General Velocity-Altitude Flight-Regime Diagram for Aeronautics and Astronautics

I. Introduction

Velocity-altitude diagrams are often employed to visualize trajectories of hypersonic flight systems in reference textbooks [1,2]. Because hypersonic phenomena are relevant only within the sensible atmosphere, the range of altitudes displayed in these diagrams is typically limited to the world of aeronautics that reigns below the edge of space or Kármán line. A more general view of those diagrams is portrayed here by increasing the altitude range far above the fringe of the atmosphere and bridging the gap with the world of astronautics. Besides eliciting six differentiated flight regimes, the diagram unearths a natural connection between aeronautics and astronautics, with hypersonics playing the fundamental role of connecting these two worlds through a corridor for space access and return.

II. Background: Sänger’s Diagram

In 1957, at the 7th International Astronautical Congress in Barcelona (Spain), Andrew Haley, a notable lawyer and former President of Theodore von Kármán’s Aerojet Corporation, presented a paper outlining the first foundational theory of a new emerging field: Space Law [3]. A pressing issue at the time was the establishment of jurisdictional boundaries to determine where national sovereignty of airspace ends, and where the freedom of space begins [3]. Haley pioneered the use of flight physics in Space Law and was the first to advocate for a physics-based demarcation of the end of aerial flight and the beginning of spaceflight based on a well-defined change in flight regime. This boundary, formulated by von Kármán for his collaborator Haley, corresponds to a critical altitude near the edge of space at which the air density is too small to render any significant lift. Instead, above the Kármán line, the centrifugal force is the one responsible for counteracting gravitation [4].

In the same paper, Haley highlights a side technical discussion with Dr. Eugen Sänger, a renowned Austrian aerodynamicist best known for pioneering the concept of antipodal bomber or suborbital long-range glider during WWII—a concept commonly known today as hypersonic glide vehicle (HGV) [5]. As part of the discussion, Sänger crafted for Haley the hand-drawn flight-regime diagram reproduced in Fig. 1 along with a brief explanation of the physics involved [3].

Sänger’s diagram gives an early comprehensive view of flight regimes at altitudes ranging from sea level to the nearest stars in velocity-altitude coordinates. An improved version of this diagram is shown in Fig. 2 with enhanced accuracy and technological developments of our time. Although the improved diagram has a shorter altitude range, it yields a more exhaustive collection of flight trajectories and critical boundaries that naturally lead to flight-regime demarcations, as discussed below. The improved diagram also elicits the role of hypersonics in connecting aerial flight with the carousel of sustained orbital motion in space.

III. Description of the Improved Diagram

In Fig. 2, the ordinate denotes geometric flight altitude from the surface of the Earth $Z$ (in kilometers) up to Martian orbital altitudes (55–400 million km). Indicated are 1) the Kármán line at $Z_K=100\mathrm{km}$, 2) the lowest three layers of the atmosphere, 3) the altitude equal to the Earth radius ($R=6,371\mathrm{km}$), 4) the altitude range of the inner and outer van Allen radiation belts from 1000 to 60,000 km, 5) the geosynchronous orbital (GEO) altitude at 35,786 km, along with 6) the edge of the sphere of Earth’s gravitational influence at an altitude of 917,390 km, beyond which the orbits are heliocentric [6]. The Moon is depicted at a mean altitude of 378,030 km above the Earth. Surrounding the Moon, and rotating at the angular velocity of the Moon around the Earth, are the L1 and L2 Lagrange points at 320,030 and 442,530 km of altitude, respectively.

The diagram depicts both non-Keplerian and Keplerian orbits. Non-Keplerian orbits in the diagram are powered at constant thrust. Instead, Keplerian orbits are unpowered, and each of them is therefore associated with a constant value of the specific orbital energy $E={\mathrm{v}}^2/2-g_0{R}^2/(R+Z)$. The diagram is useful in visualizing Keplerian orbits by increasing values of specific orbital energy in the clockwise direction: from ground-intercepting low-energy orbits (large negative $E$) to highly hyperbolic high-energy orbits departing to or arriving from interstellar space (large positive $E$).

The abscissa denotes flight velocity relative to Earth $\mathrm{v}$ (in km/s) up to approximately 0.1% of the speed of light (300 km/s). Below the Kármán line, the vertical axis indicates isolines of Mach 1, 5, and 10 based on the local speed of sound in the atmosphere. Those correspond, respectively, to the sound barrier, the approximate beginning of hypersonic flight, and the approximate onset of hypervelocity effects in the gas environment surrounding the flight system. Isolines of Mach 28, 39, and 59 are also shown which correspond—at the fringe of the atmosphere—to the first (${\mathrm{v}}_{\mathrm{C}1}=\sqrt{g_{0}R}=7.9\mathrm{km}/\mathrm{s}$), second (${\mathrm{v}}_{\mathrm{C}2}=\sqrt{2g_{0}R}=11.2\mathrm{km}/\mathrm{s}$), and third (${\mathrm{v}}_{\mathrm{C}3}=[{\mathrm{v}}_{\mathrm{C}2}^2+(\sqrt{2}-1{)}^2{\mathcal{V}}^2{]}^{1/2}=16.7\mathrm{km}/\mathrm{s}$) characteristic cosmic velocities, with $\mathcal{V}=29.8\mathrm{km}/\mathrm{s}$. The three characteristic cosmic velocities ${\mathrm{v}}_{\mathrm{C}1}$, ${\mathrm{v}}_{\mathrm{C}2}$, and ${\mathrm{v}}_{\mathrm{C}3}$ represent the three fundamental velocity scales for orbital flight in Earth’s proximity assuming vacuum conditions [7]. Specifically, ${\mathrm{v}}_{\mathrm{C}1}$ is the velocity for circular orbital flight grazing the surface of the Earth. Similarly, ${\mathrm{v}}_{\mathrm{C}2}$ is the velocity that needs to be imparted to an object on the surface of the Earth for it to parabolically escape from Earth’s gravitational field. Lastly, ${\mathrm{v}}_{\mathrm{C}3}$ is the velocity, in the geocentric reference frame, that needs to be imparted to an object on the surface of the Earth for it to escape on a direct heliocentric parabolic orbit along the Ecliptic from the gravitational influence of both Earth and Sun.

Six numbered sectors (from I to VI) are highlighted in Fig. 2 that represent different flight regimes in Earth’s proximity. These regions emanate radially from a corridor of space access and return situated near the edge of space at $Z\sim Z_K$ within the range of spaceflight-worthy velocities ${\mathrm{v}}_{\mathrm{C}1}\lesssim \mathrm{v}\lesssim {\mathrm{v}}_{\mathrm{C}3}$. The resulting wind-rose structure of flight regimes emanating from the corridor is not clearly appreciated in Sänger’s original diagram.

Regime I, bounded by the Kármán line and the surface of the Earth, corresponds to aerial flight at subsonic, supersonic, and hypersonic velocities. Sustained flight in the atmosphere is only possible at velocities higher than the limit of aerodynamic lifting power $\mathrm{v}={\mathrm{v}}_{\mathrm{C}1}\sqrt{\mathrm{\Lambda }{R}^2/(R+Z{)}^2}/\sqrt{1+\mathrm{\Lambda }R/(R+Z)}$ with $\mathrm{\Lambda }=2\beta /[\rho R(L/D)]$, for which the lift, gravitational, and centrifugal forces are balanced (the latter being negligible in this equilibrium below the Kármán line). Notional (system-nonspecific) flight trajectories for space capsules, HGVs, supersonic combustion ramjets (SCRAMJETs), re-entry vehicles (RVs) of intercontinental ballistic missiles (ICBMs), along with sounding and space rockets, are provided in Fig. 2. Details about the calculation of these notional trajectories are given in the Appendix. A velocity corridor for entry of Solar System meteors into the terrestrial atmosphere is also shown in Fig. 2 near the edge of space $Z\sim Z_K$ for velocities ranging from ${\mathrm{v}}_{\mathrm{C}2}$—namely, the minimum velocity in the geocentric frame for entry of a meteor captured by Earth’s gravitational field—to $[{\mathrm{v}}_{\mathrm{C}2}^2+(\sqrt{2}+1{)}^2{\mathcal{V}}^2{]}^{1/2}=72.8\mathrm{km}/\mathrm{s}$. The latter is the maximum entry velocity of a meteor, in the geocentric frame, resulting from a head-on intercept between Earth and an interplanetary dust cloud orbiting heliocentrically on the Ecliptic at the perihelion of a highly eccentric retrograde ellipse. The combination of compressibility (high Mach), turbulence (high Reynolds), and thermochemistry (high enthalpy) renders the bottom-right region of this regime (i.e., hypervelocities at low altitudes) inhospitable for flight vehicles due to excessive thermomechanical loads on their fuselage. This represents a gradual heat barrier of technological relevance for hypersonic flight [8,9].

Regime II occurs above the Kármán line and involves exo-atmospheric flight at suborbital velocities. As indicated in Fig. 2, this regime includes 1) the midcourse phase of ICBMs along elliptic orbits whose vacuum prolongation into the terrestrial atmosphere intercepts the surface of the Earth ($Z_p<0$), as well as 2) the powered flight of rocket upper stages for insertion into orbit. In $\mathrm{v}-Z$ coordinates, elliptic orbits are negatively sloped curves given by the vis-viva equation $\mathrm{v}={\mathrm{v}}_{\mathrm{C}1}\sqrt{2R[1/(R+Z)-1/(2R+Z_a+Z_p)]}$.

Regime III consists of orbital flight of satellites and spacecrafts along circular, elliptic, and parabolic orbits around the Earth. At a given altitude, this regime requires a relatively narrow range of flight velocities bounded by the escape velocity $\mathrm{v}={\mathrm{v}}_{\mathrm{C}2}\sqrt{R/(R+Z)}$ corresponding to parabolic orbits ($E=0$) and the minimum velocity at apogee $\mathrm{v}={\mathrm{v}}_{\mathrm{C}1}\sqrt{2R^2/[(R+Z)(2R+Z)]}$ obtained by substituting $Z_p=0$ and $Z_a=Z$ into the vis-viva equation. Sustained circular orbital motion at a given altitude is possible only at the circular velocity $\mathrm{v}={\mathrm{v}}_{\mathrm{C}1}\sqrt{R/(R+Z)}$, also obtained by substituting $Z_a=Z_p=Z$ in the vis-viva equation, or by taking the vacuum approximation $\mathrm{\Lambda }\to \infty$ in the velocity limit of aerodynamic lifting power. Regime III encompasses the velocity-altitude corridor for low Earth orbit (LEO), medium Earth orbit (MEO), GEO, along with x-GEO or cislunar space. Translunar and GEO-transfer (GTO) Hohmann orbits are highlighted in Fig. 2. A high-energy GTO is included using a notional space tug or orbital transfer vehicle (OTV), along with a direct inject to GEO using an also-notional upper stage. Figure 2 also depicts a polar, highly elliptical orbit whose elements match those of the Molniya orbit.

Regime IV represents interplanetary flight in the Solar System along hyperbolic orbits given by $\mathrm{v}=\sqrt{{\mathrm{v}}_{\mathrm{HEX}}^2+{\mathrm{v}}_{\mathrm{C}2}^{2}R/(R+Z)}$, with ${\mathrm{v}}_{\mathrm{HEX}}$ being a hyperbolic excess velocity defined as the absolute value of the difference between $\mathcal{V}$ and the velocity at perihelion (aphelion) of the Hohmann transfer for departures to—or arrivals from—planets farther (closer) from the Sun than Earth (i.e., ${\mathrm{v}}_{\mathrm{HEX}}=2.5$, 3.3, 8.8, 10.2, 11.3, and 11.6 km/s for Venus, Mars, Jupiter, Saturn, Uranus, and Neptune, respectively). This regime requires velocities ranging from the escape velocity to the hyperbolic excess velocity ${\mathrm{v}}_{\mathrm{HEX}}=\sqrt{{\mathrm{v}}_{\mathrm{C}3}^2-{\mathrm{v}}_{\mathrm{C}2}^2}=12.4\mathrm{km}/\mathrm{s}$ for an orbit exiting the Solar System. The orbit of Mars, with perihelion and aphelion radii of 206.7 million km and 249.2 million km, respectively, is depicted over 3 terrestrial years in $\mathrm{v}-Z$ coordinates, along with a trans-Martian Hohmann ellipse around the Sun with arrival at Mars aphelion.

Regimes V and VI involve interstellar flight along highly hyperbolic orbits. However, these two regimes differ in their compatibility with human space exploration as follows. For reference, Fig. 2 depicts an interstellar escape hyperbola departing easterly from Earth at ${\mathrm{v}}_{\mathrm{HEX}}=40\mathrm{km}/\mathrm{s}$. Four circumferentially powered transfer orbits, calculated using the method of Tsien [10], accelerate spacecrafts from LEO at 300 km and intercept this interstellar escape hyperbola at the indicated flight times using four different levels of continuous thrust: $g_0$, $10g_0$, $100g_0$, and $\infty$. Insertion into the interstellar escape hyperbola occurs upon thrust cutoff at the intercept. These circumferentially powered transfer orbits also apply in reverse direction at the indicated thrust levels to decelerate spacecrafts returning from interstellar space for arrival to Earth. Regime VI encompasses the indicated range of in-space flight altitudes and velocities for departing or returning powered transfer orbits that render too high accelerations—larger than $10g_0$—incompatible with human space exploration.

IV. Conclusions

A velocity-altitude flight-regime diagram is presented in Fig. 2 that improves a previous version made by Sänger [3]. Six flight regimes are elicited by the improved diagram: 1) endo-atmospheric and trans-atmospheric flight; 2) exo-atmospheric flight at suborbital velocities; 3) orbital flight around Earth (circular, elliptic, and parabolic orbits); 4) interplanetary flight (hyperbolic orbits); 5) interstellar flight (highly hyperbolic orbits); and 6) unmanned interstellar flight (too high accelerations). Spaceflight regimes are connected with aerial flight through a corridor for space access and return located near the edge of space within an interval of velocities ranging from approximately the first to the third characteristic cosmic velocities. The diagram also extends previous ones by illustrating trajectories, bridging the gap with astronautics, and providing a comprehensive view of flight regimes up to Martian orbital altitudes.