Aerodynamic versus Ballistic Flight

We consider, compare, and contrast various aspects of aerodynamic and ballistic flight. We compare the energy efficiency of aerodynamic level flight at a given altitude versus that of ballistic flight beginning and ending at this same altitude. We show that for flights short compared to Earth’s radius, aerodynamic level flight with lift-to-drag ratio 2 L D > is more energy-efficient than ballistic flight, neglecting air resistance or drag in the latter. Smaller L D suffices if air resistance in ballistic flight is not neglected. For a single circumnavigation of Earth, we show that aerodynamic flight with 4π L D > is more energy-efficient than minimum-altitude circular-orbit ballistic spaceflight. We introduce the concept of gravitational scale height, which may in an auxiliary way be helpful in understanding this result. For flights traversing N circumnavigations of Earth, if 1 N  then even minimum-altitude circular-orbit ballistic spaceflight is much more energy-efficient than aerodynamic flight because even at minimum circular-orbit spaceflight altitude air resistance is very small. For higher-altitude spaceflight air resistance is even smaller and the energy-efficiency advantage of spaceflight over aerodynamic flight traversing the same distance is therefore even more pronounced. We distinguish between the energy efficiency of flight per se and the energy efficiency of the engine that powers flight. Next we consider the effects of air density on aerodynamic level flight and provide a simplified view of drag and lift. We estimate the low-density/high-altitude limits of aerodynamic level flight (and for comparison also of balloons) in Earth’s and Mars’ atmospheres. Employing Mars airplanes and underwater airplanes on Earth (and hypothetically also on Mars) as examples, we consider aerodynamic level flight in rarefied and dense aerodynamic media, respectively. We also briefly discuss hydrofoils. We appraise the optimum range of air densities for aerodynamic level flight. We then consider flights of hand-thrown projectiles that are unpowered except for the initial throw. We describe how aerodynamically efficient ones (i.e., with large L D ) such as Frisbees, Aerobies, and boomerangs not only can traverse record horizontal distances, but (along with discuses) also can—since lift exceeds weight at achievable throwing speeds—maintain How to cite this paper: Denur, J. (2019) Aerodynamic versus Ballistic Flight. Open Journal of Fluid Dynamics, 9, 346-400. https://doi.org/10.4236/ojfd.2019.94023 Received: January 17, 2017 Accepted: December 15, 2019 Published: December 18, 2019 Copyright © 2019 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/

L D suffices if air resistance in ballistic flight is not neglected. For a single circumnavigation of Earth, we show that aerodynamic flight with 4π L D > is more energy-efficient than minimum-altitude circular-orbit ballistic spaceflight. We introduce the concept of gravitational scale height, which may in an auxiliary way be helpful in understanding this result. For flights traversing N circumnavigations of Earth, if 1 N  then even minimum-altitude circular-orbit ballistic spaceflight is much more energy-efficient than aerodynamic flight because even at minimum circular-orbit spaceflight altitude air resistance is very small. For higher-altitude spaceflight air resistance is even smaller and the energy-efficiency advantage of spaceflight over aerodynamic flight traversing the same distance is therefore even more pronounced. We distinguish between the energy efficiency of flight per se and the energy efficiency of the engine that powers flight. Next we consider the effects of air density on aerodynamic level flight and provide a simplified view of drag and lift. We estimate the low-density/high-altitude limits of aerodynamic level flight (and for comparison also of balloons) in Earth's and Mars' atmospheres. Employing Mars airplanes and underwater airplanes on Earth (and hypothetically also on Mars) as examples, we consider aerodynamic level flight in rarefied and dense aerodynamic media, respectively. We also briefly discuss hydrofoils. We appraise the optimum range of air densities for aerodynamic level flight. We then consider flights of hand-thrown projectiles that are unpowered except for the initial throw. We describe how aerodynamically efficient ones (i.e., with large L D ) such as Frisbees, Aerobies, and boomerangs not only can traverse record horizontal distances, but (along with discuses) also can-since lift exceeds weight at achievable throwing speeds-maintain

Introduction
We consider, compare, and contrast various aspects of aerodynamic and ballistic flight. In Section 2 we compare the energy efficiency of aerodynamic level flight at a given altitude versus that of ballistic flight beginning and ending at this same altitude. We show that for flights short compared to Earth's radius, aerodynamic level flight with lift-to-drag ratio 2 L D > is more energy-efficient than ballistic flight, neglecting air resistance or drag in the latter. If air resistance in ballistic flight is not neglected, then smaller L D suffices for short aerodynamic level flight to be more energy-efficient than short ballistic flight. For a single circumnavigation of Earth (the longest possible flight whose purpose is to reach a destination on Earth, with the destination being the starting point after a round-the-world trip), we show that aerodynamic flight with 4π L D > is more energy-efficient than minimum-altitude single-circular-orbit ballistic spaceflight, neglecting the very small air resistance in the latter. We introduce the concept of gravitational scale height, which may in an auxiliary way be helpful in understanding this result. If the very small air resistance at minimum circular-orbit spaceflight altitude is not neglected, then L D very slightly smaller than 4π suffices for single-circumnavigation aerodynamic flight to be more energy-efficient than minimum-altitude single-circular-orbit ballistic spaceflight. For flights traversing N circumnavigations of Earth, owing to air resistance or drag being very small even at minimum circular-orbit ballistic spaceflight altitude, L D must exceed 4πN for aerodynamic flight to be more energy-efficient than minimum-altitude circular-orbit ballistic spaceflight. But 100 L D ≈ may represent the practicable limit that can be achieved even with the most advanced aerodynamic technology. Hence if 1 N  even minimum-altitude circular-orbit ballistic spaceflight is much more energy-efficient than aerodynamic flight. For higher-altitude spaceflight drag is even smaller and the energy-efficiency advantage of spaceflight over aerodynamic flight traversing the same distance is therefore even more pronounced. In Section 2.4 we distinguish between the energy efficiency of flight per se and the energy efficiency of the engine that powers flight. In Section 3 we consider the effects of air density on aerodynamic level flight and provide a simplified view of drag and lift. We estimate the low-density/high-altitude limits of aerodynamic level flight (and for comparison also of balloons) in Earth's and Mars' atmospheres. Employing Mars airplanes and underwater airplanes on In this paper our main goal is to elucidate more conceptually than mathematically some fundamental ideas concerning energy efficiency and a number of other aspects of aerodynamic versus ballistic flight, and to provide comparison with surface transportation. Some elucidations of this type have, of course, been provided elsewhere. But to the best knowledge of the author, this paper provides a range of such elucidations, and some new ones, not found elsewhere, at least not in a single work. We do not attempt the mathematically complex and detailed fully-quantitative analyses based on rigorous application of fluid dynamics, e.g., computational fluid dynamics, as is required in the actual design of aircraft, or the analyses required in the actual design of spacecraft or surface vehicles. Thus our analyses are qualitative to semiquantitative.

Short Flights: Aerodynamic Flight Wins
We compare the energy efficiency of aerodynamic level flight at a given altitude versus that of ballistic flight beginning and ending at this same altitude, at first neglecting air resistance or drag in the case of ballistic flight. We define a short flight as one traversing horizontal distance X much smaller than Earth's radius 6370 km R  . [The dot-equal sign (  ) means "very nearly equal to".] Hence for a short flight Earth's curvature can be neglected. Let A h be the altitude above mean sea level of aerodynamic level flight, and also the beginning or initial alti- The local acceleration due to gravity is g and drag is neglected. An aerodynamic level flight of an aircraft of mass m and weight mg subject to aerodynamic lift L and aerodynamic drag D traversing any horizontal distance X, short or long, costs energy Of course, for any flight, short or long, "level" and "horizontal" mean that A h and therefore A A r R h = + is constant, a long flight following the curvature of Earth.
The first step of Equation (2) is justified because to maintain aerodynamic level flight through horizontal distance X air resistance (most generally, fluid resistance) or drag D must be overcome through distance X. The third step of Equation (2) is justified because to maintain aerodynamic level flight lift L must equal the weight mg of an aircraft [2]- [15]. 2 The lift-to-drag ratio L D of a translational-lift aircraft is the ratio (horizontal distance traversed) ÷ (altitude lost) in gliding flight without engine power relative to the air (also relative to the ground if the wind is calm) [2]- [15]. 2 For any translational-lift aircraft L D is maximized if the aircraft is flown at its most energy-efficient angle of attack. In this paper unless otherwise noted we always assume L D to be thus maximized. [For auxiliary information, which may also be helpful in some cases wherein Refs. [2]- [15] 2 are cited later in this paper, see Supplementary Notes 1-6. Also, we discuss these points more thoroughly in Section 3. Wherever helpful, refer to Decker, J.S. (2014) See How It Flies at https://www.av8n.com/how/.] Comparing Equations (1) and (2), if 2 L D > , then . Hence short aerodynamic level flight with 2 L D > is more energy-efficient than short ballistic flight beginning and ending at the same altitude, neglecting air resistance in the latter. This requirement 2 L D > is met by practically all aircraft (including all birds and flying insects), even by aerodynamically inefficient ones [2]- [15]. Thus short aerodynamic level flight is practically always more energy-efficient-in most cases considerably more energy-efficient-than short ballistic flight, even neglecting air resistance in the latter. But air resistance is not always negligible for short ballistic flight, especially in the lower atmosphere. If it is not neglected, then the requirement is even milder, namely 2 L D n > with 1 n < . Hence not neglecting air resistance in short ballistic flight, short aerodynamic level flight is all the more energy-efficient than short ballistic flight.
By Equations (1) and (2), if air resistance in short ballistic flight can be neglected, then both ,short B E and A E are directly proportional to mg. Hence reducing mg reduces the energy cost of both short ballistic flight and short aerodynamic level flight traversing given horizontal distance X equally and in direct proportion to the reduction in mg, but does not alter the ratio of energy costs between these two modes of flight. If air resistance in short ballistic flight cannot be neglected, then reducing mg reduces the energy cost of short ballistic flight less than in direct proportion to the reduction in mg.
Most typically, mg is reduced by reducing m. But we can also consider reduction of g. Two examples: (i) Aerodynamic level flight on Mars is at lower g.
(Of course, for Mars 3390 km R  .) (ii) An aircraft of mass m a fraction f ( 0 1 f < < ) of whose weight mg is offset by buoyancy can be construed as either being of effective mass ( ) in a gravitational field g or as being of mass m in a gravitational field of effective strength ( ) 1 g f − . Such partial offset of weight by buoyancy obtains, for example, for a dirigible or blimp that relies on buoyancy for only part of its lift, with the balance obtaining aerodynamically, or for a hydrofoil that cruises so slowly that it must rely on buoyancy for a The first step in the last line of Equation (3) The energy cost establishing of circular orbital speed at r, which equals the circular-orbital kinetic energy at r, is half as great: 4 Thus at all r R ≥ , escape E from radial distance r from Earth's center to infinity equals the energy required for lifting through vertical distance r from radial distance r to radial distance 2r from Earth's center if g had remained constant and equal to its value at r rather than decreasing with increasing distance from Earth.
And kin,orbit E for circular orbit at r is half as great. Hence also at all r R ≥ the free-fall velocity through vertical distance r from radial distance 2r to radial distance r from Earth's center if g had been constant and equal to its value at r equals the escape velocity from radial distance r from Earth's center. Thus the "gravitational scale height" at all r R ≥ is equal to r itself-specifically, at r R = it is equal to R itself [16] [17] [18]. 4 The " 0 r = gravitational scale height" corresponding to escape through a borehole from the center 0 r = of a spherical gravitator of uniform density (which Earth is not) is 3/2 times that from R. Hence the energy cost for escape through a borehole from the center 0 r = of a spherical gravitator of uniform density is 3/2 times that from R. This is perhaps most easily understood if one observes that within a uniform-density spherical gravitator, i.e., at 0 r R ≤ ≤ , g r ∝ . Therefore the average value of ( ) g r within R equals ( ) 2 g R . Thus the portion of an escape trip from 0 to the " 0 r = gravitational scale height", the portion at and beyond R contributes R itself, total 3 2 R . Thus the portion of an escape trip from 0 r = within R costs 1/2 as much energy (1/3 of the total) as the portion at and beyond R (2/3 of the total). Thus Escape velocity from R is ( ) Comparing Equations (3) and (6) . Hence single-circumnavigation aerodynamic level (fixed-altitude) flight with 4π L D > is more energy-efficient than single-circumnavigation minimum-altitude circular-orbit ballistic spaceflight, neglecting the very small air resistance in the latter. This requirement 4π L D > is met by many, perhaps most, but not all aircraft. It is met by soaring birds such as albatrosses [2]- [15]. It is very easily met by sailplanes [2]- [15]. Thus the longest possible (single-circumnavigation) aerodynamic level flight whose purpose is to reach a destination (the starting point after a round-the world trip) on Earth is in many, perhaps most, but not all cases more energy-efficient than the corresponding (single-circumnavigation) minimum-altitude circular-orbit ballistic spaceflight. This requirement 4π L D > neglects air resistance in minimum-altitude circular-orbit ballistic spaceflight; if it is not neglected, then the requirement is weakened to minimum-energy ballistic spaceflight is a circular orbit just above appreciable atmosphere at altitude B h , it cannot begin and end at the altitude A h of aerodynamic level flight, but .
Generalizing the third-to-last paragraph of Section 2.1 in light of this Section 2.2, by Equations (1)- (3) and (6) E remains fixed at the value given by Equation (3) for 1 N = . For, even at minimum-orbital spaceflight altitude, air resistance is almost negligible, i.e., space is almost frictionless; thus the energy cost of launching a spacecraft is one-time. Hence for flights traversing N circumnavigations of Earth L D must exceed 4πN if aerodynamic level (fixed-altitude) flight is to be more energy-efficient than minimum-altitude circular-orbit ballistic spaceflight. But for even the best currently existing sailplanes L D values are between 70 and 80 [2]- [15], and 100 L D ≈ may represent the practicable limit that can be achieved even with the most advanced aerodynamic technology [2]- [15]. Hence even minimum-altitude ( B h R  ) ballistic N-circular-orbit ( 1 N  ) spaceflight about Earth is incomparably more ener-gy-efficient than N-circumnavigation aerodynamic level (fixed-altitude) flight within Earth's atmosphere. This is even more strongly true with respect to high-altitude spaceflight for which air resistance is even more negligible, even with the one-time energy cost of launch being larger than for minimum-altitude circular-orbit spaceflight (recall Section 2.2). 4 In return for a one-time energy expenditure, spaceflight 2π X R→ ∞ even for minimum-altitude circular-orbit spaceflight, spaceflight 2π X R→ ∞ even more strongly for high-altitude, say geosynchronous, circular-orbital spaceflight, and spaceflight 2π X R→ ∞ even more strongly yet for spaceflight exceeding escape velocity. This of course is simply owing to space being essentially frictionless, and increasingly frictionless with increasing altitude, thus allowing spacecraft but not aircraft to take full advantage of Newton's first law of motion (inertia). 5 The energy cost of speed in spaceflight is one-time; the energy cost of speed in aerodynamic flight is never-ending. 5 Spaceflight is thus the only mode of transportation that can achieve mi gal km l ∞ = ∞ of fuel (or the equivalent thereof)-Spaceship Earth (whose fuel for its orbital and rotational motions was part of the solar nebula's kinetic energy) is a good example. 5 To save time in spaceflight continuous energy expenditure can be employed, for example employing solar, laser, or on-board nuclear energy. But in spaceflight continuous energy expenditure buys acceleration; in aerodynamic flight it buys only (constant) speed. 5

Flight Energy Efficiency versus Engine Energy Efficiency
The energy efficiency of flight per se should not be confused with the energy efficiency of the engine that powers flight. If an aircraft engine is a heat engine operating in a cycle with heat input at temperature hot T and heat exhaust at temperature cold T , then its thermodynamic efficiency even assuming perfect reversibility is limited by the Carnot bound ( ) cold hot 1 T T − . 6 Of course any real cyclic heat engine is less than perfect and hence its actual thermodynamic efficiency is less than the Carnot 6 bound. But the actual thermodynamic efficiency of any real cyclic heat engine under any given conditions, while less than the Carnot bound ( ) cold hot 1 T T − for any given cold hot T T , nonetheless, all other things being equal, like the Carnot 6 bound increases monotonically with decreasing cold hot T T . For example, the Curzon-Ahlborn efficiency at maximum power output assuming endoreversibility (irreversible heat flows directly proportional to finite temperature differences but otherwise reversible operation), ( ) See also the following article (most recently revised in 2019) at https://www.wikipedia.org: "Endoreversible thermodynamics". sound, etc. (While Curzon and Ahlborn derived the Curzon-Ahlborn efficiency independently [19], it had been derived previously [20]. 7 ) Thus if an aircraft engine is a cyclic heat engine, then this engine will operate most efficiently at the altitude where the atmosphere is coldest, most typically at or near the tropopause, but commonly as close to the surface as is safe in polar regions in winter. This of course assumes that the engine, if air-breathing and operating at altitude, is supercharged, and that the supercharger requires only a very small fraction of the engine's power output. The oxygen available to an air-breathing engine is of course directly proportional to air density ρ , so intuitively it would seem that so would be the engine's power output. But actually with increasing altitude A h the power output of an unsupercharged air-breathing engine decreases slightly faster than ρ . 8 Even a decrease in power output directly proportional to ρ in almost all cases more than offsets any increase in power output owing to increased thermodynamic efficiency on account of lower temperatures that typically obtain at higher altitudes. 6,7 (A real cyclic heat engine may be difficult to start in cold weather, and its efficiency immediately at starting may be reduced by the high viscosity of still-cold lubricants, but upon attaining steady-state it will operate more efficiently than in hot weather.) In any case, the energy that must be supplied to an engine whose efficiency is  in order to facilitate aerodynamic flight requiring energy A E is of course A E  . And likewise in the case of ballistic flight it is of course B E  . Of course, neither nonheat engines such as electric motors and birds' flight muscles nor noncyclic (necessarily single-use) heat engines such as rockets are limited ultimately by the Carnot bound, nor are they limited at maximum power output assuming endoreversibility by the Curzon-Ahlborn bound. (Even if a rocket engine is refurbished, each launch represents a separate single use.) Their Carnot efficiencies and even their Curzon-Ahlborn efficiencies can in principle approach 100% irrespective of cold hot T T and indeed of temperature at all. But, more often than not, in practice these engines face other limitations. Electric motors typically in practice rather than merely in principle exceed 90% efficiency. But birds' flight muscles are in practice typically considerably less efficient, typically in the range of 25% to 40%. And even if noncyclic, single-use rocket heat engines can in practice rather than merely in principle approach 100% efficiency irrespective of cold hot T T and indeed of temperature at all, typically most of their work output must be expended in accelerating exhaust gases, with only a small fraction available for accelerating payloads. 9 We should note that: (a) Even if rocket heat engines operated in a cycle, owing to their very small cold hot T T ratio their Carnot and even Curzon-Ahlborn bound would be nearly 100%. But it would still obtain that typically most of their work output must be expended in accelerating exhaust gases, with only a small fraction available for accelerating payloads. 9 (b) Not all rocket engines are heat engines, not even noncyclic ones. For example, ion-drive rocket engines are not heat engines, not even noncyclic ones, and hence (like noncyclic heat engines) are not limited ultimately by the Carnot bound, nor at maximum power output assuming endoreversibility by the Curzon-Ahlborn bound. Because of their high exhaust speeds less mass need be exhausted to achieve a given spacecraft speed, and hence less of their work output need be expended on the exhaust. 9 (c) Nonrocket spacecraft propulsion, 9 e.g., via solar sails or laser sails, or via the EM, MEGA or related drives if they are verified, 9 is not limited ultimately by the Carnot bound, nor at maximum power output assuming endoreversibility by the Curzon-Ahlborn bound.
As an aside, we mention that the ratio of the Curzon-Ahlborn efficiency to the Carnot efficiency, , decreases monotonically with increasing cold hot T T from unity in the limit cold hot 0 T T → to 1/2 in the limit cold hot 1 T T → . Thus this ratio is never greater than 1 or less than 1/2. The former limit is obvious. The latter limit is most easily verified by setting Of course, determination of how closely any given engine approaches to its theoretical maximum efficiency requires detailed analyses of the properties of that particular engine (e.g., bypass ratio of jet engines, battery and circuit design for electric motors, metabolic chemistry of birds' flight muscles, etc.) We have not considered such detailed analyses in this Section 2.4.

Functional Dependency
Drag is given by [2]- [15] 2 2 frontal,geom frontal,eff where D C is the coefficient of drag, frontal,geom A is an aircraft's geometrical frontal cross-sectional area, frontal,eff A is its effective frontal cross-sectional area (the frontal cross-sectional area that it effectively presents with respect to air resistance or drag), ρ is the air density, and v is the airspeed (also the ground speed if the wind is calm) of flight. [Note: The symbol A denoting surface area should not be confused with the subscript A, introduced in the first paragraph of The drag owing to an aircraft's frontal cross-sectional area per se is pressure drag. Pressure drag times frontal cross-sectional area is the force that an aircraft must impart to push air in front out of its way, and to overcome the pull of the partial vacuum behind it ensuing because adjacent air cannot move in behind it instantaneously. But there are two other components of drag: induced drag-a penalty that must be paid for lift (see Supplementary Notes 6 and 9), and skin-friction drag-owing to the viscosity of air rubbing against surfaces parallel to the airflow (see Supplementary Note 10). The contributions of induced drag and skin-friction drag are included along with that of pressure drag within frontal,eff A and hence within D C . Thus our more complete definitions of frontal,eff A and hence of D C : frontal,eff A is an aircraft's effective frontal cross-sectional area-the frontal cross-sectional area that it would effectively present with respect to total air resistance or drag-as if total drag had been subsumed within pressure drag, i.e., as if induced drag and skin-friction drag had been converted to and incorporated within pressure drag. Since frontal,geom A is fixed for any given aircraft flying at any given angle of attack, by Equations (7) and (8) thus construing induced drag and skin-friction drag as incorporated within pressure drag modifies D C in direct proportion to the modification of frontal,eff A from its value with respect to pressure drag alone. Accordingly, we thus construe Equations (7) and (8)  with that understood, Equations (7) and (8) where L C is the coefficient of lift, wing,geom A is an aircraft's geometrical wing area, and wing,eff A is its effective wing area-the area that it effectively presents with respect to downward deflection of air required by Newton's third law of motion 11 as the price for the upward force that is its lift. The first step of Equation (9) recognizes that to maintain aerodynamic level flight L must equal the weight mg of an aircraft. We use "wing area" as shorthand for an aircraft's entire lifting-surface area. A typical airplane obtains most but not all of its lift from its wings; its fuselage and elevators contribute some lift. The spectrum of airplane design ranges from flying wings with little or no fuselage to lifting bodies that are fuselage with little or no wing. 12 Flying wings are based on the principle that the wing has a higher L D ratio than any other part of an airplane, while lifting bodies seek to avoid structural stresses on wings, especially at high airspeeds. 12 Thus the coefficient of lift L C is given by Lift is a complex phenomenon, perhaps even more so than drag [2]- [15].
There seems to be no universal agreement concerning which explanation or combinations of explanations of lift is most correct [2]- [15]. (See also Supplementary Notes 1-6.) Equations (9) and (10)  the revolving airfoils. The revolving airfoils' geometrical area with respect to lift is their geometrical wing (not frontal) area, their effective area with respect to lift is their effective wing (not frontal) area; v with respect to lift is the root-mean-square average, taken over the geometrical wing area of the airfoils, of airspeeds of the airfoils, be they blades of a helicopter, or wings of a hovering hummingbird or insect.] The forms of Equations (7)-(10), in particular the 2 v ρ functional dependency in Equations (7) and (9) that is paramount for both drag and lift, can perhaps in some measure be most easily physically understood via the following very simplified qualitative to semiquantitative arguments. First, note that both drag and lift are forces, and that is the only combination of ρ , v, and A that has dimensions of force, or equivalently that 2 v ρ is the only combination of ρ and v that has dimensions of force per unit area (= pressure). Construe, as discussed three paragraphs previously, induced drag and skin-friction drag as converted to and incorporated within pressure drag, so that total drag is construed as pressure drag. Pressure drag times effective frontal cross-sectional area The upward force of lift L is by Newton's third law of motion 11 equal to the downward force-the downward momentum per unit time t-that an aircraft's wings must impart to air. By similar reasoning as we employed with respect to drag, the wings must impart to air downward force equal to L, in accordance The first step of Equation (12) recognizes that to maintain aerodynamic level J. Denur flight L must equal the weight mg of an aircraft [2]- [15]. Note that the first two lines of Equations (11) and (12) are, essentially, statements of Newton's third law of motion 11 , and hence are rigorously correct. Approximations are employed only in the last two lines of Equations (11) and (12).
If L C is independent of ρ and of v, as is typically approximately true at or near the most energy-efficient angle of attack and hence at or near airspeeds v corresponding to maximization of L D , then induced drag, as pressure drag, is approximately proportional to 2 v ρ , and hence can be incorporated within Equations (7) and (8) (7) and (8) via a contribution to the functional dependence of D C on ρ and on v. This is in accordance with Equations (7) and (8) The fifth steps of both lines of Equation (13) recognize that to maintain aerodynamic level flight L must equal the weight mg of an aircraft [2]- [15]. Note that A P is the power required for aerodynamic level flight, not to be confused with the power available. 13  Because it is necessary to fly at faster v in thinner air to maintain L equal to the weight mg of an aircraft and hence to maintain aerodynamic level flight despite smaller ρ , an aerodynamic level flight traversing given horizontal distance X requires less time t X v = in thinner air, so the energy cost of aerodynamic level flight traversing given X, or equiva- if L is to equal the weight mg of an aircraft: higher 2 v required for L to equal mg despite smaller ρ increases D as much as smaller ρ itself decreases D.
Hence to maintain aerodynamic level flight the required power A P , as the required airspeed v, is proportional to But flight time . Therefore in accordance with Equations (2) and (13)  In the four immediately preceding paragraphs we did not explicitly consider the effects of changing mg. But (recall the third-to-last and second-to-last paragraphs We note that Equation (14) is consistent with A E being directly proportional to mg in accordance with Equation (2). For in accordance with Equation (14) ( )

Functional Dependency
Lift and drag are in general not exactly proportional to But again at or near the most energy-efficient angle of attack and thus at airspeeds v at or near that corresponding to maximization of L D C C and hence of L D , typically these contributions to nonconstancy are small. Also for pressure drag the proportionality to 2 v ρ is usually approximate rather than exact.
Thus in general L C and D C vary with ρ (and hence with A h ) and with v, and hence with Reynolds number [2]- [15] and with Mach number [2]- [15], and furthermore in general not at exactly the same rate, so also their ratio L D C C is not strictly constant [2]- [15]. In general these variations are typically small compared with the paramount 2 v ρ functional dependency (see Supplementary Notes 4, 5, 6, 9, and 10). But three major exceptions, wherein there are large departures from constancy of L C , D C , and in some cases also L D C C , should be noted: (a) For aircraft, especially those with short wingspans, trying to maximize L at the slowest possible v well below that corresponding to maximization of L D by trying to maximize L C whatever the cost in D C , for example in trying to land at the slowest possible v, induced drag is likely the largest component of the total drag D. In such cases, owing to large induced drag, total drag D is likely considerably larger than would typically be expected for given plementary Notes 6 and 9). In regards to reducing induced drag, we should mention wingtip devices [21] [22] (see also the second paragraph of Supplementary Note 9). (b) As v increases through the typically small range of values corresponding to the transition from laminar to turbulent flow, the onset of turbulence helps to fill the trailing partial vacuum, thereby greatly reducing pressure drag and hence D C . This decrease in D C owing to the reduction in pressure drag is often sufficient to more than offset not only the increase in D C owing to the onset of turbulence itself, but over this typically small range of values of v also the increase in pressure drag and indeed in total drag D proportional to 2 v if D C had remained constant. Hence, over this typically small range of values of v, D C often manifests a net decrease with increasing v faster than 2 v increases, so  [26]. Assists for laminar-to-turbulent transition are also employed, for example, in fluid-dynamic modeling and in small aircraft (see Supplementary Note 11). (c) As Mach 1 (the speed of sound) is approached from below, D C typically manifests a sharp peak, followed by a sharp dip at values of v slightly above Mach 1. The extra drag due to shock waves at and in the vicinity of Mach 1 is referred to as shock-wave drag (or sometimes simply as wave drag). (See Supplementary Note 12.) It should be noted that, in the design of aircraft, even departures from the paramount 2 v ρ functional dependency of lift and/or drag smaller than those discussed in Items (a), (b), and (c) of the immediately preceding paragraph can yield modest but still significant improvements in aircraft energy efficiency [27] [28]: 14 See the two immediately following paragraphs.
Not uncommonly, aerodynamic level flight at a given v is more energy-efficient at lower ρ and hence at higher angle of attack than ρ decreases. 14 Of course this obtains only up to a limit: with continued increase in angle of attack L D C C and hence L D decreases ever more rapidly until stalling occurs. Higher-altitude aerodynamic level flight while maintaining the (smaller) angle of attack that maximizes L D would be at sufficiently faster v to decrease the flight time t more than it increases the required power A P , and hence would increase energy efficiency But the engine(s) may not be capable of the required increase in A P . Even if they are, in some cases increased v may be detrimental [e.g., owing to encountering shock-wave drag if Mach 1 is approached too closely (see Supplementary Note 12), or to excessive aerodynamic heating]. A specific example of this (owing to encountering shock-wave drag if Mach 1 is approached too closely) is discussed in the immediately following paragraph.
As a specific example, the energy efficiency (distance X per unit of fuel relative to the air, also relative to the ground if the wind is calm) of older commercial jet airliners was ≈43% higher at 35,000 ft to 40,000 ft (≈5 mi/100 lb fuel) than at 20,000 ft (≈3.5 mi/100 lb fuel) [27]. Assuming Curzon-Ahlborn [19] [20] engine efficiency 7 ( ) the figure is only ≈5%, but the Curzon-Ahlborn [19] [20] engine efficiency 7 is a more realistic estimate for real-world engines.] Thus improved aerodynamic efficiency rather than improved engine efficiency must have contributed a factor of ≈1.43/1.07 ≈ 1.34 to the improved energy efficiency at 35,000 ft to 40,000 ft over and above that at 20,000 ft. Perhaps improved L D ratios at given angles of attack at lower ρ and hence at higher A h might contribute somewhat. But the major contribution to this improved energy efficiency-the only contribution if, as is usually at least approximately true, L D ratios at given angles of attack are independent of ρ and hence of A h -is the reduced power A P and hence reduced energy A E required to traverse a given distance X at a given v at lower ρ and hence at higher A h . This is not uncommon, despite the required increase in angle of attack to above that which maximizes L D C C and hence L D , consequently decreasing L D C C and hence L D , as the penalty for increasing L C itself and hence L itself sufficiently to maintain L mg = in the face of decreased ρ at fixed v [recall Equations (9) and (10)]: up to a limit, L D C C and hence L D decreases more slowly with the required increase in angle of attack than ρ decreases. 14 Of course this obtains only up to a limit: with continued increase in angle of attack L D C C and hence L D decreases ever more rapidly until stalling occurs. Higher-altitude aerodynamic level flight while maintaining the (smaller) angle of attack that would ordinarily maximize L D would ordinarily also be at sufficiently faster v to decrease the flight time t more than it increases the required power A P , and hence would ordinarily increase energy efficiency Open Journal of Fluid Dynamics A A A E X P t X P v = = even more. But the engine(s) may not be capable of the required increase in A P . Even if they are, in this case increased v would be sufficiently close to Mach 1 to encounter shock-wave drag and consequently increased D and thus decreased L D even at the optimum angle of attack, and hence also an increase in A P required to overcome the shock-wave drag over and above that owing to increased v. [See Item (c) of the first paragraph of this

The Low-Density/High-Altitude Limit of Aerodynamic Level Flight in Earth's Atmosphere
If ρ is so small that even the minimum airspeed required for aerodynamic level flight equals or exceeds the speed The minimum airspeeds and hence also (since this power must be frictionally dissipated) the consequent frictional aerodynamic heating, is less by a factor of (1/4) 1/2 = 1/2 with the help of centrifugal force than without its help. But, at any rate, in Section 3.2 and the last three paragraphs of Section 3.3.1, we give only in-the-ballpark estimates.  15 Hence all flight about Earth must then be ballistic rather than aerodynamic. 15 This estimate of the low-density/high-altitude limit for aerodynamic level flight about Earth is an ultimate limit that neglects all practical difficulties. To re-emphasize, the most obvious and most general of practical difficulties are the required power A P and especially the consequent frictional aerodynamic heating, the latter being equal to A P because A P is ultimately thermally dissipated via frictional aerodynamic heating. Other practical difficulties, which we do not consider, include the reduction of the power available from air-breathing engines with decreasing ρ and hence with increasing A h (obviously aerodynamic level flight is possible if and only if the maximum available power exceeds, or at the very least equals, the required power A P ), 13 and practical difficulties that are specific for given types of aircraft, e.g., maximum airspeeds for propeller airplanes, 16 and minimum and maximum airspeeds for various types of jets. 17 By the last five paragraphs of Section 3.1.1, A P , and hence also the consequent frictional aerodynamic heating and the required rate of heat dissipation, is, at least approximately, proportional to 1 2 ρ − [2]- [15]. Thus the practical low-density/high-altitude limit of aerodynamic level flight is considerably more conservative than the ultimate limit. The altitude record for aerodynamic level flight in Earth's atmosphere as of this writing, 124000 ft 38 km ≈ ≈ , corresponds to ρ slightly smaller than . (This is also approximately the practical low-density/high-altitude limit of balloons in Earth's atmosphere as of this writing. 18 ) Note that, by Equation (2) and the last six paragraphs of Section 3.1.1 (neglecting exceptions to the paramount 2 v ρ functional dependency as per Section 3.1.2), the energy A E required for aerodynamic level flight of an aircraft of weight mg traversing given distance X is not greater at the practical-or even the ultimate-low-density/high-altitude limit of aerodynamic level flight than at sea level. Difficulties arise only because A E must be expended faster and hence thermally dissipated faster (frictional aerodynamic heating!) in thinner air-  14). But this density is nevertheless substantial enough for Mars airplanes to be within the practical, not merely ultimate, limit of aerodynamic level flight, especially given that Mars' surface gravity is only ≈0.38 that of Earth. By the last six paragraphs of Section 3.1.1, if all other things except ρ were equal (e.g., same airplane, same angle of attack, etc.), then a Mars low-altitude airplane must fly ( ) times as much power to sustain aerodynamic level flight. Even this is within the practical, not merely ultimate, limit of aerodynamic level flight. But we must still consider one important factor that is unequal in favor of Mars-Mars' weaker gravity: g on Mars is only ≈0.38 of g on Earth [35] [36] [37]. Thus by the first line of Equation (14), all other things except ρ and g being equal (e.g., same airplane, same angle of attack, etc.), to maintain aerodynamic level flight airspeed ( )  Aerodynamic level flight at lower ρ as on Mars while maintaining the (smaller) angle of attack that maximizes L D would be at sufficiently faster v to decrease the flight time t more than it increases the required power A P , and hence

Level Flight in Rarefied and
At

density is approximately equal in Mars' and
Earth's atmospheres. At all higher altitudes, Mars' atmosphere is denser than Earth's, and in increasing ratio  with increasing altitude h.
Thus the practical low-density limit of aerodynamic level flight in Earth's at- This is also approximately the practical low-density/high-altitude limit of balloons in Mars' atmosphere. The practical low-density limit of buoyant flight Open Journal of Fluid Dynamics is equal in Earth's and Mars' atmospheres because lower g on Mars reduces both weight and buoyancy by the same factor of ≈0.38. Indeed for this reason it is 3 3 10 kg m ρ − ≈ in any atmosphere of any planet. This is strictly true for buoyant flight via vacuum airships, which obtain buoyant lift equal to atmospheric density ρ per unit volume of (essentially) perfect vacuum, or Since for a lifting gas  equals the ratio of the molecular weight of the lifting gas to that of the ambient atmosphere, unlike for vacuum airships (or partial-vacuum airships such as hot-air balloons)  is larger for a given lifting gas in a higher-molecular-weight atmosphere such as Mars' atmosphere than in a lower-molecular-weight one such as Earth's. (Obviously, 0 1 <   is required if the practical low-density limit of buoyant flight  This practical low-density/high-altitude limit considers A P and consequent frictional aerodynamic heating: A P must be thermally dissipated. (Note that while the practical high-altitude limits of aerodynamic level flight and of buoyant flight are approximately equal in Earth's atmosphere, 18 the former is considerably higher than the latter in Mars' atmosphere. As noted in the immediately preceding paragraph, the practical high-altitude limit of buoyant flight corresponds to    It is perhaps worthwhile to re-emphasize that, by Equation (2)  Earth as anywhere-   (2) and the last six paragraphs of Section 3.1.1, our Earth underwater airplane requires the same energy A E as our standard (atmospheric) Earth airplane to traverse a given horizontal distance X.

Underwater Airplanes on Earth and (Hypothetically) on Mars
Recalling the last six paragraphs of Section 3.1.1 and that g on Mars is ≈0.38 of g on Earth, if atmospheric pressure on Mars was high enough for liquid water to exist, then both the required flight speed and the required power for aerodynamic flight of a Mars underwater airplane would by Equation (14)  for liquid water to exist in the past [37] but perhaps by a smaller margin than generally believed [38]. 19 ) Also, by Equation (2)

Dual-Density Flight: Hydrofoils
Hydrofoils should be classified as aircraft because their lift obtains primarily if not essentially entirely aerodynamically rather than via buoyancy, even though the density ρ of their aerodynamic medium (water) is ≈800 times that of air at sea level. By lifting the hull out of the water into the air, drag on the hull at any given speed is reduced ≈800 times; only the wings need suffer water resistance as opposed to air resistance. (We define "hull" as incorporating all parts of a hydrofoil that are lifted out of the water, e.g., including most or all of the struts.) Thus, probably uniquely among aircraft, for hydrofoils two values of aerodynamic-medium density ρ are pertinent: drag ρ , the density pertinent to drag, and lift ρ , the density pertinent to lift. Also, probably uniquely among aircraft other than hovercraft, for hydrofoils unless there is neither wind nor water current two values of velocity v are pertinent: drag v , the velocity pertinent to drag, 19 Our knowledge of Mars is increasing rapidly. See in addition to Ref. [38] itself relevant articles that cite Ref. [38]. For an overview, see the following article (most recently revised in 2019) at https://www.wikipedia.org: "Mars". and lift v , the velocity pertinent to lift. Let air ρ be the density of air, water ρ be the density of water, air v be the velocity of a hydrofoil relative to the air, water v be the velocity of a hydrofoil relative to the water, ( ) This average value 2 drag v ρ can be employed in Equation (7). Note that, in Equation (18), frontal,eff,wing A is the effective frontal cross-sectional area of a hydrofoil's wings with respect to drag as per the first two paragraphs of Section 3.1.1, not the effective surface area of its wings with respect to lift as per the third and fourth paragraphs of Section 3.1.1. Since typical speeds of hydrofoils are ~5 times typical wind speeds, 20 water v and air v by and large differ by ~20%. (Water currents are usually much slower than winds.) In contrast with Equation (7) for drag, Equation (9) for lift requires no reinterpretation for hydrofoils other than setting water ρ ρ = and Because water is ≈800 times as dense as air at sea level, the wings of a hydrofoil need only have ≈1/800 of the surface area with respect to lift as the wings of a (low-speed) airplane in order to obtain the same lift (if L C is the same for both the hydrofoil and the airplane, and if water v for the hydrofoil equals air v for the airplane).
Any surface watercraft experiences an additional form of drag that we do not consider in this paper: wave drag, the energy cost of generating waves (of course not to be confused with shock-wave drag experienced by aircraft at and in the vicinity of Mach 1). 21 But, because only the wings and at most only the lower part of the struts of a hydrofoil, which have minimal surface area, intersect the surface of the water, this form of drag is minimal for a hydrofoil if, as we assume, it cruises at sufficient speed to lift all but its wings and at most the lower part of its struts completely out of the water.

The Optimum Range of Air Densities for Aerodynamic Level Flight
Hydrofoils excepted, the range of aerodynamic-medium (air) densities on Earth, from ≈1 kg/m 3 at sea level to ≈10 −3 kg/m 3 at the approximate practical (as opposed to ultimate) high-altitude limit of aerodynamic level flight, seems to be optimum for aerodynamic level flight, indeed for any aerodynamic flight. Much denser aerodynamic media, such as water in the case of underwater airplanes discussed in Section 3.3.2, allow aerodynamic level flight with much less power, but owing to the great resistance of a very dense medium such as water the speed of aerodynamic level flight will then be relatively slow. (Even hydrofoils are much slower than most airplanes.) Much more rarefied aerodynamic media allow higher speeds but entail difficulties of large required power and consequent dissipation of this power via equally large frictional aerodynamic heating, as discussed in Section 3.2 and the last paragraph of Section 3.3.1. As discussed in Section 3.3.1, lower g such as on Mars reduces the practical low-density limit somewhat from ≈10 −3 kg/m 3 as obtains on Earth. But if g is too small, appreciably smaller than on Mars, then an atmosphere cannot be retained at all, thus precluding aerodynamic level flight, indeed precluding any aerodynamic flight (except in pressurized indoor facilities). The considerations of the immediately preceding paragraph are modified in the case of hydrofoils, because they obtain lift via a dense medium (water) but the vast majority of their frontal cross-sectional area (both geometrical and effective) suffers resistance or drag only from a much more rarefied one (air). Of course, if g is too small, then an atmosphere cannot be retained, and with vanishing atmospheric pressure liquid water-indeed any liquid-cannot exist (except in pressurized indoor facilities).

Farther against the Wind Than with it
All flights of hand-thrown projectiles that are unpowered except for the initial throw are obviously short flights as defined in Section 2.1. Thus it is not surprising that the record traversed horizontal distances for hand-thrown projectiles obtain for those executing aerodynamic flight, e.g., Frisbees, Aerobies, and boomerangs, as opposed to those executing flight that is at least primarily ballistic, e.g., sports balls [23] [24] [25] [26] [39] [40] and even more so javelins [41] (see also Supplementary Note 15). For all flights of hand-thrown projectiles: (a) the flight distance assumes that the flight is above a horizontal (level) surface, most typically level ground but possibly a smooth water surface, and (b) the flight distance is taken as the total horizontal distance traversed along the flight path, not the straight-line distance between the beginning and ending points of the flight. The latter can be short (or even zero) even for a long return-boomerang flight.
From among hand-thrown projectiles, we define as hand-thrown aircraft We should also mention the discus [42] [43] and hand-thrown (e.g., paper or balsa) gliders [44]. The discus is a hand-thrown aircraft, whose aerodynamic lift can equal or exceed its weight at achievable throwing speeds [42] [43], even though because discuses are considerably more massive than Frisbees, Aerobies, and boomerangs their distance records are much less. Indeed their distance records are less than those of sports balls and even of javelins. As of this writing, the official discus-throw flight-distance records are We use the term "paper glider" rather than "paper airplane" (or "paper plane" for short) because such hand-thrown aircraft are unpowered except for the initial throw and hence are gliders, not airplanes.] We now describe what may be the simplest example of how a hand-thrown aircraft capable of flight with aerodynamic lift exceeding weight at achievable throwing speeds, such as a discus, Frisbee, Aerobie, or boomerang, can maintain its altitude farther if thrown horizontally against the wind than with it. Let a discus, Frisbee, Aerobie, or boomerang of mass m and weight mg be thrown horizontally in calm air ( wind 0 v = ) at altitude Since a hand-thrown aircraft such as a discus, Frisbee, Aerobie, or boomerang is unpowered except for the initial throw, immediately after being thrown aerodynamic drag D will have reduced air v (and given calm air also ground v ) to less than air,min v and hence lift L to less than weight mg.
with the wind. By Section 3.1.1, especially Equations (7) and (8) and the associated discussions, typically D is greater for a discus, Frisbee, Aerobie, or boomerang thrown against the wind than with it by a ratio is a finite number, typically only moderately larger than unity. Moreover, by Equations (9) and (10) and the associated discussions, typically L is also greater for a discus, Frisbee, Aerobie, or boomerang thrown against the wind than with it by a comparable ratio . Therefore typically L D and hence L D C C is at least approximately equal with the wind and against the wind (and also with no wind). Hence drag  typically being moderately larger than unity does not contravene our result that it is possible for a discus, Frisbee, Aerobie, or boomerang to maintain altitude farther against the wind than with it. In our specific examples, altitude is maintained for finite horizontal distance 0 X > against the wind, as opposed to infinitesimal horizontal distance 0 X = with no wind and not even infinitesimal horizontal distance 0 X = with the wind-in all three cases the initial horizontal throw being at ground air,min   least not relative to the ground. For example, consider throwing a discus, Frisbee, Aerobie, or boomerang into a Category 5 extreme-hurricane-force or an EF5 extreme-tornado-force head wind. 23 It will still traverse finite horizontal thrown horizontally against the wind-but relative to the air, not relative to the ground. Such an extreme wind will reverse the hand-thrown aircraft's direction of motion relative to the ground almost instantaneously, well before it traverses this finite horizontal distance 0 X > (peaking at altitude to the ground. By contrast, hand-thrown projectiles whose flights are primarily ballistic, such as javelins [41] or even sports balls (e.g., golf balls, baseballs, etc.) [ [40], cannot maintain altitude farther against the wind than with it at achievable throwing speeds, irrespective of wind speed. For upon such primarily-ballistic hand-thrown projectiles, air imposes drag but, at achievable throwing speeds, provides at best insufficient lift to equal, let alone exceed, weight and perhaps in some cases no lift at all (see also Supplementary Note 15).
Thus far in this Section 4, we considered hand-thrown projectiles on Earth.  times that for identical hand-thrown projectiles on Earth. Thus aerodynamic flight of hand-thrown discuses, Frisbees, Aerobies, and boomerangs is at the very least much more difficult to achieve on Mars than on Earth, and may be impossible to achieve on Mars. In order for low-altitude achievable throwing speeds of hand-thrown discuses, Frisbees, Aerobies, and boomerangs on Mars to equal those on Earth, by the first line of Equation (14) their masses would have to be reduced by a factor of ( ) 0.38 1 70 0.38 70 27 ≈ ÷ = × ≈ while still maintaining their same aerodynamic lifting areas and hence their same sizes and shapes as on Earth. Thus, with respect to hand-thrown aircraft, lower g on Mars is insufficient to compensate for lower air density there.

Energy Efficiency of Surface Transportation versus That of Flight
It may be of interest to compare the energy efficiency of both aerodynamic level flight and ballistic flight with that of horizontal surface (land and/or water) transportation. If the frictional force opposing horizontal motion of a surface vehicle of mass m is a fraction F of its weight mg, then the energy cost of its traversing horizontal distance X is .
(A single-circumnavigation journey is the longest possible one whose purpose is to reach a destination on Earth, with the destination being the starting point after traveling around the world.) Typical values of the coefficient of surface friction S C for land vehicles range from ≈0.01 to ≈1 for sliding friction (≈0.005 for some maglev trains), and as low as ≈0.001 for rolling friction of hard wheels on hard surfaces-for example low-speed to moderate-speed traditional (not maglev) railroad transportation.
For land vehicles at speeds low enough that air resistance is small compared to surface (e.g., sliding, rolling, or maglev) friction, S F C = and is at least approximately independent of speed. For land vehicles at higher speeds S F C D mg = + , D being given by Equation (7) with ρ being the density of air. (While traditional railroad transportation is more energy-efficient at low to moderate speeds, maglev trains almost completely abolish wear on the tracks, owing to their lack of mechanical contact with the tracks.) See Supplementary Note 17.
Since even minimum-altitude circular-orbit ballistic spaceflight must be above any appreciable atmosphere, B r for even minimum-altitude circular-orbit ballistic spaceflight must exceed R. But for simplicity, as in Section 2. ) surface journeys to be more energy-efficient than short ballistic journeys neglects air resistance in ballistic flight. But air resistance is not always negligible in short ballistic flights, especially in the lower atmosphere. If it is not neglected, then the requirement is weakened to 1 2 F n < with 1 n < . The requirement 4π F < for single-circumnavigation surface journeys to be more energy-efficient than single-circumnavigation minimum-altitude circular-orbit ballistic journeys neglects air resistance in minimum-altitude circular-orbit ballistic spaceflight; if it is not neglected, then the requirement is weakened to 1 4π F n < with 1 n < . But it is weakened only very slightly, because air resistance even at minimum-circular-orbit spaceflight altitude is very small. Not neglecting air resistance in ballistic flight, for journeys of intermediate length (X ranging from much smaller than R to approaching 2πR ), the value that F cannot equal or exceed if surface transportation is to be more energy-efficient than ballistic flight decreases monotonically from ( ) 1 2n X towards ( ) 1 4πn X as X increases from very small values towards 2πR : ( ) 1 n X < but increases monotonically towards very nearly 1 as X increases from very small values towards 2πR . Air resistance in minimum-altitude circular-orbit ballistic flight is very small and hence also ( ) For, even at minimum-circular-orbit altitude, air resistance is almost negligible, i.e., space is almost frictionless; thus the energy cost of launching a spacecraft is one-time. Thus for multi-circumnavigation ( 2π X RN = , 1 N  ) journeys, the energy efficiency of even minimum-altitude circular-orbit ballistic spaceflight surpasses that of surface transportation by an arbitrarily large margin, just as it does that of aerodynamic flight by an arbitrarily large margin, the margin being even larger if spaceflight is high-orbit and even larger yet if it exceeds escape velocity. With respect to both aerodynamic flight and surface transportation the reason is that stated in Section 2.3: Space is essentially frictionless, and increasingly frictionless with increasing altitude, thus allowing spacecraft but neither aircraft nor surface vehicles to take full advantage of Newton's first law of motion (inertia). 5 The energy cost of speed in spaceflight is one-time; the energy cost of speed in aerodynamic flight and in surface transportation is never-ending. 5 Spaceflight is thus the only mode of transportation that can achieve mi gal km l ∞ = ∞ of fuel (or the equivalent thereof)-Spaceship Earth (whose fuel for its orbital and rotational motions was part of the solar nebula's kinetic energy) is a good example. 5 To save time in spaceflight continuous energy expenditure can be employed, for example employing solar, laser, or on-board nuclear energy. But in spaceflight continuous energy expenditure buys acceleration; in aerodynamic flight and in surface transportation it buys only (constant) speed. 5 We note that lighter-than-air craft, for example dirigibles and blimps, should be included within the category of surface transportation rather than within the category of aerodynamic flight, because their lift obtains typically at least primarily and often entirely via buoyancy rather than via aerodynamics. Similarly the lift for surface (land and/or water) transportation vehicles obtains via support of the ground for land surface vehicles and via buoyancy for ships and submarines, rather than via aerodynamics. In this regard, we construe maglev and air-cushion vehicles as being surface (land and/or water) transportation vehicles rather than aircraft, because their lift obtains via support of the surface through the intermediary of a magnetic-repulsion or an air cushion. [Aircraft very near the ground obtain some extra lift from the air-cushion "ground effect" (see Supplementary Note 18).] For a dirigible or blimp in level flight F is the ratio of air resistance to the unbuoyed weight of the dirigible or blimp, i.e., F D mg = , D being given by Equation (7) with ρ being the density of air at flight altitude. For a fully-submerged submarine F is the ratio of water resistance to the unbuoyed weight of the submarine, i.e., F D mg = , D being given by Equation (7) with ρ being the density of water. For a ship or surface-cruising submarine, or for a hydrofoil, F is the ratio of combined water and air resistance to the unbuoyed weight of the vehicle, i.e., F D mg = , D being given by Equation (7) with 2 drag v ρ taken as for hydrofoils as per Section 3.4 [see especially Equation (18) and the associated discussions]. (F for land vehicles was discussed in the third paragraph of this Section 5.) By contrast, as previously noted [see especially Section 3.4 but also the second paragraph of Section 1, the second paragraph following that containing Equation (2), and Section 3.5], hydrofoils should be classified as aircraft, because their lift obtains primarily if not essentially entirely aerodynamically rather than via buoyancy, even though the density ρ of their aerodynamic medium (water) is ≈800 times that of air at sea level. (By lifting the hull out of water into air, drag on the hull at any given speed is reduced ≈800 times; only the wings need suffer water resistance as opposed to air resistance.) Any surface-cruising watercraft experiences wave drag, i.e., the energy cost of generating waves (of course not to be confused with shock-wave drag experienced by aircraft at and in the vicinity of Mach 1), 21 which we have not considered in this paper. Wave drag is minimal for a hydrofoil cruising at sufficient speed to lift its hull completely out of the water, but for ships and surface-cruising submarines it is typically the largest component of drag. 21 While this is obvious, perhaps it is worthwhile to note that there is a minimum flight speed for any (nonhovering) aircraft, but no minimum speed for land vehicles, dirigibles, ships, or submarines. Increased induced drag is imposed on (nonhovering) aircraft at minimum flight speed (see Supplementary Notes 6 and 9). By contrast: (a) For land vehicles, as speed is reduced to zero, S F C = remains at least approximately constant. (b) For dirigibles, for ships, and for both fully-submerged and surface-cruising submarines, as speed is reduced to zero, D and hence also F D mg = is reduced to zero. Of course, the remarks of the last paragraph of Section 2.1, and especially the remarks of Section 2.4, distinguishing between the energy efficiency of aerody-namic or ballistic flight per se and the energy efficiency of the engine that powers aerodynamic or ballistic flight apply equally in distinguishing between the energy efficiency of surface transportation per se and the energy efficiency of the engine that powers surface transportation. The energy that must be supplied to an engine whose efficiency is  in order to facilitate surface transportation requiring energy S E as per Equation (19) is of course S E  .
Generalizing the third-to-last paragraph of Section 2.1 and the last paragraph of Section 2. the balance obtaining aerodynamically, for a hydrofoil that cruises so slowly that it must rely on buoyancy for part of its lift, or for an underwater surface vehicle that is denser than water. Because all solids are hundreds to thousands of times as dense as air at sea level, f is negligible for surface vehicles on land as it is for aerodynamic vehicles (aircraft) in air. But f is not negligible for underwater surface vehicles (and for underwater airplanes as discussed in Section 3.3.2) that are denser than water, because even the densest solids are little more than 20 times as dense as water. [An evenhanded comparison, in Section 3.3.2, between a standard (atmospheric) airplane's aerodynamic level flight in air and an underwater airplane's aerodynamic level flight in water was facilitated by an extra 10 3 kg/m 3 of density for the latter to offset the buoyancy provided by water.] In this paper in general we have not considered the energy cost of building and maintaining vehicles. In this Section 5 in particular we also have not considered the energy cost of building and maintaining pathways for surface transportation. Concerning the latter, we have not, for example, considered the energy cost of building and maintaining roads, railroads, and canals. Of course, Open Journal of Fluid Dynamics for transportation that does not require artificially-built pathways, such as transportation in air, on or in water (except via canal), on land and/or water via hovercraft or other surface vehicles that do not require roads, or via spaceflight, this latter energy cost is zero.

Brief Concluding Remarks
Hopefully, we have provided at least somewhat helpful insights concerning energy efficiency in aerodynamic versus ballistic flight, concerning aerodynamic lift and drag, concerning selected aspects and examples of flight, in distinguishing between the energy efficiency of flight per se and the energy efficiency of the engine that powers flight, and via considering the relation between the density of an aerodynamic medium and aerodynamic level flight. Also, hopefully, our comparison with the energy efficiency of surface transportation and our discussion of surface transportation have been helpful.
While we have focused mostly on Earth, with some consideration of Mars, our results are easily generalizable to any planet or other astronomical bodies on which aerodynamic flight and/or surface travel is possible, i.e., to any planet or other astronomical bodies with an atmosphere, and/or a solid and/or liquid surface. Also, they are valid irrespective of the values of M, R, g, m, ρ , and (except as for simplicity we assume A [15].
We should emphasize the limitations of this present work. In this paper our main goal was to elucidate more conceptually than mathematically some fundamental ideas concerning energy efficiency and a number of other aspects of aerodynamic versus ballistic flight, and to provide comparison with surface transportation. We did not attempt the mathematically complex and detailed fully-quantitative analyses based on rigorous application of fluid dynamics, e.g., computational fluid dynamics, as is required in the actual design of aircraft, or the analyses required in the actual design of spacecraft or surface vehicles. We also neglected many details required in the actual operation of vehicles: to mention just one example of many, we neglected reduction of vehicle mass m as fuel is consumed. Moreover, we focused mainly on the paramount  Thus we concealed the difficult and complex physics underlying these departures within D C and L C , specifically, within departures of D C and L C from constancy. Thus again our analyses were qualitative to semiquantitative.
But hopefully they may still have been helpful.
In closing, we note that, even given all of the advances in aerodynamics, new discoveries are still being made, e.g., see Ref. [46].
gliders is accessible at https://www.leichtwerk.de. The Eta glider probably has the highest L D ratio of any sailplane built thus far. For additional related information see the following articles (all most recently revised in 2019) at https://www.wikipedia.org: "Lift-to-drag ratio", "Lift coefficient", "Drag coefficient", "Gliding flight", and "Eta (glider)".
Supplementary Note 5: The most energy-efficient angle of attack for aerodynamic flight, which maximizes L D , is always assumed in this present paper unless otherwise noted. Angle of attack is defined on pp. 114-115 and 139-140 of Ref. [10], and on pp. 30-33 of Ref. [13]. maximizes L D and which we always assume in this present paper unless otherwise noted, is not the optimum angle of attack for all facets of flight. As per Supplementary Note 5, maximum endurance in engine-powered level flight and minimum sinking speed in gliding flight without engine power obtain at a larger angle of attack and less-than-maximum L D . In this Supplementary Note 6 we consider landing. In order to land at the slowest possible speed, a still larger angle of attack than that corresponding to maximum endurance in engine-powered level flight and minimum sinking speed in gliding flight without engine power is optimal, namely the maximum practicable angle of attack, as near to a stall as is safe, which maximizes L for a given airspeed v at the expense of more-than-minimum sinking speed and even-lesser-than-maximum to Earth's radius R, whereas in this present paper it is stated with respect to general r R ≥ (and also considering a borehole with respect to 0 r = in a spherical gravitator of uniform density). The term "gravitational scale height" is analogous to "atmospheric scale height", the height of the top of are cited therein. For simplicity, we construe these to be subsumed within pressure drag. A similar approach is taken, for example, on pp. 108-109 of Ref. [10] concerning interference drag, where interference is mentioned, but not classified as a separate type of drag. [Shock-wave drag (discussed in Chap. IV of Ref. [9], on p. ix of Ref. [10], and on pp. 58-60 and 182-183 of Ref. [14]), due to shock waves at and in the vicinity of Mach 1, does not occur for aircraft that do not reach speeds approaching Mach 1.] Supplementary Note 9: Induced drag is a penalty that must be paid for lift. By Chap. VI in Ref. [10], and the Wikipedia article "Lift-to-drag ratio" (most recently revised in 2019). Thus applying Equations (9) and (10): , and the last equality is justified because L mg = in aerodynamic level flight. (Span, chord, and aspect ratio are defined at pp. 112-114 and 134-135 of Ref. [10] and pp. 55-58 of Ref. [13].) Typically at airspeeds v at or near that corresponding to maximization of L D , L C is approximately independent of air density ρ and of v. But for an airplane or bird, especially one with short wingspan b and small aspect ratio AR , trying to land at the slowest possible speed v well below that corresponding to maximization of L D and hence trying to maximize L in the face of this slow v whatever the cost in total drag D, induced drag induced D is likely the largest component of the total drag D. In such cases, owing to large induced D , total drag D is likely considerably larger than would typically be expected for the given with rectangular wings. And tapered wings were in actual use considerably earlier than the 1970s (see for example Sects. VI.4 and IX.7 of Ref. [10], p. 61 of Ref. [14], and pp. 63-64 of Ref. [15]). [By contrast, the aerodynamic fence, also known as the wing fence, blocks airflow towards the wingtips and serves a different purpose; hence it is not a wingtip device. See pp. 61-62 of Ref. [14], and the following articles: "Which is more effective a wing fence or winglet?" (date of most recent answer is 2018) at https://www.quora.com/Which-is-more-effective-a-wing-fence-or-winglet; . Indeed only if 1 1 a a = ⇔ − = − is skinfric D no function of ρ . But neither C nor a need equal 1. As per Chap. IV and especially Section V.5 of Ref. [10], more accurate consideration of skin-friction drag, even for the simple case of a flat rectangular plate of span b perpendicular to the airflow and chord c in the direction of (parallel to) the airflow assuming a laminar boundary layer, yields a result, as per Equations (5) and (6) on p. 106 of Ref. [10], that almost certainly could not be anticipated intuitively: ( )  [13]. Boundary layer is discussed in some detail on pp. 55-58 of Ref. [9], and on pp. 87-96 and in Section IV.5 of Ref. [10].) Note that air density ρ appears in the more accurate equations for skinfric D but not in the intuitive one skinfric || boundary D A v µ =  , that intuitively skinfric D is directly proportional to µ but not so according to more accurate equations, and that only the magnitude of || A matters in the intuitive equation for skinfric D while both the magnitude and the shape of || A matter in more accurate ones. Also note that neither the intuitive result for skinfric D nor the more accurate ones are even approximately proportional to increases with increasing v only as v 9/5 rather than as v 2 . By contrast, given the paramount 2 v ρ functional dependency of pressure drag, and also-recall Supplementary Note 9-of induced drag, they are disposed to increase with increasing v as v 2 . Hence skin-friction drag becomes less important relative to pressure drag and to induced drag with increasing v, or more fundamentally with increasing Reynolds number Re. The coefficient of skin-friction drag is Open Journal of Fluid Dynamics typically small, a few times 10 −3 . Hence skin-friction drag is often only a small fraction of the total drag D as given by Equation (7) of this present paper. See pp. 87-97 (especially pp. 93-94) of Ref. [9], and most especially Sects. V.5 and VI.2 of Ref. [10]. Often the functional dependence of skin-friction drag is not stated explicitly, and skin-friction drag is instead merely incorporated into Equations (7) and (8) of this present paper via a contribution to D C that is often small, and even if not small can still be thus incorporated within D C , in accordance with the second and eighth paragraphs of Section 3.1.1 of this present paper. For underwater airplanes as discussed in Section 3.3.2 of this present paper, substitute "water" for "air"; most generally, substitute "fluid" for "air". See also pp. 172-174 of Ref. [2], Refs. [3]- [8], and Sects. 7.10, 12.11, and 14.10 of Ref. [11].
Supplementary Note 11: The transition from laminar to turbulent flow is discussed on pp. 78-79 of Ref. [9], in Section V.3 of Ref. [10], and in Figs. 7.7 (a) and 7.7 (b) on pp. 400-401 and Sects. 14.10.5-14.10.6 of Ref. [11]. A concise survey of golf-ball aerodynamics is provided in Ref. [23]. For more detailed discussions see Refs. [24] [25], which are cited in Ref. [23]. See also the excellent, easily-understandable discussion of golf-ball aerodynamics in Ref. [26]. Assists for laminar-to-turbulent transition as employed, for example, in fluid-dynamic modeling and in small aircraft are discussed on pp. 107-108 and 208-209 of Ref. being much less pronounced than for (streamlined) airplane wings, and perhaps in some cases even nonexistent. See Section 15.10 of Ref. [11].
[Shock-wave drag (discussed in Chap. IV of Ref. [9], on p. ix of Ref. [10], and on pp. 58-60 and 182-183 of Ref. [14]), is due to shock waves at and in  [27]. Meticulously detailed data, including discussions, tables, and charts, concerning atmospheric density and other atmospheric properties as a function of altitude are provided in Refs. [29] [30] [31]. Somewhat less recent data, but also meticulously detailed, including discussions, tables, and charts, concerning atmospheric density and other atmospheric properties as a function of altitude are provided in Ref. [32]. Somewhat abbreviated but still more than adequate discussions, tables, and charts concerning atmospheric density and