# The Edge of Space: Revisiting the Karman Line

Jonathan C. McDowell<EMAIL_ADDRESS>Harvard-Smithsonian Center for
Astrophysics, 60 Garden St, Cambridge, MA 02138, USA

(2018 Jul 6)

###### Abstract

In this paper I revisit proposed definitions of the boundary between the
Earth’s atmosphere and outer space, considering orbital and suborbital
trajectories used by space vehicles. In particular, I investigate the inner
edge of outer space from historical, physical and technological viewpoints and
propose 80 kilometers as a more appropriate boundary than the currently
popular 100 km Von Kármán line.

###### keywords:

astronautics , Karman line , atmosphere , mesosphere , perigee , boundaries

††journal: Acta Astronautica

##  1 The Edge of Space

###  1.1 Introduction

The argument about where the atmosphere ends and space begins predates the
launch of the first Sputnik (e.g. [1]). The most widely - but not universally
- accepted boundary is the so-called Karman Line, nowadays usually set to be
100 km altitude, but boundaries ranging from 30 km to 1.5 million km have been
suggested, as summarized in a 1996 book by Goedhart [2].

Although the subject has not been much addressed in the physics literature,
there is an extensive law/policy literature on the subject - see e.g. [3] [4],
[5], [6] [7]. Hansen [7] notes that COPUOS has wrestled with the issue
continuously since 1966 [8] without a conclusion. COPUOS, the Committee on
Peaceful Uses Of Outer Space, was established in 1959 and is the UN body
dealing with astronautics. In COPUOS the USSR repeatedly proposed either 100
or 110 km but the US rejected any definition.

As early as 1957 Robert Jastrow ([1], cited in [6]) suggested that the air
space boundary should be at 100 km. Goedhart (p. 3) lists almost 30 different
proposals from the 1951-1962 period for an altitude boundary ranging from 20
to 400 km; most values are in the 75-100 km range. A number of these authors
suggest that the large variations with time of atmospheric properties make it
futile to locate a true boundary of space based on physical arguments. In this
paper I will argue the contrary: there is a moderately-well-defined boundary
of space, it coincides with the Karman line as originally defined, and that
line is close to 80 km, not 100 km.

###  1.2 The Functionalist Objection

There have been objections (particularly in the United States) to defining any
legal boundary of space on the grounds that it could cause disputes about
airspace violations below the boundary, or that too high a boundary could
inhibit future space activities. Those advocating this position, beginning
with McDougal and Lipson [9], are sometimes referred to as ‘functionalists’
(see also [6],[5]). The functionalist approach would ensure that long range
ballistic missiles were not made subject to international agreements on ‘space
objects’, which may explain part of its appeal to the US establishment.

The general tenor of these objections, however, seem applicable to any law
about anything. Functionalists also suggest that space law would apply to an
orbital rocket even while it was within the atmosphere, or possibly on the
ground. This seems unnecessary as national and international law would already
apply. Suggestions that the purpose of a vehicle, not its location, should
determine the legal regime may be appropriate for questions of licensing, but
will not help if a vehicle classified as belonging to one regime collides or
interferes with one from another regime.

The special need for distinct laws specifically for space (and thus the need
for legal definition of space) arises from:

  * 1.

The lack of national boundaries in space (analogous to international waters)

  * 2.

Objects in space may remain in motion relative to the Earth for long periods
of time (depending on the orbit, from days to millenia) without the need to
refuel or land.

  * 3.

The large area swept out by a space object in a given time due to the large
kinetic energies involved in space travel, meaning that a given space activity
will extend over a wide area rather than be spatially localized as most Earth-
based activities are.

  * 4.

The high destructive potential of collisions, since their effects also are
felt over a wide area.

The latter two considerations in particular are specific to the region above
the atmosphere where orbital dynamics dominate, although they do not apply to
activities on the surfaces or in the atmospheres of other worlds such as the
Moon or Mars.

In any case, my main interest here is to define a boundary of space for the
use of historians of spaceflight, rather than to define a legal regime, so I
will give no further consideration to the functionalist view. To answer
questions such as ‘how many astronauts have flown in space?’ or ‘how many
European Space Agency rockets have reached space?’, we need to adopt a
definition of space, even if it is not a legal one.

I do not argue that there needs to be a single definition of ‘space’ that
applies in all contexts. Physicists, lawyers and historians may need a
boundary of space for different purposes and to address different questions;
the ’edge of space’ might be defined differently in different international
fora. Nevertheless it is useful for those definitions to be based on a common
and accurate understanding of the physical conditions at the air-space
boundary, and I hope that this article can make a positive contribution in
that respect.

###  1.3 McDowell (1994) proposal

Milt Thompson’s book [10] talks about his flights in the X-15 rocketplane to
the ‘edge of space’. In my Quest article ‘The X-15 Spaceplane’ [11] I
discussed this term and concluded that the correct choice for the edge was 80
km, not 100 km. The discussion is repeated and expanded in this paper. In
part, I argued:

> In the late 1950s the USAF decided to award ‘astronaut wings’ to pilots
> flying above 50 statute miles. This boundary was chosen as a nice round
> figure, but I want to argue that it is also the right choice from a physical
> point of view …. … it seems natural to choose the outermost [physical
> atmospheric] boundary, the mesopause, as the physical boundary which marks
> the edge of space. It turns out that the traditional value for the height of
> the mesopause, 80 km, is also within 500 metres of the 50 mile ‘astronaut
> wings’ boundary historically used by the USAF. I therefore suggest that we
> adopt as the formal boundary of space an altitude of exactly 80 km,
> representing the typical location of the mesopause.

In this paper I expand on the arguments in my 1994 article. All references to
altitudes are intended to represent geodetic height (height above reference
ellipsoid); I ignore local topography. Note that in some astronautical
contexts use is made of a ‘geocentric’ height relative to a fictional
spherical Earth, which may differ by of order 10 km.

##  2 Cultural arguments: historical definitions of the edge of space

In recent decades the 100 km Karman line has gained ascendancy as the most
commonly used boundary, notably for the Ansari X-Prize won by the Spaceship
One team.

The ‘official’ status of the von Karman line, such as it is, comes from the
undated paper ‘100 km Altitude Boundary For Astronautics’ [12] on the web site
of the Astronautics Records Commission (ICARE) of the Fédération Aéronautique
Internationale (FAI), which certifies world records for aeronautics and
astronautics. It is unfortunate that discussion in this official document in
the section ‘demonstration of usefulness of Karman line’ appears to be poorly
researched:

> In the early 1960s the U.S. X-15 Aircraft was flown up to 108 km. In that
> part of the flight it was really a free falling rocket, with no aerodynamic
> control possible. In fact, it was considered an astronautical flight, and
> the pilot got, as a consequence, his astronautical wings, i.e. the
> recognition of being an astronaut.

This is not incorrect, but the USAF considered all X-15 flights above 80 km as
astronautical flights and gave those pilots astronaut wings. So this paragraph
would argue for 80 km, not 100 km. The first non-NASA pilot to be awarded
astronaut wings was X-15 pilot R. White, as described in Life Magazine (Aug 3
1962):

> Major Bob White of the US Air Force is the nation’s newest space hero. […]
> He has […] a brand-new award on his chest that makes him a member of the
> nation’s most exclusive club. It was a special set of pilot’s wings that
> signified he had flown higher than 50 miles above the earth and thereby had
> qualified as a spaceman.

White flew to 95 km on 1962 Jul 17. If a limit of 100 km instead of 80 km is
used, White, Robert Rushworth, Jack McKay, Bill Dana and Mike Adams lose their
space traveler status (Joe Engle keeps his because he later flew on Shuttle,
and Joe Walker passed the Karman line on his X-15 flights).

In the past few years, even lower values have been proposed. The prominent
astrophysicist Alan Stern has argued (personal communication) for balloon
altitudes in the 30-35 km range as being ‘space’ or ‘near space’; Stern is
involved in the World View high-altitude-balloon near-space tourism venture.

##  3 Physical boundaries in the atmosphere

###  3.1 Atmospheric layers: the mesopause as a proposed boundary

As you leave the surface of the Earth and ascend into the atmosphere, it gets
colder - until you pass a boundary at which the temperature begins to increase
again. There are several such reversals in temperature gradient and the
traditional definition of atmospheric layers uses them to define the layers of
the atmosphere as ‘-spheres’ with boundaries called ‘-pauses’ [13]:

  * 1.

The troposphere, between the ground and the tropopause

  * 2.

The stratosphere, between the tropopause and the stratopause (about 50 km)

  * 3.

The mesosphere, between the stratopause and the mesopause (about 85-90 km).
Here CO2 cooling dominates solar heating.

  * 4.

The thermosphere, between the mesopause and the exobase (about 85 to 500 km,
variable). In the thermosphere the physical state is dominated by absorption
of solar radiation; the resulting ionized atoms have their own behaviour and
the composition of the atmosphere departs from the N2/O2 mix of the lower
layers. The thermosphere region overlaps (but has a definition which is not
quite the same as) the ionosphere, the region where ionized particles dominate
the physics. It includes the LEO region where the ISS orbits.

  * 5.

The exosphere, beyond the exobase. Here the density is so low that the atoms
don’t act like a gas.

Another relevant boundary is the turbopause, below which all the different
molecules have the same temperature, and above which they behave
independently; below the turbopause you are in the ‘homosphere’ where
everything is mixed; above it is the ’heterosphere’ where everything acts
independently. The turbopause is at about 100-120 km. In 2009, press releases
referencing a paper by Sangalli et al [14] about the Joule II rocket mission
trumpeted a measurement of the ‘edge of space’ at 118 km. This was actually
the height at which the motion of charged atoms (ions) becomes dominated by
the electromagnetic field rather than by winds in the neutral atmosphere; it
is likely a function of time and location and so their value for 2007 Alaska
should not be taken as a generic result.

The chemical composition of the atmosphere is largely constant up to the
mesopause. From a physical point of view, it is therefore reasonable to think
of the atmosphere proper as including the troposphere and stratosphere and
(with some qualification) the mesosphere, and identifying the thermosphere and
exosphere with the common idea of ’outer space’. Either the mesopause or the
turbopause are reasonable choices for a boundary, as the outermost physical
atmospheric boundaries below the region where most satellites orbit. It is
true that each of these definitions varies in height by 10 km or more
depending on solar activity, upper atmosphere dynamics and other factors. The
1976 US Standard Atmosphere value for the mesopause is a constant 86 km; Xu et
al [15] used observations with the SABER radiometer on the TIMED satellite to
study variations in the mesosphere altitude. Their data suggest a mesosphere
altitude of 97 $\pm$2 km for equatorial and winter polar regions and 86$\pm$2
km for summer polar regions; these values are higher than I had assumed in my
1994 discussion.

A reasonable alternative air/space boundary would be the base of the
mesosphere instead of its ceiling; or, one may consider the mesosphere as
neither air nor space. In 1976 Reijnen [16] and Jager and Reijnen [17]
introduced the idea of ‘mesospace’ as an intermediate legal regime between
airspace and outer space; the mesosphere is a natural candidate for mesospace.
Oduntan [18] suggested a buffer zone from 55 to 100 miles (88 to 160 km),
apparently partly based on the existing incorrect estimates of 150 km as the
lowest orbital perigee. In fact, as shown below, 55 to 100 km would be a more
suitable choice. Pelton [19] has coined the term ‘protospace’ or ‘the
protozone’ for the intermediate region, which he defines as the 21 to 160 km
range. In general, however, the idea of mesospace has not yet gained general
acceptance.

###  3.2 Outer limits of the atmosphere and boundaries in deep space

The true outer edge to the Earth’s atmosphere, or a reasonable candidate for
it, is the magnetic shock front with the solar wind. The magnetopause boundary
forms a comet-shaped region, typically around the height of geostationary
orbit on the sunward side of Earth and extending out to beyond the Earth-Sun
L2 point. One can also consider the gravitational boundary of the Earth-Moon
system with respect to the Sun, conventionally chosen to be the 1.5-million-km
radius Hill sphere marked by the Earth-Sun Lagrange points L1 and L2. While
material within the magnetosphere and/or the Hill sphere could be considered
part of the Earth’s outer atmosphere, few would argue that this region is not
‘space’. Rather, these boundaries may be used to distinguish space in the
Earth-Moon system from interplanetary space.

Indeed, one may usefully identify a number of different conventional regions
in the region of space humans and their robots have explored, listed here for
convenience of reference.

  * 1.

The boundary between Low Earth Orbit (LEO) and Medium Earth Orbit (MEO), is
sometimes taken to be a 2 hour orbital period, which corresponds to an
altitude of 1682 km for equatorial orbits, but nowadays a round value of 2000
km is relatively standard, e.g. [20].

  * 2.

The geosynchronous altitude, 35786 km above the equator [21].

  * 3.

The Earth-Moon 1:4 resonance altitude EL1:4, 145688 km altitude. I introduce
here this boundary between ‘near-Earth space’, where the effects of lunisolar
perturbations are minor and a simple Keplerian elliptical satellite orbit is a
reasonable approximation and ‘deep space’, which I take to include both
distant Earth satellite orbits (such as that of the TESS satellite launched in
2018) and lunar and planetary missions. For Earth satellite orbits in ‘deep
space’ the lunar perturbations are large enough to make big changes in the
orbital elements on month-long timescales. As a practical matter, NORAD/JFSCC
systematically monitor orbits of near-Earth space spacecraft but do not
attempt to monitor deep-space Earth satellite orbits in a comprehensive way -
this is left to astronomers who accidentally pick up satellites in such orbits
while searching for asteroids. As will all these boundaries, one could
reasonably make a different choice here - a round altitude of 100,000 km, or a
different resonance like EL1:3. I propose EL1:4 by analogy with the Sun-
Jupiter 1:4 resonance that is conventionally taken to mark the inner edge of
the asteroid belt (and thus the point inside which solar orbiting objects can
be considered as not strongly perturbed by Jupiter).

  * 4.

The Hill Sphere [22], bounded by the Earth-Sun Lagrange points, with a radius
of 1.496 million km. This is the conventional boundary between considering
objects as orbiting Earth but perturbed by the Sun, and considering objects as
orbiting the Sun but (if close to the boundary) perturbed by the Earth.
Another choice here is the so-called ‘gravitational sphere of influence’ or
Laplace sphere [23], which is at approximately 929,000 km radius; it is used
in the method of patched conics. In general the Hill sphere, which takes into
account the orbital angular momentum, better reflects the effective boundary
at which orbiting objects may be captured by or escape from the Earth-Moon
system (e.g. [24],[25]).

  * 5.

The $\nu_{6}$ secular Sun-Jupiter-Saturn resonance which marks the
conventional inner edge of the asteroid belt at 2.06 astronomical units (308
million km) from the Sun [26]; it coincides with the 1:4 Sun-Jupiter resonance
[27] and asteroid orbits near this resonance are unstable, soon perturbed to
enter the inner solar system. Although there is no generally agreed
definition, this location is a reasonable place to mark as the boundary
between the inner and outer solar system.

  * 6.

The outer edge of the Solar System itself is controversial. Plasma physicists
associated with studies by the Voyager probes have made various estimates of
the ‘heliopause’ boundary between the solar wind and the broader-scale flow of
interstellar gas, for example at 121.7 astronomical units ($1.8\times 10^{10}$
km) [28]. However, dynamical astronomers would point out that objects remain
gravitationally bound to the Sun much further out [29], of order 200,000
astronomical units ($3\times 10^{13}$ km).

##  4 Technological boundaries at the edge of space

###  4.1 The highest vehicles using aerodynamic lift

On 1973 Jul 25 a modified Mig-25 designated the Ye-266 reached 36.2 km, and on
1977 Aug 31 a Ye-266M reached 37.65 km. These altitude records for non-rocket-
powered airplanes were set at the Soviet LII Gromov flight test center by test
pilot Aleksandr Fedotov [30]; they involved brief arcs to high altitude. The
highest steadily flying non-rocket plane was the remotely piloted Helios,
which reached 29 km on 2001 Aug 14. The highest crewed balloon reached 34.6 km
in 1961. However, uncrewed scientific research balloons reach over 40 km on a
routine basis. A 1972 record of 51.8 km was exceeded in 2002 by the 54-meter-
diameter BU60-1 balloon from the Japanese ISAS team, which reached an altitude
of 53 km [31]. Despite the then-stated intent of the ISAS team to reach 60 km,
it appears that the technological limiting ceiling of vehicles which require
the atmosphere for lift is close to the stratopause at 50 km. This sets a
sensible lower limit for the boundary of space.

###  4.2 The lowest quasi-circular orbits

The much-cited FAI article [12] about the Karman line continues:

> Later in the same decade (or very early in the next; Soviet information at
> the time was very scanty) the Soviet Union put in orbit an unmanned
> satellite, in very low orbit, whose attitude was controlled by aerodynamic
> forces. The real reason of such an experiment is not yet known. It is known
> however that it successfully described a few orbits just above the 100 km
> line (how much higher I do not know), but collapsed rapidly shortly after he
> crossed, or got too much close to, the 100 km. Karman line.

Soviet information was not that scanty even at the time; the author is clearly
referring to the well known Kosmos-149 satellite, which carried an extendable
structure used to stabilize it along the velocity vector. This satellite,
whose then-classified name was DS-MO No. 1, was launched into a 245 x 285 km
orbit, low enough for the drag stabilization to work but much higher than
needed to avoid catastrophic decay. It remained in orbit from 1967 Mar 21 to
Apr 7. The last US orbital data was on Apr 5, at which time it was in a 201 km
circular orbit. But there are many well documented cases of even lower
altitude satellites. Since the idea that 200 km is the low boundary for
satellite orbits is so widespread, I consider here a number of
counterexamples.

In May 1976, the satellite GAMBIT Mission 4346 (1976-27A, US National
Reconnaissance Office) was tracked in an orbit with a perigee between 125 and
135 km for a full month; this is not unusual for this kind of satellite, which
performs frequent rocket burns to counteract decay. Its apogee was around 350
km. Empty rocket stages are frequently left in low orbits of under 200 km and
reenter after several days; the final tracked orbit is often between 130 and
140 km.

From 2016 Aug 16-19, China’s Lixing-1 satellite operated in a near-circular
orbit of 124 x 133 km for three days prior to reentry; this is the lowest
circular orbit ever sustained for multiple days.

In contrast, when the Space Shuttle lowered its perigee to 50 km as part of
the deorbit burn, it reenters within an orbit. Shuttle external tanks,
discarded at orbit insertion, often had perigees around 70-75 km and in all
cases did not complete their first orbit.

Based on circular orbit data, 125 km is a conservative upper limit for the
beginning of space.

###  4.3 The lowest perigees for elliptical orbits

A satellite in an elliptical orbit can survive a brief periapsis passage at
lower altitudes than the extended exposure of a circular orbit would permit.
Below I give examples of low perigee elliptical orbit satellites. The air
density increases rapidly, and so there is a limit below which even a highly
elliptical orbit satellite will be rapidly destroyed. This limit turns out to
be in the 80 to 90 km range except in very special cases.

Consider a satellite in an elliptical orbit whose perigee is around 80 km. Are
we to say that it is in space only for the higher parts of its orbit, and
that, for example, space law stops applying to it at each perigee passage? The
repetitive nature of an orbit makes this case different from the one-off
transition from the space to aviation environment during launch or reentry. I
therefore conclude that attempts to use ‘lowest circular orbit’ to define the
space boundary are fundamentally misguided, and ‘lowest sustainable perigee’
(for more than two revolutions, say, of an elliptical orbit) is a more
appropriate criterion.

Before considering specific examples, a detailed discussion of the pitfalls in
satellite perigee height calculations is warranted.

The Earth satellite catalog in widespread use is that currently maintained by
the US military, since the corresponding Russian catalog is not publicly
available and other sources (e.g. those from hobbyists) are relatively
incomplete. The catalog was begun in 1957 by the Smithsonian Astrophysical
Observatory [32]. The North American Air Defense Command (NORAD) collaborated
with SAO and ultimately took over the catalog. Orbital data here and below are
obtained from the Two-Line Orbital Elements (TLE) issued by the US Joint Space
Force Component Command, [33], the current inheritor of NORAD’s space tracking
responsibilities. For each satellite, there may be several orbit
determinations (‘element sets’, ‘TLE sets’ or simply ‘TLEs’) per day. These
data have been federated with spacecraft historical information from the
author’s catalog of satellites [34].

The TLEs provide mean motion and eccentricity of a fitted time-averaged orbit,
the ‘SGP4 mean elements’ [35], [36]. It is common practice (notably in the
official public satellite catalog on space-track.org, or historically in the
RAE Table Of Earth Satellites and reports derived from it) to describe a
satellite orbit by quoting the perigee and apogee height of the SGP4 mean
elements relative to a fictitious 6378 km spherical Earth. To find the actual
perigee height of a satellite above the true surface of the Earth, one must
first apply the SGP4 theory to derive the osculating elements (or,
equivalently, state vector) at perigee. For an orbit with significant
eccentricity this perigee may be different from the SGP4 mean value by of
order ten kilometers. Next, the correction to the height above the Earth
ellipsoid rather than the spherical Earth model ranges up to 22 km at the
poles. I use the WGS-84 ellipsoid for calculations in this paper.

Low perigee TLEs are common for the final element set for a satellite, but in
some cases this may represent only the final orbit where perigee is not
survived. Single element sets are suspect; for example Kosmos-168 was tracked
in a 52 x 386 km orbit on 1967 Jul 4, but data for surrounding epochs make it
clear that this was an erroneous solution with the right period but wrong
eccentricity. In particular, for elliptical orbit satellites, as the orbital
period decreases and perigee drops below of order 100 km the sequence of mean
element solutions show increasing set-to-set noise and an increasing fraction
of spurious fits. In a review of an archive of 90 million TLE sets for 43000
satellites I identified 50 satellites where the data are not severely affected
by these problems and where geodetic perigee heights of less than 100 km were
maintained over 2 or more complete revolutions of the Earth.

A few illustrative examples are shown in Figure 1. These include the Soviet
Elektron-4 satellite (SSN 748) which appears to have made 10 revolutions with
perigee at or below 85 km at the time of its reentry in 1997, and the US
Centaur AV-031 rocket (SSN 38255) which had perigee geodetic height below 110
km for 4 days prior to reentry, and between 80 and 95 km for much of that
time.

Figure 1:  Geodetic height of apogee and perigee versus time for the decay of
selected elliptical orbit satellites. Horizontal lines at 80 and 100 km are
superimposed on the perigee plots. Despite noisy fits, these satellites appear
to have survived multiple perigee passages below 100 km. (a) Satellite 748
(1964-006B, Elektron 2, 2D No. 2); (b) Satellite 12512 (1981-30A, Molniya-3
No. 30); (c) Satellite 14587 (1983-126A, Kosmos-1518, Oko 6022); (d) Satellite
22189 (1992-069A, Kosmos-2217, Oko 6059); (e) Satellite 29399 (2006-038B,
Chang Zheng 3A Y10 third stage rocket); (f) Satellite 38255 (2012-019B,
Centaur AV-031 rocket).

To summarize, the lowest possible sustained circular orbits are at of order
125 km altitude, but elliptical orbits with perigees at 100 km can survive for
long periods. In contrast, Earth satellites with perigees below 80 km are
highly unlikely to complete their next orbit. It is noteworthy that meteors
(travelling much more quickly) usually disintegrate in the 70 to 100 km
altitude range, adding to the evidence that this is the region where the
atmosphere becomes important.

###  4.4 Air-space vehicles

Goedhart [2] correctly notes that during ascent a space vehicle reaches 100 km
altitudes in quite a short downrange distance, so that usually crossing
someone else’s territory while in the atmosphere is not an issue, and normal
spacecraft landings are similar in this respect, but on reentry a winged
spaceplane can be at 60 km altitudes or lower while traversing long ground
distances. The issue of airspace violations during spaceplane reentry is
therefore something to worry about. Goedhart correctly says that the limit of
‘aviation’ and aerodynamic rather than ballistic phenomena is at 50 km (and
infers that the Paris and Chicago conventions apply up to this limit). He
immediately contradicts himself by stating that ‘aeroplanes are already
capable of flying above the 50 km limit’, which I suspect is a confusion
between a true aeroplane and rocketplanes like the X-15. Several authors
exhibit this confusion: like the later Shuttle, the X-15 functions as an
airplane (in fact, a glider) during descent through the lower atmosphere, and
so it looks like an airplane. But when it is above the lower atmosphere it
uses rocket thrusters to maneuver - it is then operating as a spacecraft
despite its exterior appearance, and makes no use of its aerodynamic surfaces.
To use it as an example of a high altitude aircraft is to miss the point.

###  4.5 Other technological considerations

Goedhart [2] also discusses what he calls the ‘biological theory’, that at
about 20 km humans cannot survive unprotected due to low air pressure, but
notes that clearly air vehicles above this line are not considered to be in
space.

de Oliveira Bittencourt Neto [6] discusses the idea of ‘effective control’
(you own the airspace you can control) and points out that’s a really bad idea
as it leads to different boundaries of space above different countries.
Satellites would cross such boundaries every few minutes.

It is clear that for a definition of space to be useful and consistent with
the generally understood meaning of the term, it should be well above typical
airplane altitudes and should be globally uniform (this last constraint does
not mean that one can’t adopt different globally-uniform definitions of space
for different purposes and situations).

##  5 The Effective Karman Line

###  5.1 Mathematical analysis of the Karman line

The ‘von Karman line’ appears to be what mathematicians refer to as a ‘folk
theorem’, arising out of a conference discussion but never formally published
by him. It was fleshed out in later publications, especially in the
influential work of Haley (1963, [3]) and there is some justification for
calling it the ‘von Karman-Haley line’. 111Note added in press: A recent paper
by Gangale [37] untangles this complicated history in detail.

von Karman’s argument was that the space boundary should be defined where
forces due to orbital dynamics exceed aerodynamic forces. A rough order of
magnitude argument was used to show that this was at of order 100 km (as
opposed to 10 km or 1000 km), but in reality the von Karman criterion defines
a line whose altitude varies with position and time (because of variations in
atmospheric density due to solar activity) and with the lift coefficient of
the spacecraft.

Haley ([3], p 78) extended the argument to satellite drag and places the line
at 84 km. The strong association of the term ‘(von) Karman line’ with a
definite 100 km value is a more recent development.

Satellite launch vehicles reach 100 km altitude in the first minutes of
flight, well before they have accelerated to orbital velocity; thus the
appropriate value of the parameter $f$ is less than one, and drag is smaller
and the gravity/drag force ratio correspondingly larger at a given altitude;
hence the effective Karman line is even lower in this phase or for suborbital
missions. I will consider only orbital flight in the following calculations.

I consider a spacecraft of mass m, cross sectional area A and lift and drag
coefficients $C_{L}$ and $C_{D}$ travelling at velocity v, which I’ll later
take to be the orbital Keplerian circular velocity $v_{c}$. The spacecraft is
travelling at geocentric radius r through atmosphere of density $\rho$ in the
gravity field of the Earth whose mass is $M_{E}$.

The lift force is

$F={1\over 2}AC_{L}\rho v^{2}$ (1)

and the drag force is expressed my the same equation with a different
coefficient $C_{D}$.

Following Haley I consider that drag forces are more relevant to Earth
satellites, so instead consider the ratio of drag force to weight (i.e. to
gravitational pull). Because atmospheric density changes by many orders of
magnitude in a few tens of kilometers, use of $C_{D}$ rather than $C_{L}$ does
not change the final Karman Line location much, as we shall see below.

The ratio of gravitational force (weight W) to aerodynamic force (F) is

$R=W/F={mg\over{1\over 2}AC_{D}\rho v^{2}}$ (2)

where the local acceleration due to gravity is

$g=GM_{E}/r^{2}$ (3)

and circular orbital velocity is

$v=\sqrt{GM_{E}/r}$ (4)

We introduce the ballistic coefficient $B=C_{D}A/m$, which is essentially the
specific drag, or drag per unit mass (warning: some authors use the term
ballistic coefficient for 1/B instead). Then the above results can be
simplified to

$R={2\over Br\rho}$ (5)

When R is much greater than unity, orbital dynamics dominates aerodynamics
(and, per the original von Karman argument, lifting flight is not possible).

Because of the rapid change of density with height, R changes by orders of
magnitude in the range of interest. It is therefore convenient to instead use
the logarithm - I define

$k(B,r,\rho)=\log_{10}R=\log_{10}\left({2\over Br\rho}\right)$ (6)

and call this logarithmic measure the Karman parameter222Reijnen [16] suggests
a slightly different parameter, the height at which drag reduces the height of
a single circular orbit by 10 percent (and thus its radius from Earth center
by about 0.2 percent). If $\delta$ is the fractional change in the semi-major
axis. Reijnen claims  $\delta r=2\pi B\rho r^{2}$ so $\delta=R$, hence
Reijnen’s criterion corresponds to a Karman parameter k of -2.7 rather than
0..

###  5.2 Reference ballistic coefficient and fiducial Karman parameter

The Karman parameter is a function of position, atmospheric properties, and
spacecraft ballistic coefficient B. With a known atmospheric profile and value
of B, one can derive $z_{B}(k)$, the altitude at which k has a particular
value. The value $k=0$ defines an effective Karman line height $z_{B}(0)$ at
which aerodynamic and gravitational forces balance.

Let us explore how much that line shifts around for a range of plausible
values. Typical values of $C_{D}$ are 2.0 to 2.4 for satellites, while typical
values of B are of order 0.006 to 0.05 m2/kg [38], [39]. The International
Space Station has an average cross sectional area of 2040 sq m and a B of
0.010 m2/kg, while the Planet Dove cubesats have a cross-sectional area
varying from 0.2 to 0.02 sq m and a B of 0.1 to 0.01 depending on flight
attitude [40].

However balloons and special high density satellites can have more extreme
values. The Echo balloon satellite had a high B of around 22 m2/kg, while the
LARES high density geodetic research satellite had a record low B of around
0.001.

As an intermediate fiducial value I adopt $B_{0}=0.01\mbox{m${}^{2}$/kg}$ and
define the fiducial Karman parameter $k_{0}$ as

$k_{0}(r,\rho)=k(B_{0},r,\rho)$ (7)

so that

$k(B,r,\rho)=k_{0}(r,\rho)-\log_{10}\left(B/B_{0}\right)$ (8)

We then define the function $z(x)$ as the geodetic altitude at which
$k_{0}=x$. The usefulness of $k_{0}$ is that its values and the corresponding
altitudes $z(k_{0}$) can be calculated for a given atmosphere independently of
the satellite properties. One may then read off the Karman line location for a
particular satellite by determining which value of $k_{0}$ is appropriate for
its B. For example if one considers a satellite with a high B = 0.05, the
Karman line $k=0$ corresponds to $k_{0}=\log(5)=0.70$.

Mathematically,

$z_{B}(x)=z(x+\log_{10}\left(B/B_{0}\right))$ (9)

with the Karman line at x=0.

Figure 2: Fiducial Karman parameter $k_{0}$ versus geodetic altitude for US
Standard Atmosphere 1976. The red error bars summarize the results of figures
3 and 4, indicating the range of variation in the altitude for a given value
of $k_{0}$ found in runs of the NRL atmosphere model for different dates,
latitudes and longitudes.

###  5.3 Numerical evaluation of the effective Karman line

I now derive the geodetic altitude z for various values of the fiducial Karman
parameter and consider how the Karman parameter changes with location and
time.

A useful reference atmosphere is the US Standard Atmosphere 1976 (hereafter
USSA76) [41], which is a single, fixed, atmosphere model. Figure 2 shows the
altitude as a function of fiducial Karman parameter for this atmosphere.
However, in practice the atmospheric density at a given altitude varies with
longitude and latitude, and also with solar activity as the atmosphere is
heated by the solar flux.

To understand the effect of these variations on the $k$ I ran models for
assorted times and geographical locations using a code which implements the
NRL MSISE-00 atmospheric model [42]. Atmospheres were calculated at 10 day
intervals from Jan 1960 to Jan 2020 to fully sample several solar cycles. At
each selected day atmospheres were calcualted for 0, 6, 12 and 18h GMT at four
latitudes (80S, 0N, 45N and 80N) and four longitudes (0, 90, 180, 270E). For
each epoch, actual or (for future dates) predicted solar activity levels from
the Celestrak space weather archive [43], [44] were used.

In Figure 3 I show the $z(k_{0})$ lines for high values of $k_{0}$, namely
$k_{0}$ = 4,5,6. There are high amplitude variations in the $k_{0}=6$ Karman
line correlated with the solar cycle, reflecting the well-known sensitivity of
atmospheric density to solar flux at the corresponding altitudes, above the
mesopause. Note, however, the much reduced amplitude of the variations at
lower altitudes and Karman parameter values. The USSA76 model values are in
all cases within the range of the NRL model variations; the range of the
variations at each modelled parameter value is summarized as an error bar on
the USSA76 curve in Figure 2.

Figure 3:  Curves showing z(4) (lowest), z(5) and z(6) (highest) as a function
of time, showing that the effects of the solar cycle are more important at
high Karman parameter. These integrations are for NRL atmosphere models
evaluated at 45 deg N, but curves for other latitudes are similar. Arrows
indicate dates of solar maxima.

In particular we are interested in fiducial Karman parameters $k_{0}$ equal to
-0.3,0.0,+0.7, corresponding to effective Karman lines $k=0$ covering the
typical range of B for satellites discussed above. The USSA1976 reference
model gives z(-0.3)=72.0 km, z(0.0)= 76.7 km, and z(0.7)=86.7 km. In Figure 4
I show corresponding NRL model atmosphere calculations as a function of time.
Seasonal variations are most prominent at polar latitudes but their amplitude
is only a few kilometres. The fact that actual historical solar flux values
for each date were used confirm that irregular solar flares do not affect the
result.

In all cases the effective Karman lines calculated from the NRL atmospheres
remain within 5 km or so of their USSA values, and the overall range of the
data (for a factor of 10 in ballistic coefficient) is from 66 to 88 km. In
other words, the region where aerodynamic forces transition from dominant to
negligible is relatively well defined despite typical variations in satellite
and atmosphere properties (at least to the extent that this atmosphere model
reflects reality). The range of ballistic coefficients considered here is
comparable to the difference between typical lift and drag coefficients, and
so this conclusion still holds if the original lift-based Karman criterion is
preferred.

It is true that for satellites with extreme properties the results do change -
for a balloon satellite like Echo, the effective Karman line z(1) is around
140 km. For a very dense satellite such as LARES, the effective Karman line
z(-1) is around 60 km.

Figure 4: Curves for z(-0.3), z(0.0), z(0.7), corresponding to the effective
Karman line for $B=0.005,0.01,0.05\mbox{m${}^{2}$ kg${}^{-1}$}$ respectively.
Each plot gives calculations for a different latitude; there is less
atmospheric variation at intermediate latitudes. At these low altitudes the
effects of the solar cycle are minimal.

It is undesirable to have a definition that will change with improving
technology, so one might argue that the correct way to define space is to pick
the lowest altitude at which any satellite can remain in orbit, and thus the
lowest ballistic coefficent possible should be adopted - a ten-meter-diameter
solid sphere of pure osmium, perhaps, which would have B of $8\times
10^{-6}\mbox{m${}^{2}$/kg}$ and an effective Karman line of z(-4) at the
tropopause. In practice z(0) seems a more realistic limit for finite orbital
lifetime (see previous section). The few high density satellites at low
orbital altitudes (e.g. LOADS 2, B of around 0.002 m2/kg) have reentered when
their perigees were around 120 km.

We can summarize the results of this section by saying that for a vehicle of
typical ballistic coefficient, z(0) represents the altitude at which gravity
will exceed aerodynamic forces for any object in steady flight at that
altitude (since such flight must always be at or less than the Keplerian
circular velocity for that altitude). That altitude lies in the 70 to 90 km
range, and 100 km is always too high.

###  5.4 Extension of the Karman argument to elliptical orbits

In section 4.3 I showed that elliptical orbit satellites can survive lower
altitudes than circular orbit ones. I now derive the ratio of gravitational to
aerodynamic force at the perigee of an elliptical orbit.

For an orbit of eccentricity e, perigee velocity is related to the circular
velocity for that altitude by

$v=v_{c}\sqrt{1+e}$ (10)

and so

$R={2\over Br\rho(1+e)},$ (11)

i.e. the ratio is lowered by a factor of one plus the orbit eccentricity, so
the effective Karman line is actually somewhat higher. The low-perigee
satellite is indeed drag-dominated near perigee, causing rapid reduction of
apogee and consquently reduction of the eccentricity, so that close to reentry
the Karman ratio tends to the circular value.

##  6 Conclusion

I have shown that for a typical satellite ballistic coefficient the effective
Karman line is close to (within 10 km of) 80 km independent of solar and
atmospheric conditions, rather than the currently popular 100 km value; and
that historical orbital data for actual artificial satellites confirms that
orbiting objects can survive multiple perigees at altitudes around 80 to 90
km. This altitude range is consistent with the highest physical boundary in
the atmosphere, i.e. the mesopause, and with the 50-mile ‘astronaut wings’
boundary suggested by the United States during the first years of the Space
Age.

On the basis of these physical, technological and historical arguments, I
therefore suggest that a value of 80 km is a more suitable choice to use as
the canonical lower ‘edge of space’ in circumstances where such a dividing
line between atmosphere and space is desired.

### Acknowledgements

I thank Brian Weeden, Joe Pelton and Isabelle Rongier for critical comments on
an earlier version of this paper. They will still disagree with its
conclusions. I also thank Martin Elvis for his careful reading of the
manuscript.

This research did not receive any specific grant from funding agencies in the
public, commercial or not-for-profit sectors.

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