fext = ∗σ + Fgrav @(R+B) where B is a smooth, compact submanifold of the spatial foliation N of spacetime M, with boundary @B, enclosing the fuel, ρ is the density of the fuel ρ 2 C1(M), u 2 X(M) is a velocity vector field for the fuel, with its spatial coordinate components denoted as u, voln 2 Ωn(N) is the spatial volume form on N, feig is a (coordinate) frame on M, t 2 R is time, Fext is an external force on the total system, R is a smooth, compact submanifold of the spatial foliation N with boundary @R that encloses the rocket, ∗ ∗ σ 2 Γ(T M ⊗ T M) is the Cauchy-stress tensor on M, and Fgrav is the force due to gravity. The fluid dynamics of the fuel motivates the form of its dynamics on the left-hand side (LHS) of (1.1) (cf. (6.7)). dm_ 2 Ωn(N) is a n-form on N describing the mass transport of an infinitesimal piece of the fuel (cf. (6.6)), which I'll recap as well here: @ρ _ n n n dm = vol + diuρvol = L @ +uρvol @t @t R i _ R _ In (1.1), if one subtracts from both sides the term B u dm ⊗ ei ≡ B dm ⊗ u, then the so-called thrust, due to the ejection of fluid out of the rocket R, pushing on the rocket (Newton's third law!), force is obtained as a contribution to the acceleration of the rocket a: Z − dm_ ⊗ u B 2 R j @ui n The term B ρu @xj vol ⊗ei in (1.1) also informs us that the geometry of the nozzle can push the rocket forward. One should also note that the expression for the time derivative of the momentum of the fluid Π_ at its boundary can also be useful in evaluating rocket performance (cf. (6.7)): Z i Z Z Z @u n @ρ n i n i n @ei Π_ = ρvol ⊗ ei + vol ⊗ u + ρu iuvol ⊗ ei + ρu vol ⊗ + [u; ei] B @t B @t @B B @t and in particular this term: Z Z @ρ n i n vol ⊗ u + ρu iuvol ⊗ ei B @t @B where the geometry of the surface from which the fluid exits from comes into play in the quantity n iuvol . In (1.1), on the RHS, I claim that the term Z ∗σ @(R+B) which is the Hodge star operator applied to the (0; 2)-type symmetric tensor σ, representing the Cauchy stress tensor, is a general, manifestly covariant, expression for the sum of all the forces on a system due to stress and strain at the system's boundary. What one then does is plug into σ a particular form of the stress tensor. In this note, I apply one example, the perfect fluid with fluid pressure (cf.(7.2)). 2. Energy Assuming a steady state flow, from calculating the time derivative of the total energy E of a fluid inside a smooth, compact body B with boundary, there is a term that describes the change in energy due to fluid leaving the boundary surface (cf. (9.5)) Z Z 1 2 p n 1 2 p n (2.1) d( u + w − )iuvol = ( u + w − )iuvol B 2 ρ @B 2 ρ where Stoke's law was used and where ρ is the mass density and p the pressure. It should be noted that no conditions on the thermodynamic process was made as the definition of internal energy per molar mass w was solely employed to express the time change in energy. Part 2. Perspectives and Prospects 3. Why? I became interested in rocket propulsion for two reasons: Elon Musk's rationale for advancing humanity to become a truly spacefaring civilization so to make (intelligent) life multi-planetary 1 and I wanted enable the observation and creation of physical phenomenon - black holes, worm holes, anti-deSitter (AdS) space - seen in (the best movie ever made) Interstellar (Nolan, 2014) possible. Apparently, Musk began, after the successful IPO (initial public offering) of PayPal, to have any expertise on rocket propulsion by reading books and talking with experts 2. Reading books was something I could immediately do after my completion of my Master's thesis in Physics (2014). Books on rocket propulsion he read included Rocket Propulsion Elements by Sutton and Biblarz [2] and Aerothermodynamics of Gas Turbine and Rocket Propulsion by Oates [3]. 1A Conversation with Elon Musk https://youtu.be/vDwzmJpI4io 2Richard Feloni \Former SpaceX Exec Explains How Elon Musk Taught Himself Rocket Science", Business Insider, http://www.businessinsider.com/how-elon-musk-learned-rocket-science-for-spacex-2014-10 3 Also, I did not find any MOOCs (Massive Open Online Courses) for graduate-level rocket propulsion, but I did find one MIT OCW (Open CourseWare) for 16.512 Rocket Propulsion, taught by Prof. Martinez- Sanchez in the Fall 2005 [4],a course suggested by Loretta Trevi~noon quora.com 3, which also, in turn, led me to this suggested reading [5]. One small endeavor I would like to help in is open, online education at a high-level, the graduate or professional level, for aerodynamics and propulsion because I want to encourage physicists, particularly theoretical physicists, to interface and help with engineering problems and bring their unique expertise to be applied there. I would point to Dr. Feynman's role in investigating the Challenger disaster (1986) as an example of how theoretical physicists can help engineers in aerospace 4. So one very small part that I do is to write (a lot of) notes on my readings of these engineering texts and catalogue resources I find online, sharing this educational path for (hopefully) others with a physics or pure math background to try the same. 4. Applications I'll allow myself, here, to speculate on some possible applications from math and physics to rocket propulsion.