ARTICLE IN PRESS
Journal of Theoretical Biology 224 (2003) 9–26
Physical theory, origin of flight, and a synthesis proposed for birds
Charles A. Longa,*, G.P. Zhangb, Thomas F. Georgec, Claudine F. Longa
a
Department of Biology and Museum of Natural History, University of Wisconsin-Stevens Point, Stevens Point, WI 54481, USA
b
Department of Physics, Indiana State University, Terre Haute, Indiana 47809, USA
c
Office of the Chancellor/Departments of Chemistry and Physics & Astronomy, University of Wisconsin-Stevens Point,
Stevens Point, WI 54481-3897, USA
Received 4 April 2002; received in revised form 25 February 2003; accepted 7 March 2003
Abstract
Neither flapping and running to take-off nor gliding from heights can be disproved as the assured evolutionary origin of selfpowered flight observed in modern vertebrates. Gliding with set wings would utilize available potential energy from gravity but gain
little from flapping. Bipedal running, important in avian phylogeny, possibly facilitated the evolution of flight. Based on physical
principles, gliding is a better process for the origin of powered flight than the ‘‘ground-up’’ process, which physically is not feasible in
space or time (considering air resistance, metabolic energy costs, and mechanical resistance to bipedal running). Proto-avian
ancestors of Archaeopteryx and Microraptor probably flapped their sparsely feathered limbs synchronously while descending from
leaps or heights, with such ‘‘flutter-gliding’’ presented as a synthesis of the two earlier theories of flight origin (making use of the
available potential energy from gravity, involving wing thrusts and flapping, coping with air resistance that slows air speed, but
effecting positive fitness value in providing lift and slowing dangerous falls).
r 2003 Elsevier Ltd. All rights reserved.
Keywords: Origin vertebrate flight; Physics; Synthesis; ‘‘Flutter-gliding’’; Archaeopteryx
1. Introduction
A feathered, four-winged ‘‘dromaeosaur’’ from Jurassic beds in Liaoning Province in northeast China was
recently discovered and given the name Microraptor gui
(Xu et al., 2003). It superficially resembled the famous,
bipedal, primitive bird Archaeopteryx in having teeth,
clawed fingers, a long feathered tail, and even some
asymmetrical feather vanes. Comparably small (1 m
length), it differed from anything seen previously in
having a fringe of flight feathers on the extended hind
limbs that allowed them to form a continuous air foil
with the front limbs [plus the fringed tail]. The pelvic
connections to the hind limbs seem irreversibly adapted
to a gliding niche, and not conducive to running or to
flapping flight.
A local news reporter asked us a curious question.
Could we make a sketch translating our mathematical
equations and biological and physical findings to depict
*Corresponding author. Tel.: +1-715-346-4208; fax: +1-715-3463624.
E-mail address: [email protected] (C.A. Long).
0022-5193/03/$ - see front matter r 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/S0022-5193(03)00116-4
a proto-bird ancestral to the early bird Archaeopteryx
and resembling the newly discovered four-winged fossil
reptile? Voila! However, M. gui is not ancestral to
Archaeopteryx, although remarkably similar, and our
depicted ‘‘ancestor’’ would be somewhat intermediate,
have more down feathers, fewer specialized flight
feathers, shorter wings, and a quadrupedal or bipedal
stance, and we could not predict whether the teeth
would resemble those of dromaeosaurs or Archaeopteryx.
Certainly the newly discovered glider Microraptor
presents a great deal of evidence that proto-birds were
arboreal and did not run and leap into flight. However,
there are troublesome observations on the importance of
bipedal locomotion, flapping wings, and highly specialized wing feathers in proto-birds that disparage a
transition of fixed-wing gliding to flapping flight. M. gui
could never have evolved flapping flight. One important
analogy shows that most gliding mammals differ in
function profoundly from the aerial bats, and never the
twain shall meet. The important survival values of
slowing air speed, creating lift, and lessening dangerous
impact with the ground have been seldom addressed in
explaining flight evolution.
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C.A. Long et al. / Journal of Theoretical Biology 224 (2003) 9–26
Therefore, the century-old controversy continues to
this day, on the question whether flight in vertebrate
animals evolved in bipedal running forms (from the
‘‘ground-up’’) or in arboreal, gliding forms (from the
‘‘tree-down’’). We compare the physical and biological
attributes of the two hypotheses, and present a synthesis
(described below as ‘‘flutter-gliding’’) to combine them.
Advocates of the ‘‘ground-up’’ theory for pterosaurs,
bats, and especially birds included Nopcsa (1907, 1923),
Ostrom (1974a, b, 1976, 1979, 1986), Caple et al. (1983),
Gauthier (1986), Padian and Chiappe (1998), and
Burgers and Chiappe (1999). Proponents of the ‘‘treedown’’ theory included Marsh (1890), Heilmann (1927),
Spurway (1955), Bock (1965, 1969, 1983, 1985, 1986),
Bock and Buhler
.
(1995), Savile (1957), Yalden (1971a),
Feduccia and Tordoff (1979), Norberg (1985, 1990),
Rayner (1991), Feduccia (1996), Simmons and Geisler
(1998), and Chatterjee and Templin (2003). Related
studies treated the possible origin of birds from
dinosaurs (Ostrum, 1973; Gauthier and Padian, 1985;
Hou, 1995; Chatterjee, 1995, 1997; Martin, 1983, 1991;
Normille, 2000; Prum, 2003; and others). Many
ornithologists referred to ‘‘phylogenetic analysis,’’ and
considered the flight origin settled in favor of the
running and leaping theory. In addition to the bipedal
stance, the dinosaur to bird evidence rests on analyses of
character complexes of flight adaptations in related but
not ancestral reptiles. In one typical review, Dingus and
Rowe (1997), who suggest modern birds are [feathered]
dinosaurs, mention ‘‘physics’’ supporting the ground-up
theory, but discuss hardly any physical evidence.
Ostrom (1973) listed an imposing array (>20 skeletal
characters) linking Archaeopteryx to small, bipedal
coelurosaurian dinosaurs. Bipedal dinosaur fossils
found recently in China show avian characters, including feathers (Xu and Wang, 1998; Xu et al., 2000, 2002),
and the aforementioned glider Microraptor gui had four
feathered wings (Xu et al., 2003).
Neither classical school of thought concedes that both
theories, gliding versus running and flapping, might be
partly right. Herein, we review some of the known
features of proto-flyers, and especially for primitive
Archaeopteryx. Since archaic flight form and behavior
were inferior to modern flight adaptations and flight, we
discuss the struggles of some modern birds having
ineffective flight. We develop mathematical principles
dating from Sir Isaac Newton to analyse forces affecting
primordial flight. Ballistic forces affecting a projectile’s
lift-off and trajectory include gravity, lift, thrust, and air
resistance. Running forces include these too, as well as
pressure friction and mechanical limitations (Long et al.,
2002).
The aforementioned forces have been discussed
previously for flying animals as aerodynamics (Pennycuick, 1968, 1975; Pennycuick et al., 1988; Caple et al.,
1983; Rayner, 1988; Tucker, 1987; Tennekes, 1997;
Burgers and Chiappe, 1999; Padian, 1982). There was
little attention to air resistance and its effects on time in
flight, range, and take-off of living organisms. Norberg
(1985, 1990) focused primarily on the development of
thrust in falling animals, increased lift due to expanded
wingspan, and extension of the gliding flight path. She
appreciated that drag retarding speed and gravity work
against lift-off and increase energy expense of a running,
flapping bird (especially in comparison to a gliding
bird). She recognized the utilization of potential energy
from gravity in the tree-down theory during glides with
fixed wings. Perhaps she glossed over the falling animal’s
need to utilize the air resistance to lessen the dangerous
impact from forcibly falling. A paradox in the gliding
theory is the dangerous downward pull of gravity,
following the initial velocity of leaping from a
perch (e.g., Martin, 1983), becomes the prevailing
force in early evolution (providing distance, air time,
energy, and opportunity) for the evolution of upward
flight.
Homologs of feathers, which probably appeared as
body down for heat conservation in several early reptiles
(Zhang and Zhou, 2000), evolved as larger feathers in
serial follicles along the posterior margin of the
forelimb. Ancestral birds possibly flapped and fluttered
while descending before fixing their wings for glides, and
Microraptor gui is proof of gliding (Xu et al., 2003). The
flutter-glide synthesis presented here incorporates bilaterally synchronous flapping, thrust, air resistance, speed,
and gravity together. It defines ‘‘flutter-gliding’’ as a
flapping descent from a high prominence or tree canopy.
Glissading is an analogous term from mountaineering,
describing rough mountain descents and lessening of the
final impact. Flutter-gliding fits well with the derived
mathematical proofs presented herein.
2. Methods
2.1. Some comparisons from biology
Many biological problems of flight origin were
discussed in the aforementioned papers. The several
works of Bock, Yalden (1971b), and Caple et al. (1983)
emphasize biological functions, and even mention the
problems of impact. Problems of take-off are analysed
herein of some modern, poorly flying birds and
fledglings, using classical observations from natural
history. Flight performance was observed of three
clipped parrots that learned to fly as their primaries
grew out, over a year period of observations. A
dendrogram was generated using a MacClade Version
4.0 (Maddison and Maddison, 2000), based on a
character matrix of biological and paleontological
characters.
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2.2. Physical comparisons
Aerodynamic forces affect any projectile with mathematically predictable results. Take-off and sustained
flight from the ground or utilization of gravity leaping
from some height are different processes easily compared in physically precise terms. Apart from obvious
forces of gravity Mg and ground friction ðF Þ; air
resistance, also called total drag Dto is important:
Dto ¼ induced (Di )+frictional (Df ) drag components.
Also, there may be added drag from inclined flight
ðDclimb Þ; which we do not consider, except recognizing
fitness for the gliders that gain potential energy at a
gradual expense of climbing heights, and the observed
use of it in swooping upward at the end of a flight
to lessen impact. Combined, Dto ¼ 1=2dV 2 SCd þ
KL2v =ð1=2dV 2 SÞ; where d is the air density, Cd is the
drag coefficient based upon shape, lift is perpendicular
to the air speed V ; and K is a coefficient that varies from
animal to animal, from 0 to 10, and even more when
flapping (Lighthill, 1977). The vector Lv is the same
magnitude as gravity Mg; supported by lift and drag
(Fig. 1). An important drag component indirectly
retarding thrust is derived from the resultant R between
Lv and thrust tr. The velocity is lessened by Dto and
increased by tr: The wing-stroke herein is only the downstroke wing speed w; which with Vrun creates the air
speed V : S is the cross-sectional, not the wing area,
which in this paper we would call A: If Cd ¼ 1; then the
drag resistance is determined, instead of defining Cd :
From Rankine-Froude, we know the ‘‘induced drag’’
Fig. 1. Forces of flight for take-off, gliding, or flutter-gliding.
11
(Di ) in level flight decreases with high speed and
increases by slowing speed. [Long et al. (2002) observed
that running is akin to level flight, as the runner does not
move up or down much, and only one or even none of
the four feet may be in contact with the ground.]
Induced drag primarily comes from the displacement of
a flux of air about the wings and running hind limbs,
and is profound at slow speed because Di ¼
2Mg2 =pdV 2 b2 ; where we note the square of V in the
denominator.
Prior to powered flight, we assume the animal gained
its air speed velocity V by running or jumping upwards
ðVrun or Vleap Þ or, alternatively (Martin, 1983; Chatterjee
and Templin, 2003), leaping downwards from heights
ðVglide Þ: Due to the finite volume of the animal with mass
M (in grams), it will experience wind resistances f
proportional to the velocity V ; but having opposite
direction, namely f ¼ lV ; where l is called the
resistance factor. This factor depends on the animal’s
shape, the area of forelimb or wing, the cross-sectional
area S; and the direction of flight. The l can be
mathematically written as l ¼ CZS sin f; where C is a
conversion constant for systems of units of measure; f is
the angle between the direction of movement and the
wing surface; and the variable Z reflects the body
curvature, viscosity, and air density d: It was similar to
that used to compute air resistance (Stoke’s law) and
used by Nobel Laureate Robert Andrews Millikan in his
experiment on oil droplets.
The air density would have been about the same for
all arboreal or terrestrial proto-birds. The aerodynamic
drag, which rigorously and physically should be called
the resistance force from air, is approximated by lV at
the low Reynolds number for birds. The reasoning is as
follows: During the collision between the proto-bird and
the air molecules, the momentum change of the bird is
DP ¼ MðVf VI Þ; where Vf is the final velocity and VI
is the initial velocity, P is the momentum, and M is the
mass. The force is the momentum change over the time
elapsed during the collision. With a given velocity, with
the same speed and direction, the collision time is
constant, and then the force is directly proportional to
the velocity. That is why ‘‘drag’’ is proportional to
velocity. This approximation for force works in actual
practice, and predicts a terminal speed for the parachutist. Such an elegant prediction from physics shows
how the pressure changes along the flight path according
to the Bernoulli equation as velocity increases. This
leads to a nonlinear term (e.g., for high-speed rockets)
although we do not expect higher order terms for
primordial birds. We expect lV to vary from 0 to a
proportion of V . A greater exponent (V 2 or V 3 ) would
strengthen our argument. Theoretically and experimentally our treatment is justified.
To compare running take-off from land or water,
gliding from high places, and theoretical flutter-gliding
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C.A. Long et al. / Journal of Theoretical Biology 224 (2003) 9–26
from high places, ten mathematical equations derived
from the principles of space aeronautics and ballistics
are presented for ground-up or tree-down scenarios. The
numerous differential equations written to derive these
ten are on file at the University of Wisconsin—Stevens
Point. A dendrogram similar to that based upon
biological information was created, based on ten
ballistic features proved in our equations, and we
compared it to the four flight models. These features
were leaping against air resistance, with change in x,
height attained against air, with change in y, weight
limits, maximal glide, flight body form and function (not
fitness), large wings, small wings, bipedal running
facilitating climbing or a possible take-off, and wing
lift and thrusts. These were scaled from 0 to 2.
Usage of somewhat overlapping words for the use of
energy for flapping or running is somewhat confusing,
often called ‘‘biological power,’’ ‘‘active flight’’ (compared to gliding with set wings), body metabolic energy
(E) mostly spent as heat, ‘‘induced power’’ apparently
for that arising within the flyer (in contrast to gravity
force), or muscular power, where attention to muscles
often leads to overlooking much elastic power from
joints, tendons, and ligaments. Biological ‘‘powers,’’ by
any name, may work against gravity, air resistance and
ground friction, and may even work with these forces.
The functions resulted in the evolution of flight.
3. The primitive Archaeopteryx
The Jurassic bird Archaeopteryx lithographica has
provided much information on proto-birds, even if it
was or was not capable of self-powered flight. Olson and
Feduccia (1979) doubted that Archaeopteryx, lacking a
keeled sternum and triosseal canal for a specialized
supracoracoideus flight muscle, could possibly have
flown up from the ground. That muscle, however, raises
the wing and is not used in the critical downthrust. A
furcula was present, perhaps providing an indication
of flight muscle. Short coracoids and small sternum
suggested small muscle mass, but Yalden (1971b)
concluded flapping flight was possible. Padian and
Chiappe (1998) doubted that Archaeopteryx was even
a glider, or even arboreal, because: the foot is not
‘‘fully’’ modified to grasp; many animals can climb trees
but are not arboreal (and perhaps there were no tall
trees); there is a ‘‘swivel wrist’’ (showing lateral flexion);
the bipedal gait leaving the forearms free to catch prey
or flap; and the complete absence of lateral patagia.
These points are hardly convincing, except the bipedal
gait, which is assumed to be the mode of ground
locomotion in Archaeopteryx; it becomes necessary if
the forelimbs were used as wings. Some workers (Bock,
1965, 1969; Norberg, 1985, 1990) believe the clawed
digits on Archaeopteryx, which resemble those of the
extant hoatzin (Opisthocornus) and those in bats
(Chiroptera), suggest clambering about in the branches
of trees. Heptonstall (1970) probably overestimated the
mass (0.5 kg>0.2–3 kg) and wing-loading of Archaeopteryx according to Yalden (1971b), but both recognized the difficulty and danger for a fast moving
projectile to alight. Yalden suggested lessening impacts
was accomplished by merely swooping upward against
gravity ðDclimb Þ when approaching tree branches, confirmed by many of our observations of clipped parrots
(see below). Heavy wing loading probably applies to
some reptilian forebears of Archaeopteryx. Bock (1985,
1986), Bock and Buhler
.
(1995), and Norberg (1985)
reviewed morphological and paleo-physiological evidence and developed a reasonable tree-down hypothesis.
The long, flattened but well-feathered tail, airfoil
wings, and archaic sternum suggest gliding, although
flapping flight seemed likely in Archaeopteryx (Yalden,
1971a; Olson and Feduccia, 1979; Bock and Buhler,
.
1995). It may have fluttered and flapped its wings while
descending. Physically, a synthesis between gliding and
wing-flapping seems reasonable (see below), suggesting
an arboreal or cliff-dwelling origin of flight. Such a
synthesis for fluttering birds would lead inevitably
toward self-powered flight (Caple et al., 1983), rather
than the efficient, gravity-powered gliding in volant
mammals (Scheibe and Robins, 1998). Recently, Chatterjee and Templin (2003) discounted running take-offs
in Archaeopteryx and proposed ‘‘phugoid gliding,’’ i.e.,
dropping to gain airspeed and flitting to a nearby tree
(cf., Tennekes, 1997, for modern birds).
The Solnhofen region is of great interest to those
regarding Archaeopteryx as a proto-bird (Viohl, 1985).
It apparently was a xeric habitat, but there was a salty
lagoon ideal for preservation of limestones. Some
workers noted there were no tall trees for either gliding
or perching. The absence of streams may account for the
absence of logs in this lagoon. Terrestrial xeric plants,
such as cacti, suggest that the highlands were not barren.
The most abundant plants were low conifers, Brachyphyllium, Palaeooyparis, and several shrubs not exceeding 3 m in height. These formed a dense canopy.
Archaeopteryx, its fingers adapted to clambering, may
have frequented low shore-line bluffs. It may have fed
on washed-up carrion, beach organisms, and possibly
fishes. There are a few beach pebbles associated with the
fossils. Viohl (1985) discounts ‘‘trunk climbing’’ for
Archaeopteryx. If it clambered in bushes, it may have
fluttered from bush to bush. Yalden (1997a, b) and
Feduccia (1993) consider the foot to be adapted to
arboreal niches. Archaic proto-birds fluttering with
poorly feathered wings could live in shrubland or might
swoop up abruptly ðDclimb Þ to decelerate and alight after
fluttering or gliding from tree to tree.
Since there are no known fossils directly ancestral to
birds, although Microraptor seems closely related, one
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may presume in the true phylogeny of Archaeopteryx
that proto-bird ancestors lived in habitats unfavorable
for preservation. Modern birds dying in forestlands are
seldom preserved because the bodies are small with light
bones easily decomposed, and woodland habitats do not
create sedimentary rocks. Fortunately, some arboreal
reptiles that possess bird-like traits have been found in
China and Texas.
4. The primitive Protoavis, Microraptor, Sinovenator,
and Longisquama
The famous Late Triassic find of Protoavis texensis in
Texas was a bird-like fossil appearing much earlier than
Archaeopteryx. This fossil had numerous features that
seemed pre-adapted to flight, such as a keeled sternum
and several avian features of the shoulder girdle not
even observed in Archaeopteryx. Chatterjee (1995)
suspected the animal had feathers and could fly, and it
was bipedal. He considered its foot to have a scansorial
form. Its habitat was forest.
The small dinosaur Microraptor zhaoianus reportedly
had feathers and climbed trees. Like Archaeopteryx or
Protoavis, this little dinosaur had a ‘‘cursorial ancestor’’
(Xu et al., 2000). Another recent find in northern China
(Xu et al., 2002), of a small, feathered dinosaur
apparently contemporary with Archaeopteryx, may shed
light on proto-bird ancestry. Belonging to the dinosaur
family Troodontidae, this early Chinese species,
Sinovenator changii, and other troodontids share some
distinctive characters, but this little feathered reptile had
a small size, long hind limbs, and an avian coracoid and
furcula. Peter Mackovicky discounts the furcula as a
distinctively avian feature (personal comm.). The skull
and ischium extension resemble that of Archaeopteryx.
Separated from Archaeopteryx from China to Germany,
their joint possession of feathers is important. Fluvial
deposits indicate riparian habitat.
The tree-down theory of flight in arboreal forms is
probably strengthened by these finds. However, the
bipedal stance in some suggests running or leaping
locomotion, and according to some workers substantiate a dinosaur relationship to Archaeopteryx (Mackovicky, pers. comm.). Sinovenator apparently did not
evolve flight; its feathers may have been for homoiothermy. From this little predator little can be
determined on how flight evolved. The old classification
of maniraptoran dinosaurs by Gauthier (1986) is
strengthened because the basal Troodontidae is close
to early birds and dromaeosaurs.
From central Asia, lacustrine deposits from the Late
Triassic Period have yielded a peculiar Archosaur (Jones
et al., 2000). This quadrupedal and gliding reptile
(Longisquama insignis) suggests that an archosaur line
led to both coelurosaurian dinosaurs and birds, and had
13
appeared by the Late Triassic. Longisquama possessed
feather-like epidermal extensions of several kinds. These
included some on the postaxial forelimbs that closely
resemble flight feathers. The feathers seem homologous
with bird feathers, which is perhaps not true (Alan
Feduccia, pers. comm.; also see Zhang and Zhou, 2000)
of so-called feathers of the small dinosaurs (their
filaments may not be derived from scales as are true
feathers). The feather-like structures of Longisquama
suggest that birds may not be descended from the
hypothesized small theropod dinosaurs after all, but
from an archosaurian glider that perhaps gave rise later
to some feathered forms with bipedal locomotion. The
fluffy dorsal outgrowths may have enhanced a parachuting role in these arboreal reptiles. They certainly
would hinder a running take-off. The free arms may
have controlled landings, lessening impacts.
The four-winged dromaeosaur Microraptor gui is
described above. Whether or not it was a flutter-glider
or a fixed-wing glider is unknown, but without a
patagium and by free use of the forelimbs, one mode
seems as likely as the other. We submit that a fourfooted parachutist with primordial feathers suggests
flutter-gliding, and a four-footed ancestor of M. gui may
have had preadapted preconditions (i.e., die Anlagen) for
both kinds of gliding leading to two divergent styles in
evolving efficient use of gravity for energy (in Microraptor and Archaeopteryx). Four-limbed fixed-wing
gliding, with laterally extended hind legs, probably
could not succeed to flapping flight by means of the
forelimbs, as it probably did in bats (in which each
possessed a billowing patagium). The elongate hind
limbs of M. gui might have lost their fringes of feathers,
but could never revert to a bipedal stance.
5. Fluttering the wings
If an animal falls from a high perch, generally
disadvantageously, its forelimbs are presented downward or horizontal (‘‘spread-eagled’’), as often observed
in falling tree squirrels or cats. The spread limbs and
body push against the air and slow the momentum of
the fall. Bats, which hang in tree branches by their hind
feet; flying squirrels, which launch head-first from the
head-up or head-down position; and poorly flying birds,
such as nestlings, chickens and clipped parrots, all
deflect the vertical fall. They create some lift opposing
the free fall by spreading the limbs outward and pushing
against the air. With a bony (firm) leading-edge of an
evolving wing, and a flexible membrane or feathers
posterior to it, fluttering uses air resistance to resist
falling and also to naturally thrust forward. The
behavior is bilateral, the paired limbs flapping in
synchrony. Although the gliding of these animals,
especially any with patagia, might be disrupted by
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muscular flapping exertions, there are statistical advantages in righting the animal and braking the momentum
of descent. Therefore, the limitations against vertical
leaping from flapping favor falling animals. We prove
below that vertical velocity is retarded by air resistance,
which restricts either leaping upward or falling too fast.
Flapping the wings (with bilaterally synchronous
downstrokes) by a partially feathered bird can be
observed in any falling nestlings or fledging birds,
except a few modern species capable of true fledging
flight. In some observed fledglings, and in our clipped
parrots, the wings beat approximately six times during
short falls, more if the bird manages to slow its fall and
descend away from the vertical. Therefore, any advantage from a wing-beat is multiplied by a factor of 12
or greater, and much greater when the descent is
prolonged. This gives extra survival value to this trait
when appraising its functional use, although we did not
include this in our tally (Fig. 3). Other workers have
mentioned ‘‘extending the flight path.’’
An altricial young bird or proto-bird likely would
never fall from a height of higher than 20 m, unless
leaping off a high cliff (which requires fluttering, design
of the wings, and plumage to prevent too much impact).
Such a vertical fall from tree height would likely last less
than 2 s (the known speed of falling objects regardless of
mass). Flapping against air while pushing the wings
downward is obviously adaptive. A fact from sailing
suggests even an upward push from the wings by a fast
falling bird adds negligible downward force (because the
air is emptied from above the wings by rapidly falling
into the wind). Therefore, there is potential energy to
spare, and much to gain in increasing escape distance
or food finding, while lessening impact upon striking
the ground. Some horizontal thrust may be gained,
especially useful for increasing range.
Several morphological characters of fossil birds
supporting the theory of running and flapping fit well
into the present synthesis of flutter-gliding. There is no
need of a patagium useful for gliding, or keeled sternum
enabling flapping with great force. The long tail for
steering would be adaptive, and other morphological
fossil evidence discussed as adaptive for flapping
(Padian and Chiappe, 1998) likewise would seem as
useful for descent. What we prove herein as limitations
for leaping upward are obviously limitations on falling
too forcibly, favoring either the flutter-gliding or gliding
hypotheses.
Norberg (1985, 1990) discussed the advantages of
slow flapping creating some thrust without loss of any
lift in gliders. She hypothesized that, by ‘‘slightly
flapping,’’ proto-flyers might show ‘‘vorticity wakes’’
transitional between gliders and self-powered flyers.
Rapid wing strokes, instead of a powerful and sweeping
wing stroke, might be beneficial and much more likely in
unspecialized flyers. Of course, any loss of lift by
flapping (Caple et al., 1983) must be compensated by
loss of vertical velocity, gain in thrust, or something
directly or indirectly useful for survival.
Analysis of the Norberg (1985, 1990) diagram of
forces for gliding and partially flapping flight applies to
this new synthesis. Especially interesting is the likely
evolutionary adaptation of lengthening the wing-tips to
increase both thrust and lift. What she claims to be
superior to a running take-off, by her gliding and slowly
flapping form, shows benefits in energy conserved and
length of flight (see also Eqs. (9) and (10)). What was not
discussed is an obvious need for parachuting, i.e.,
development of lift (lVy ) preventing too forcible an
impact at the landing. Fluttering may have preceded
fixed-wing gliding, which hardly slows the velocity to
retard impact.
5.1. Natural history and ontogeny
An insight into the primordial flight problems of
proto-birds might be obtained by observing modern
birds that develop self-powered flight from flightlessness. Anecdotes appear in the news media, usually with
reference to the chicken (Gallus gallus). Some report that
they run and fly up from the ground, while others say
they cannot. One of the authors observed that some do
leap upwards (e.g., if a dog attempts seizing it), but the
flight is little more than a bound. Unpenned chickens
often develop enough power to flutter up into branches
of small trees or onto henhouses to roost. Chickens have
been observed (same author) thrown from the roof of a
12-story Kansas building to onlookers at a parade; the
chickens flew less than 50 m horizontally but fluttered
enough while falling to avoid injury.
Every newly fledged bird that flies developed that
function from flightlessness. It likely ontogenetically
recapitulated some of the phylogenetic stages (e.g.,
down feathers, to short feathers, to elongate wing
feathers; improving motor skills and muscular development, etc.). Fledglings of primitive birds probably stray
from recapitulation less than highly specialized birds,
such as hole-nesters or hummingbirds. Furthermore,
when some advanced birds show evolutionary convergence of form and function to hypothesized adaptations,
such as running take-offs (e.g., cursors, roadrunner),
their similar problems with aerodynamic forces may
have been solved by similar, even if convergent,
morphological evolution. It would be a paradox if the
fast-striding North American roadrunner Geococcyx
californianus had evolved elongate wings. It has not.
Even though it occasionally uses an extended wing while
running to counterbalance while swerving, and flits
between trees or to a nearby thicket, it is primarily a
glider that must clamber ‘‘limb by limb’’ into a shrub or
tree. No one expects a ledge-nesting sea bird, such as the
murre, to have well-developed, elongate hind limbs for
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running, and the fledglings leap and flutter down from
coastal, rocky heights.
Generally, most fledged, altricial birds flutter down
from well-constructed nests, tree branches, or rocky
crevices, and hide from predators in ground vegetation.
Then they, and precocial ground nesters as well,
scramble about and feed on food supplied by the
parents or found opportunistically (Gill, 1990; E. H.
Forbush, in Pearson et al., 1917). Many fledged birds
within a shrub or tree canopy flutter from branch to
branch. Most adult birds utilize gravity and a downstroke of the wings to obtain enough air speed for lift.
Ontogeny suggests that fluttering downward instead of
fixed-wing gliding generally precedes flight, although the
somewhat cursorial rails and roadrunners run with
bipedal strides, and bound or glide in their open
environments. Thus, both of the classic models of flight
origin have some representatives in modern vertebrates,
although fluttering down is far more general compared
to either of them.
Bats and pterosaurs, with membranous patagia,
probably always glided from heights, and judging from
the structure of their hind limbs never ran at all. Birds
developed the forelimb as an airfoil by evolving patterns
of feathers. Fluttering became important to them if they
fell from dangerous heights. Even fuzzy feathers, also
useful for body warmth, would soften ground impacts
by retarding vertical velocity of a small bird and
cushioning its body. When fledging murres flutter down
from cliffs, they have been observed to fall 150 m, some
striking the rocky walls, but they were seldom injured
(H. Job, in Pearson et al., 1917). Adult murres also
‘‘learn’’ to use the sea wind to lift upwards into the air,
fluttering their wings rapidly to gain air speed; they
either circle around returning to alight or plop down
into the sea. Adult thick-billed murres fly from cliffs or
the sea surface, but reportedly never from land. For
most ledge nesters, fluttering while descending precedes
upward flight, and running is impossible.
The primitive ostrich (Struthio) and emu (Dromiceius)
cannot fly at all (and have relatively short wings), but
they are specialized for running (with relatively large
hind limbs). The primitive loons (Gavia) and some other
water birds (swans Cygnus, coot Fulica, mergansers
Mergus, storm-petrels Oceanites, Pelagodroma), even
the greater flamingo (Phoenicopterus rubus rosea, see
Ryan, 2001), run with feet extended and flap their wings
over the open water before launching into shallow flight.
Young waterfowl may ‘‘skitter’’ quite a remarkable
distance with the parents, or dive into the water; when
old enough, some can be induced to fly (pers. obs.).
Even as adults they do not ascend well, but this observed
running take-off does suggest that flight possibly may
have developed from bipedal running on water (hardly
likely in birds with an airfoil tail such as that in
Archaeopteryx). Water may present less ground friction
15
to the hind feet stepping upon it, if the bird does not bog
down in it. Cursorial birds such as roadrunners
(Geococcyx), some quail (Callipepla, Oreortyx), most
grouse (Dendragapus), rails (Coturnicops, Laterallus),
and purple gallinules (Porphyrula) are reluctant to take
off, flap rapidly, glide often, and even dangle the legs.
They seldom fly far when flushed (Pearson, 1917;
Hughes, 1996; Gill, 1990). Once air speed is attained,
distant flight becomes possible for some of these, but
then alighting and replenishment of the body energy
become problems. All modern birds have some bipedal
locomotion, and observations on fledglings cannot
entirely predispose one theory on flight origin over the
other. They do show that fluttering down from heights is
general. Leaping into flight following a take-off run is
avoided or impossible in most modern cursorial birds.
Some modern vertebrates do glide or flutter-glide after
rapidly and strenuously climbing or leaping to an
elevation, and the height attained is not necessarily
more than a meter or two to attain flight speed from
gravity.
5.2. Clipped parrots
Observing problems of clipped parrots flying at fast or
slow speed (with wings shortened by cut-off primary
feathers) provided us confidence in our flutter-glide
theory. Almost daily records were summarized for
observations on a wild-caught Nanday conure
(Nandayus nenday, body M ¼ 285 gm), hand-reared
sun conure (Aratinga solstitialis, M ¼ 105 gm), and
hand-reared African gray parrot (Psittacus erithacus,
M ¼ 465 gm). Clipped more than a year previously, all
three parrots were incapable of flight. They could only
flutter and fall more or less directly to the floor until
their primaries had partially grown out. The two handreared parrots, one heavier and the other lighter than
the experienced flyer, were taught to fly, practicing daily
during the year. All three initially tried to ‘‘flutter-glide’’
from a height, such as the finger or their perch,
eventually flapping across the room and down to the
floor, from perch to perch or from one’s lap or shoulder
to their perch. The heavier African gray could flutterglide, but not fly upwards to its perch until after 14
months. The sun conure occasionally hit its perch, but
usually instead grabbed the cage below it. The parrots
often flapped their wings rapidly while hanging to the
perch, prior to leaping into the air. Such flutter-gliding
from perch to perch, as well as to the ground, would
seem reasonable behavior for a primitive bird like
Archaeopteryx and its proavian ancestors (Yalden,
1971a, b).
Comparing the sun conure with the heavier African
gray parrot, the conure flapped rapidly but had a
shallow wing-beat, used gravity for dropping pitch,
turned slowly with poor control of yaw (apparently
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C.A. Long et al. / Journal of Theoretical Biology 224 (2003) 9–26
using only the tail to turn), and its synchronous,
bilateral wing-beats handled roll poorly. The African
gray now uses its wings to stall and decrease pitch, turns
fast with a controlled roll, and uses a forceful, deep
wing-beat. Both sun conure and gray clambered, seldom
flying on their own except when urged or frightened;
they both landed forcibly and clumsily on the floor.
Descending from greater heights of 20–30 m, the
speedier and heavier gray still will not alight, pulls up
and soars away, or seeks a high tree canopy and pulls up
to alight, as hypothesized for proto-birds by Caple et al.
(1983), Yalden (1971b), and Norberg (1985). The
African gray now readily descends 2–3 m to alight on
the floor. It never does so on flights of 6 m or greater,
but for those flights seeks a low perch from which to
descend to the floor or take off. The African gray has
learned to stall and hover above a prospective perch
after short (2–3 m) flights, but not well after longer
flights, usually grabbing at it with feet and beak.
However, it can now fly fast and far, for miles. The
sun conure makes descents (usually 5–7 in) without any
pause, i.e., with too much force, but never pulls up or
turns away. Its longest ‘‘flight’’ is about 10 m from a
height of 1.5 m to a perch about the same height. It tries
to lift when landing by strenuous fluttering of the wings.
J. Devine confirms our observations, in many
additional trials on parrots. In an aviary
15.3 m 9.15 m 6.1 m (height), usually 30–40 macaws
are maintained. Including some newly fledged ‘‘chicks’’
with parrots of mixed pet and wild behaviors, these
birds were rotated in and out of the flight facility. Most
of his 100 macaws had to be taught to fly. After clipped
wings grew out, Devine ‘‘would toss them (macaws) in
the air, gently at first, and they would automatically
flutter their wings for balance. Soon they were soaring
across the aviary as if they had been flying all their lives’’
(Roberts, 2000).
6. Physical theory
In running for lift-off, a profound retardation of
speed and force is created by the product of the crosssectional area of the projectile and speed divided by
mass. Forelimbs of a bipedal reptile extended into the
wind and a light body both would be detrimental,
physically retarding the speed and costly in energy spent
as extra work (Long et al., 2002). Although animals that
leap or fall from tree branches pump both forelimbs
downward simultaneously, no evidence indicates that an
ancient, bipedal runner would; to the contrary, they
likely would not. Quadrupedal runners usually swing
the hind limbs counter to the opposite forelimb. The
forelimbs counteract and counterbalance the hind limbs,
even in the evolution of bipedal runners. Only in birds
and gliders do animals leap and spread their wings or
patagia with bilaterally synchronous thrusts, and few of
them are runners.
Differing from runners, saltatorial (hopping) animals
do thrust both forelimbs forward and downward
synchronously. However, in their evolution these
animals tend to diminish their forelimbs in size and
even in functions. They tend to hold them together
instead of extending the forelimbs outward.
While running and flapping to obtain lift and air
speed, the exertion results in expending a tremendous
output of energy, especially by the hard-working hind
limbs. Running faster significantly increases both air
drag and ground friction. In archaic flyers, how long
might thrusting with the hind limbs and wings working
together continue before all metabolic energy was spent
(see Pritchard and Pritchard, 1994; Cavagna et al.,
1964)? In the evolutionary history of bipedal runners,
only the hind limbs were hypertrophied to work fast and
efficiently. Instead of increasing wing thrust (see Burgers
and Chiappe, 1999), which was lessened anyhow (Long
et al., 2002), an angle of attack combined with a
powerful downstroke would function better for a
running or leaping take-off (thereby increasing air
speed). Considering air resistance (both frictional and
induced drag against a bipedal strider), gravity, and
ground friction, a running take-off seems impossible
without highly developed wings contributing both lift
and some forward thrust (Long et al., 2002).
Lighthill (1977) did suggest that downstrokes, even
with induced drag from flapping and with a maximal lift
coefficient ðC1; max Þ; might enable a bird to fly at slow
stalling speed, with an energy penalty only 1.5 of gliding
(denominator ð12dV 3 SCd Þ; dividing again by V to
determine the least power spent). However, he calculated a huge penalty when the speed approached zero.
His penalty does not take into account the extra energy
cost of a bipedal runner with body stance nearly vertical
in a take-off run (Long et al., 2002, also see Body Form
below), nor reflect the significant cost of energy to
overcome ground friction (Von Mises, 1959; Long et al.,
2002). Furthermore, Lighthill’s slow flyer no doubt had
modern, streamlined avian form.
Flapping would seem adaptive for the fluttering
glider, because every downthrust increases lift even at
slow air speed. Best of all, there is no new expense of
unreplenished energy costs to be expended while already
exerting at full speed; indeed, much available potential
energy from gravity is available to either gliders or
flutter-gliders.
Flapping flight requires in modern birds or bats
coordinated functions of numerous muscles, tendons,
and bones. Appropriate motion of the wings into the
wind through both up and down strokes must be
controlled. Propulsion by specialized wings, said to
resemble propeller drive, integrates lift and thrust forces
while minimizing drag. The outer part of the wing
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thrusts as the inner maintains a suitable angle of attack
for lift. Archaeopteryx possibly could have leaped into
flight, as Burgers and Chiappe (1999) suggest, but not
necessarily by running and thrusting with wings.
Specialized wing feathers of this bird, with two rows
overlapping at the bend of the wing, suggest separate
inner and outer wing functions and an extensible
wing length. Yalden (1971a) discussed wing flexing in
Archaeopteryx, the primary (thrust) and secondary (lift)
feathers, and wing area appropriate for flight.
Burgers and Chiappe (1999) suggest some proavian
reptiles were preadapted to ground-up flight, having
effective flapping wings. In the tree-down scenario,
however, there is greater opportunity and likelihood of
powered flight gradually arising by adaptive and
preadaptive transitional stages (Spurway, 1955; Bock,
1965; Norberg, 1985, 1990), all of which would
contribute to continued evolutionary change. Wing
thrusting might develop eventually only if it steadily
(over many generations) increased momentum. With a
wing creating lift in descending flight, how relatively
simple it would have been to evolutionarily lengthen the
wing tips and use them for thrust. How could the protoflyer leap upward into flight before attaining complex
flight muscle and wing specialization, even before simply
lengthening the wings?
Air resistance, lV ; also called total drag ðDto Þ; is a key
force in a ballistics analysis. It should be considered as
the sum of two distinct forces, induced drag (Di ), and
frictional drag (Df ) having somewhat contrary effects.
Generally, a projectile shows an increase of drag
proportional to its velocity squared (V 2 ). A doubled
speed equals a dramatic fourfold increase of drag, and
remarkable deceleration also. Flapping projectiles have
extra resistance (Norberg, 1985, 1990; Tennekes, 1997;
Pennycuick, 1975) called ‘‘induced drag.’’ Powerful
wings pushing downward upon an air mass create a
flapping force ðK ¼ wqÞ: When air density is d; and the
wing length is b; the air mass (defined by the dimension b
as the radius of a cross-section) becomes q ¼ dCd V pb2 .
Such a product of air density and a given volume of air
allows the weight (W ¼ Mg) to be accounted for as
W ¼ dwV pb2 and the downward push to become w ¼
W =dV pb2 : An ‘‘induced power’’ sufficient to raise W
results from flapping with flight muscles.
Then the power becomes Ww ¼ ðW ÞW =dV pb2 : If the
projectile’s speed is that of a flapping bird, Di from such
flapping becomes W 2 =dV 2 b2 p because the power must
be reduced by V 2 : Often this relation is given as
L2v =ð12dSV 2 Þ; where S is the surface area. The denominator shows that flapping projectiles that accelerate are
decreasing the induced drag (but not the frictional drag,
which increases with V 2 ).
While running slowly on the ground, a projectile
suffers both induced and frictional drag. The sum of the
drags, increased somewhat by gravity on the climb,
17
attains minimal value (optimum speed) on the power
curve at greater speed than at the estimated stalling or
take-off.
Any slowing of speed dramatically increases Di :
Halving the speed increases induced drag fourfold. Slow
running must be hindered by powerful drag, but such a
drag retardation certainly would seem useful for a
fluttering and descending bird attempting to avoid
dangerous impact. Drag is of negative value to a leaping
and flapping projectile that must decelerate at every
bound, lose purchase with the ground, and with each
upward inclination work against gravity. Bipedal
runners or leapers often have no contact at all with
the ground, and suffer the attendant problems retarding
slow flight prior to lift-off.
Drag problems were solved necessarily in the evolving
ontogeny of proto-bird fledglings, and continued
throughout the phylogeny of the adult proto-birds as
flapping flight evolved. Using gravity as a helpful force
was an easy way to obtain air speed and lift to overcome
drag.
Furthermore, the squared velocity shows that by
evolutionarily increasing the wingspan, flying at slow or
moderate speeds would decrease the induced drag and
thereby create more lift. This is a reason for flyers,
gliders and fluttter-gliders (but certainly not runners) to
lengthen their wing tips (Norberg, 1985; Tennekes,
1997). Although the aerodynamic equations for gliding
and powered flight are basically the same, the flapping
of archaic wings suggest Di would have been an
important problem working against proto-birds, especially if taking off after a bipedal (mostly airborne) run
(Long et al., 2002). As mentioned above, even modern
birds have problems with such take-offs and shallow
flight (Norberg, 1990).
Archaic birds with poorly formed wings, possibly long
tails, and long (heavy, energetically expensive) hind
limbs doubtless had slow speed either in the air or on the
ground, with or without wing flapping. These protobirds would suffer enormous induced drag by flapping
(Norberg, 1985). If the speed were increased on the
ground, lift-off is not assured; the frictional (Df ), as well
as ground friction, would increase. A leap upward
would retard the forward velocity and increase adverse
gravity, indirectly retarding momentum. Thrust, if it
were regularly increased by flapping (Burgers and
Chiappe, 1999), hypothetically was combined with legdriven speed to create an ‘‘ascentional’’ force lifting the
bird from the ground. However, in acceleration for takeoff, thrust wanes instead of decreasing (see Von Mises,
1959, p. 470; Long et al., 2002). For thrust to overcome
the profound drag, both induced and ordinary, a
powerful, complex wing-driven force would be necessary. Both an angle of attack and aerodynamic lift seem
essential for a running take-off. If the proto-flyer was a
glider using gravity as its key force (Norberg, 1985,
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C.A. Long et al. / Journal of Theoretical Biology 224 (2003) 9–26
1990), lift needed for take-off would be unnecessary, but
might prove useful for lengthening range or alighting.
Tennekes (1997) discusses and even graphs variable
induced and total drag versus flying speed. He found
induced drag to dramatically increase with slowing (less
than optimal) speed. The proto-bird or Archaeopteryx
running on the ground prior to take-off would suffer
even more resistance and greater energy cost than he
suggested. Von Mises (1959, p. 469) described the
ground to air take-off in terms of Dto and V 2 ; involving
also some negative pressure forces ðF Þ between the
projectile and the ground. This ground friction increases, and the total drag retarding runners increases
exponentially as discussed previously. At the slow speed
of bipedal running, the induced drag plays a significantly adverse role. Another relevant problem is the
hindrance by an appendage or outward extension, such
as a primitive forelimb. Any lift derived notwithstanding, the resultant drag of a running body with two
appendages extended into the wind exceeds (by 30–50%)
the sum of the drag forces of all three parts. Such
‘‘interference’’ (Von Mises, 1959, pp. 108–109) of a
minor appendage or extension may more than double
the total drag of a streamlined body. Induced drag, total
drag, and drag from extended appendages all benefit the
fluttering parachutist, lending support for the parachutist-glider-flyer scenario, or just as reasonably for the
flutter-gliding synthesis. We emphasize again that the
induced and frictional drags in a fluttering fall may
be regarded as lift against the dangerous force of
gravity.
7. Space problems in leaping
To the physicist, who separates drag into vertical and
horizontal directions, air drag against the vertical force
of gravity may be regarded as a type of lift. Air
resistance (lV ), also called total drag ðDto Þ; is a force
that combines with gravity against upward velocity. But
either extended gliding or flutter gliding may soften
landings or even raise trajectories to extend the flight
paths. Putting aside for the moment the ground friction
that affects running and leaping, we analyse motion
problems (both horizontal and vertical velocities) with
regard especially to air resistance, which powerfully
affects either running or flight.
[Note: With the air drag of lV ; the differential
equations are naturally independent along different
directions, but this is not an assumption. The displacement (i.e., change in distance) is a vector and has two
components. The velocity has components Vx and Vy :
The force is also a vector. Therefore, the air drag has
two separable components (lVx and lVy ). This proves
that the drag along the x-axis is proportional to Vx and
the drag along the y-axis is proportional to Vy :]
For an airborne escaping or attacking animal, the
critical quantity may be how far and high it can move.
Such a distance also might provide fitness in movement
to food sources or finding mates. The aforementioned
forces of total air drag and ground pressure (F ), and
even the retardation by the machines themselves (see
Long et al., 2002) combine to oppose running and
leaping in archaic proto-birds. In focusing on air
resistance alone, the following equations make physical
comparison easy for leaping upward or downward
against air resistance. The values Vx and Vy are
velocities along the x (horizontal), and y (vertical) axes
with the initial values Vxo and Vyo : The simplest example
is l ¼ 0; and yðtÞ ¼ Vx0 t 12gt2 ; where t is time in units
of seconds and g is gravity (9.8 m/s2). For the leaping
process, the maximal horizontal distance is xmax ¼
2
2
V02 =g; where Vo2 ¼ Vxo
þ Vyo
and the maximal height
2
is ymax ¼ V0 =2g: One cannot obtain both maximal
distance and height. The initial value of the ascent,
then, based on Pythagoras’ theorem, is Vo : When la0;
the distance x and height y become
xðtÞ ¼ Vx0 ðMÞ=lð1 elt=M Þ;
ð1Þ
yðtÞ ¼ Mg=l þ ðM 2 g þ lMVy0 Þ=l2 ð1 elt=M Þ:
ð2Þ
We see that both x and y are reduced by l: This can be
seen more clearly in expanded form as follows:
xðtÞExðtÞ lVx0 t2 =2M;
ð3Þ
yðtÞEyðtÞ lVy0 t2 =2M:
ð4Þ
From Eqs. (1) to (4), the air resistance significantly
reduces not only the horizontal distance but also the
height from the ground. A quantitative example will be
given below. For a larger initial velocity, the reduction
becomes even larger. This limits the speed that the
animal can achieve by jumping or running, which
certainly retards take-off and inhibits self-powered
flight.
Norberg (1985) asked why an animal needs to expend
so much energy to leap fast, except to escape predators,
and running is usually faster than jumping. The
instantaneous power output or rate of work by powerful
leaps is tremendous. Aside from the loss of purchase
during leaps, with acceleration and consequent deceleration at every bound, an often ignored fact for the
leaping process is that horizontal running does not
necessarily contribute much at all to vertical velocity.
Only after the wing became specialized might the
situation improve. The hypothesized appearance of the
preadaptive wing as a functional organ to catch prey,
e.g., insects in the air (Ostrom, 1974a, b, 1976) is
physically questionable. The proponent has now disclaimed this theoretical function. Although perhaps
adding aerodynamic lift and drag due to the significant
air resistance, the forelimbs likely would be spread apart
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19
by the wind instead of closing on potential prey. Most
probably the forelimbs for bipedal runners would
diminish in size during further evolution.
Another effect of air resistance for the leaping process
is an upper limit on the animal’s weight. Even among
modern birds, which have evolved a suite of adaptations
to flight, a goose-sized bird can hardly leap into the air
(D’Arcy Thompson, 1961; Pennycuick, 1968; Yalden,
1971a). In hypothetical proto-birds, resembling the
dinosaur Deinonychus (Ostrom, 1974a), powerful hind
limbs must have evolved for leaping, as the forelegs were
relieved of body weight. It must have been cantilevered
by a heavy tail, with robust pelvis and sacral vertebrae in
between (D’Arcy Thompson, 1961). No known flyers or
gliders have heavy tails; they may be elongated and
flattened. If more air speed was necessary, it would be
much more economical for the available energy E to
gradually lengthen the legs than to increase the rate of
striding for bipedal runners (Taylor, 1977; Long et al.,
2002).
We can estimate the reduction of the leap (height) due
to the air resistance as
Burgers and Chiappe (1999) had overcome the force of
drag. A presumption is not a postulate. Their hypothesized counterclockwise rotation of the power resultant,
while indeed predictable, indicates that the force tr
wanes toward zero, instead of increasing as they claim.
Their diagram and scenario are no more convincing
than those given by Norberg for partially powered
gliding. Our equations show that spatially the process of
leaping from the ground is not favorable, but partially
powered gliding is feasible.
y* max ¼ ðM 2 g=l2 Þ ln ð1 þ lVy0 =MgÞ
Thus, due to air resistance, the time is shortened, and
for the above parameters, it is reduced to 0.52 s.
Reduction of time is certainly unfavorable to flight
evolution.
After T* up ; the vertical velocity is zero and the animal
starts to descend. The falling time from a height y with
zero initial velocity ðVy0 ¼ 0Þ is determined by the
equation
y ¼ ðMg=lÞT* down M 2 g=l2 ð1 expðlT* down =MÞÞ: ð8Þ
3
=3Mg2 :
þ M=lVy0 Eymax lVy0
ð5Þ
Suppose Vy0 ¼ 10 m/s, M ¼ 0:2 kg (an estimated
value for Archaeopteryx) and l ¼ 0:1 kg/s; then
ymax ¼ 5:10 m becomes y* max ¼ 3:36 m. Consequently, a
reduction of 1.74 m is made in the height from the
ground. Flight requires that ymax > 0; which follows
from the constraint on the ratio l between the initial
force f0 and weight
f0 ¼ lVy0 o32Mg or l ¼ f0 =Mgo32:
ð6Þ
With air resistance, an increased velocity does not
always yield flight. Two options for the animal are: (1)
to reduce the surface area A of the wing, and (2) to
adjust the flight trajectory by reducing the angle of
attack, f: The first choice is inappropriate as the animal
eventually will need wings (especially the wing-tips) in
the evolution of flight. For the second choice, a finite
angle is required to generate sufficient aerodynamic
force. To increase the angle f would decrease the time in
flight and shorten the escape distance, unless the lift to
drag increased by some means such as lengthening, not
shortening, the wings. The animal can hardly change f
except to make it smaller. Although Burgers and
Chiappe (1999) suggest Archaeopteryx and its proavian
ancestors attained lift and increased velocity of several
m/s by running and sculling the wings, they did not
consider the profound deceleration of such flapping
caused by air drag. Their diagram of forces for lift-off is
nearly the same as the diagram of forces given by
Norberg (1985) for a slowly flapping glider transitional
in a tree-down model, except there is no vector for drag.
Perhaps this presumes that the thrust described by
8. Time for leaping
We calculate how long the animal remains aloft.
Without air resistance, the time for upward and downward paths is tup ¼ tdown ¼ Vy0 =g: The total time is their
sum, tto ¼ 2Vy0 =g: For the initial velocity Vy0 ¼ 10 m/s,
tto ¼ 2:04 s. If la0; we should treat them separately.
For the upward process, or leap against l;
T* up ¼ M=l ln ð1 þ lVy0 =MgÞEtup lV 2 =Mg2 :
ð7Þ
y0
Upon expansion to second order, this equation takes
the form
yEgðT* 2down =2 l=6MðT* 3down Þ or T* down
¼ ½2y=g þ lT* 3down =3Mg1=2 :
ð9Þ
If l ¼ 0; then tdown ¼ ½2y=g : Thus, T* down is always
larger than tdown : This extra time certainly would be a big
advantage in evolution, permitting manipulation and
experimentation toward flight. However, such an
advantage immediately diminishes in the leaping process
since y cannot be too high due to the air resistance, not
to mention the limitations of the leaper’s muscular
power (Norberg, 1985; Long et al., 2002). The l cannot
become a larger value, in order to satisfy Eq. (6). These
factors greatly limit the duration in the air. If we use
y ¼ 3:36 m and the same resistance as before, then
tdown ¼ 0:828 s, and T* down ¼ 0:834 s: The difference
0.006 s is insignificant. Hence, the air resistance again
plays a negative role, even in the duration time of
leaping.
These apparent difficulties in the leaping process
become advantages for the gliding process, in time
gained, space, geometry and energy conserved. For a
horizontal escaping distance x, when the variable l ¼ 0;
1=2
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C.A. Long et al. / Journal of Theoretical Biology 224 (2003) 9–26
20
to simplify the comparison, the maximal distance xmax is
xmax;glide ¼ Vx0 =gðVy0 þ
2
½Vy0
þ 2gy
1=2
;
ð10Þ
where y is the animal’s vertical distance measured
upward from the ground or between two different
heights. If we set y ¼ 0; we then go back to the leaping
process xmax;leap ¼ 2Vx0 Vy0 =g: With the same initial
velocity, xmax;glide > xmax;leap :
9. Energy for leaping
Interestingly, xmax;glide becomes even greater with
larger y: This increment has not been studied much,
but in principle y has no limit since, as in leaping, it is
not determined by the running velocity, but by how high
the animal climbs above the ground. Falling from
heights by arboreal animals might provide time to
practice using their limbs, tail, feathers, or membranes
while gliding. Gliding is energetically ‘‘cheap’’ unless a
great deal of it were expended in climbing, which
empirically is readily and steadily replenished. Physically, an accumulation of potential energy can be easily
gained by either the relative height difference (such as
from a cliff) or by climbing upward prior to dropping
from a tree canopy. For the leaping process, the minimal
2
energy is MVy0
=2; and it would cost 10 J of energy for
an animal of M ¼ 0:2 kg to reach the instantaneous
velocity V0 ¼ 10 m/s.
Norberg (1985) and Long et al. (2002) mentioned the
remarkable output of power needed by a bipedal protobird running on the ground and concomitantly flapping
to increase speed. Norberg and Tennekes (1997) added
that gravity also enters into the problem at lift-off.
Empirical results on flight energetics of small parrots
(budgerigars, Melopsittacus undulatus) in wind tunnels
show that the minimal energy used in flapping flight is in
a fluttering descent, and the most expensive flight is
flapping upward (Tucker, 1968, 1987). When a bird
flutters upwards (as proposed by Burgers and Chiappe,
1999), its energy expenditure probably is greatest,
whereas a hypothetical flutter glider would require the
least expenditure of energy.
Wind tunnel studies obviously account for air
resistance, whereas most aerodynamic analyses have
emphasized the other forces of gravity, lift and thrust.
The energy expenses of modern flight are of great
interest, but they explain little on the theoretical costs
for a proto-bird evolving flight. However, the economy
of energetics in evolution likely parallels that observed
in the wind tunnel, as both phenomena relate to
underlying physics. Adaptations for physiological economy in evolution probably follow a similar sequence
(e.g., fluttering down, horizontal flight, then fluttering
upward, which seems probable for avian flight evolution). Even a modern bird with relatively larger and
superior wing musculature, high metabolism, light hind
limbs and tail, fast cruising speed, and with slow flight
possibly as low as 8 and perhaps up to 20 mph, expends
something like 33% more power needed for ascending
rather than moving horizontally. When a fluttering bird
descends, the gravity supplies some of the energy.
A budgerigar gliding at 15 , with a glide slope of
26%, travels 4 m with a height loss of 1 m. A heavier
bird such as a seagull may glide 10 m for every m of
altitude lost. A bird with many adaptations to flight and
body weight as heavy as a seagull requires for take-off
four times as much power as in ordinary flight. It could
not sustain that expenditure many seconds. ‘‘Most birds
prefer to take off from a tree or some elevated object.
Starting with a brief dive, the bird gains the necessary air
speed by letting gravity do the work’’ (Tennekes, 1997;
Chatterjee and Templin, 2003). The experimental
evidence for avian expediency agrees with Eq. (9)
and (10).
10. Body form
Some of the following information relates to increasing the length of the glide path, discussed also by
Norberg (1985, 1990). Gliders do this by increasing
speed or obtaining lift from elongated wings. We discuss
the resistance of air to the dimensions of the flying
animal, the vertical versus the horizontal. In the
horizontal plane, the resistance l is very small, while
in the vertical plane, l is large. This disparity works
against deep (high) and narrow bodies for any
individual, or streamlining body form for the species
in ages of evolution. A bipedal runner with its vertical
stance presents its entire frontal plane to the wind, and
during evolution lengthens the hind limbs for superior
striding (Long et al., 2002). A glider avoids much
resistance by its flattened shape. From Eq. (1), we know
the range or escape distance is inversely proportional to
the resistance lx: Smaller lx certainly helps the animal
reach a longer distance. Secondly, a long duration time,
Tdown ; is needed, which is determined by the height y and
resistance ly [see Eq. (9)] requiring a large value for ly:
The prolongation helps the animal attain more opportunity to manipulate its rudimentary wings. A leaping or
flying animal actively and evolutionarily faces increased
vertical air resistance, but may decrease the horizontal
resistance by flattening its projectile shape (leaping
spread-eagled, extending the limbs outward, and pushing against the air). Not only the increased wingspan
and lift-to-drag ratio, but the deep breast (sternum
keel and massive pectoral musculature) of modern birds
was absent in Archaeopterx (Ostrom, 1971; Ostrom
1974a; D’Arcy Thompson, 1961). A broad and
shallow form suggests tree-down evolution of airborne
animals.
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C.A. Long et al. / Journal of Theoretical Biology 224 (2003) 9–26
Fig. 2. Physical theory does not favor the ground-up origin of flight in
space and time. The initial velocities are Vxo ¼ Vyo ¼ 10 m/s, and the
mass of the bird is M ¼ 0:2 kg. (a) With an increase of l; the horizontal
distance x is reduced. (b) With an increase of l; the vertical distance y
is reduced. (c) and (d) The maximal distances for x and y are decreased
as the resistance l increases. (e) The duration time is decreased
monotonically with increased l:
In summation (Fig. 2), we plot how the resistance l
influences the horizontal xðtÞ and vertical yðtÞ distances
as well as the duration time in air. The initial horizontal
and vertical velocities Vx0 and Vy0 are 10 m/s. The
theoretical mass of the bird is M ¼ 0:2 kg. Fig. 2(a)
shows that with an increase in resistance, the horizontal
distance is decreased while the maximal height y is also
reduced (see Fig. 2(b)). Their explicit dependence on l
becomes more obvious if we plot the maximal xmax and
ymax as a function of l: Spatially, the leaping process is
not favorable. As one sees from Fig. 2(e), the duration
time in air is also monotonically reduced. In terms of
time, the ground-up process for lift-off is unfavorable.
Norberg (1985, 1990), Tennekes (1997), Long et al.
(2002), and others arrived at similar conclusions.
11. Discussion
The basic dialectics on the origin of flight are: (1)
running and leaping into flight and (2) gliding to flight.
(1a) A biologically powered, ballistic bipedal form ran
and leaped and thrust itself into its trajectory, with
distance, maneuverability, and lift evolving later.
Physically, this evolution seems unlikely. (1b) A
biologically powered, ballistic bipedal form leaped from
the ground after running and lifted itself by hypothetical
wings, with maneuverability and thrust subsequently
evolving. Physically, this also seems unlikely, but has
occurred in some modern waterfowl. (2a) Some arboreal
animals mutated patagia and became gravity-powered
parachute-gliders, with the potential energy related to
21
the height climbed into trees. The biological fitness
improved indirectly with lengthened escape distance or
attack distance. This scenario fits the principles of
physics, and for pterosaurs and certainly bats has
become the accepted explanation. Specializations of
modern gliders seem extreme for them to possibly evolve
to powered flight; their expense of energy drops
abruptly, remaining low for long glides (Scheibe and
Robins, 1998). (2b) An evolutionary process proposed
herein, for birds, initially involved fluttering during
descent. It required a scansorial body form, such as
having clawed fingers or prehensile feet. It required
feathers. Flutter-gliding could apply to ledge-nesting,
piscivorous, bipedal reptiles and altricial proto-birds.
Biological fitness was indirectly increased both by the
height climbed (potential energy) and the slowing by air
resistance of the dangerous momentum of vertical fall
(Eq. (9)). Fluttering the paired wings (with firm leading
edges) lengthened the glide by horizontal thrust,
certainly decreased momentum of the vertical fall, and
led in time to the evolutionary expansion of distal areas
at the wing-tips (Bock, 1969; Norberg, 1985, 1990;
Tennekes, 1997). These improved maneuverability,
permitted an upward pitch, and significantly raised the
lift to drag ratio. The bird’s wing plausibly evolved in
this fashion from the forelimb in either small arboreal
dinosaurs or archosaurs ancestral to birds. The ‘‘flutterglide’’ explanation is a synthesis of the two earlier
theories, in that the potential energy of gravity was
available to feathered arboreal clamberers, and the air
resistance that limits leaping or running was used for
several advantages by vigorously flapping feathered
wings.
Fluttering as a prelude to flight was suggested for bats
(Caple et al., 1983) which, considering the patagial
function, likely never used this process. It was described
better by Rayner (1991) as a fluttering ‘‘model’’ where
‘‘erratic’’ flapping causes useful ‘‘aerodynamic forces’’
and flight evolved almost by ‘‘saltation.’’ According to
Rayner, such a process must ‘‘fail’’ because it takes no
account of the extreme morphological, physiological
and behavioral specializations required for flight, and
‘‘need not be considered.’’ However, (a) in proto-birds
only archaic preadaptations, not specializations, were
available for the early phases of flight evolution, (b) in
gliding theories there usually is accepted an early
adaptive zone called ‘‘parachuting,’’ and (c) fluttering
wings synchronously is not ‘‘erratic.’’ Not only leaping
for some benefit from a height, but also avoiding
dangerous impact at landing would increase survival.
Fluttering certainly functions to brake a fall, may permit
alighting in a tree, and may incline the trajectory from
free fall, increasing the horizontal distance and time in
the air.
From several considerations (Heptonstall, 1971;
Bramwell, 1971; Yalden, 1971a, b; Norberg, 1985) of
ARTICLE IN PRESS
22
C.A. Long et al. / Journal of Theoretical Biology 224 (2003) 9–26
possible flight of Archaeopteryx, where its estimated
weight varies (200–500 g), the estimated area varies for
lift (including with or without the body strip, and even
that of the elongate tail), and the estimated velocity also
varies (V or V 2 ), but there seems little doubt that this
bird could flap, glide, and alight in trees. Archaeopteryx
could easily fly from tree to tree; ‘‘seems poorly adapted
for landing on the ground,’’ when landing in a tree with
‘‘all four limbs’’ would not require ‘‘particularly well
developed stability and movement control;’’ its wings
were ‘‘assumed to supinate during the upstroke so that
the air meets their ventral sides over the entire wing
stroke and net thrust during the downstroke [probably]
exceeds net drag during the upstroke;’’ and ‘‘from the
beginning, flapping may have been used to increase the
glide-path length.’’ In our equations, the conventional
V 2 instead of the use of V alone would only strengthen
these predictions.
A proposed ‘‘new’’ stooping theory by Garner et al.
(1999) and Taylor (1999) is neither feasible nor new (the
dropping into a glide and subsequent flapping flight is
the widely accepted explanation for flight origin in bats,
which do not have in Recent or fossil species any
possible running capabilities). Their deviant form of the
‘‘tree-down’’ theory overlooks the role of air resistance
for a poorly feathered archaic form, and, indeed, they
proposed that the proto-birds accelerated by virtue of
body mass to stoop at prey. Only after the danger of
impact is overcome might proto-birds or proto-bats
practice hawking predation utilizing ‘‘wing drag’’ and
‘‘lift’’ that obviously were later specializations. Lessening impact may be observed in modern birds of inferior
flight, including observations discussed herein, and their
own observations on newly fledged, tree-nesting ducks.
Another peculiar but interesting deviation of the treedown model is based on the recent observation of a
behavior of partridge chicks (Alectoris chukar) rapidly
and strenuously ascending nearly vertical surfaces, to
which the running feet were applied by thrust from the
flapping wings, before falling into flight (Dial, 2003). To
the problems of air resistance, surface friction, and great
energy expense (see Long et al., 2002), another form of
drag becomes paramount ðDclimb Þ: Not only must the
climber have in reserve sufficient energy for great
vertical acceleration, but also a small body mass. The
usual understanding of the tree-down model is that the
climbing is a separate energy cost not expended so
suddenly and so close to the time of gliding. We have
proved in our paper that Vymax limits Vxmax ; so that we
would predict that a rapidly ascending partridge chick is
limited from moving far or fast in a horizontal direction
without continuous induced power. Were it not for what
we have described as flutter-gliding, to soften impacts on
landing, an accelerating vertical climb and subsequent
fall would confer limited survival and evolutionary
fitness. If these partridges are relevant to flight
evolution, they do not exemplify either of the classical
theories but do fit in our synthesis.
In comparing four possible modes of evolution of
flight, a tabulation (Table 1) of suggested fitness
attributes and their negatives was made for Archaeopteryx, albeit the data are based on some arbitrary
assumptions. They reveal a marked preponderance of
positive values for our flutter-gliding synthesis. The total
scores tell only part of the comparison, because both
ground-up hypotheses had numerous negatives (one or
some of which might falsify these hypotheses entirely).
For example, the competition of body energy E for both
running and flapping is a serious problem, not only for
the ascending proto-bird itself, but for subsequent
evolutionary modification. A possible negative attribute
of falling forcibly in the arboreal hypotheses is lessened
by either flutter-gliding or possibly by extended gliding,
and further compensation is likely in survival from
predation, increased mobility, or obtaining a new food
source. Although climbing tree trunks and crags may
have been impossible (a negative attribute?) by use of
the hind feet, which, nevertheless, are reportedly
adapted to an arboreal niche, flitting from branch to
branch or rock to rock and clambering with the clawed
wings seem plausible. This seems characteristic even for
early birds such as Archaeopteryx. Their leaping
capability, moreover, increases the initial velocity of
any necessary descent and might facilitate leaping from
the ground into low branches of trees, as observed in
roadrunners and tree squirrels (Sciurus sp.). Other
attributes in the table are discussed in the text above
and in the cited literature. The results supplement our
ten physical equations favoring both tree-down theories
as parsimonious and logical.
A dendrogram (Fig. 3) was generated based on a
character matrix (from Table 1) using the biological and
paleontological information (scaled 0–2, with a few
queries). The branches obtained represent hypothetical
models on the origin of flight, not the organisms (i.e.,
phyletic lines) evolving flight themselves. This parsimonious tree, containing only 19 steps, supports our
synthesis. The two kinds of tree-down models branch
highest on the tree, and running take-off seems most
disparate. If one combines the gliding with either of the
running branches of the tree, it would add 9 steps. It
adds 10 steps to combine fluttering (i.e., flapping) with
running.
We generated a second matrix to include data
published on the reptilian, bird-like fossils from Texas
and China, including such traits as feathers, a supracoracoideus, and the keeled sternum in Protoavis. Exactly
the same dendrogram was created. When we combined
the running take-off with running on water in different
ways, or especially when gliding was combined with
flutter-gliding, we again determined only 19 steps. This
suggests a general resemblance of running to take-off on
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C.A. Long et al. / Journal of Theoretical Biology 224 (2003) 9–26
23
Table 1
Attributes of fitness for Archaeopteryx in four models on the origin of flight
Fitness attributes
Small pectoral muscle mass
(a) ? supracoracoideus
(b) No keel on sternum
(c) Small sternum
(d) Short anterior coracoids
(e) Furcula
Swivel-wrist in relatives
Feathered wing
Homoiothermy
Body feathers
Air resistance (drag)
Body energy (E), flight vs. running
Energy from gravity (Mg)
Lateral patagia lacking
Elongate and flattened tail
Large and elongate hind limbs
Ground speed
Ground friction
Parachuting, reduced danger
Unencumbered take-off runs or descents
Weight (0.2–?0.5 kg)
Too heavy
Too light
Foot structure appropriate for the hypothetical function
Clambering by clawed manus
Evidence/some modern adult birds
Evidence/clipped birds or fledglings
Totals
Ground-up theory
Tree-down theory
Running on land to take-off
On water
Gliding
Flutter-gliding
+
+
+
+
?
+
+
+
?
+
+
+
+
?
+
+
?
?
+
+
OK
OK
OK
OK
+
+
+
+
+
OK
+
+
+
N/A
N/A
?+
+
OK
OK
OK
OK
+
+
+
+
+
OKa
+
+
+a
+
+a
N/A
N/A
?+
+
b
+
?
?OK
7
b
?
?
+
4
OK
b
+
+
?OK
?OK
+9
OK
OK
+a
+
+
+
+16
N/A means not applicable and OK means approximately neutral. These supplement the 10 physical equations herein, also useful in comparing the
several models.
a
See text.
b
Retards speed.
land or water, as well as that for both tree-down models.
All other combinations increase the number of steps by
7 or more. If one hypothesizes that running take-offs
and flutter-gliding are somewhat similar owing to
flapping and thrusting of the wings, which seems
reasonable, such an arrangement becomes less parsimonious. This result surprisingly points up our conclusion
that fluttering can be, on the one hand, disadvantageous
against air drag during flight, but, on the other hand,
advantageous in providing lift and safe landings in the
tree-down models. The result disparages alternate
leaping (i.e., hopping) and flapping or leaping and
gliding, which is a deviant form of the ground-up
running to take-off model. The poorest results arise
from double splitting of the two kinds of gliding and the
two kinds of running. The most feasible pairing seems to
be of the two kinds of gliding. Both are tree-down
models. The data for them seem to reflect evolutionary
fitness.
From a similarly created matrix based on ballistic
features and analysis of the aforementioned ten equations, eight features were negative for either running
take-off or running on water, and there was a net gain of
five (for gliding) and six for flutter gliding. Flutter
gliding might even multiply advantages by multiple wing
strokes, in extending the length of the maximal glide,
and in providing lift and power by multiple thrusts. This
multiplication might be seen also in the running models,
except that both air resistance and ground friction
retard such slow flight. Bipedal running with only one
foot or none on the ground, and with wings extended,
resembles slow flight (Long et al., 2002). Running
significantly increases the induced drag, especially in
lightweight forms running with their bodies exposed to
the force of the wind. The ballistics dendrogram
generated is exactly the same as either dendrogram
based on biological information (Fig. 3); manipulating
the branches produces similar results. Splitting the
dichotomy of gliding and flutter gliding adds five steps
to 12. This gives the same result as splitting the two
kinds of running. Taken together (see Fig. 3) for
biological or ballistics information appraised objectively
and quantitatively, the three dendrograms show that the
flutter gliding synthesis is at least as robust as the gliding
ARTICLE IN PRESS
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C.A. Long et al. / Journal of Theoretical Biology 224 (2003) 9–26
especially Eqs. (1)–(10), provides merit for hypothesized
‘‘flutter-gliding.’’
12. Summary
Fig. 3. Dendrogram comparing four models of flight origin, based on
biological and paleontological characters (either of two matrices, see
Methods) in Table 1. The same tree also resulted from a third matrix
based on ballistics features (see Methods). Splitting either the two
gliding models, or the two running models, adds steps to a
parsimonious tree (seven or more for the biological analyses, five or
six for the ballistics analysis). Combining running with gliding or
flutter-gliding adds a significant number of steps in either analysis. The
synthesis for flutter-gliding is at least as robust as the gliding model,
more so than are the hypotheses for running, which are paired on the
basis of numerous negative features.
model. Both seem superior to the running models, which
are also closely paired, because in the biological
analysis, theoretical considerations of fitness over long
time spans are included. The ballistics analysis appraises
immediate effects on a projectile (mostly of the
important force, air resistance). All three dendrograms
show pairing, and these affinities result from traits
perceived as obviously energy-saving or contributing to
fitness. One may expect physical principles to underlie
both modern-day performance and evolution.
Limitations of running and flapping from the ground
into flight, e.g., ground friction and loss of purchase
(costing loss of distance, height, and ground speed, and
requiring coordinated synchrony of both wing strokes
also working against bipedal striding), with air resistance against the archaic wings, hind limbs, and erect
body favor an arboreal tree-down origin. But air
resistance was the only force softening the dangerous
fall of the proto-bird. Some of the evidence for and
against the running and leaping theory becomes applicable in a flutter-glide scenario, whenever falling was
beneficial but falling forcibly was not. The evidence,
The running and leaping (ground-up) origin of flight
in vertebrates is not feasible physically, in space or time
considering air resistance or forces against bipedal
running. Air and mechanical resistance and energy
allocation from hind limbs to wings all favor the treedown models over running and leaping in the origin of
flight of vertebrates. Gliding or flutter-gliding makes use
of the potential energy from height, and the air
resistance in falling may have been useful both to lessen
the momentum of the fall (especially in flutter-gliding)
and lengthen the descent. Avian features were considered in other reptiles, both Archosaurs and theropod
dinosaurs. Interpretations of the energetics, body form,
and behavior of some modern birds and the relation of
Archaeopteryx to its habitat support the synthesis of
flutter-gliding. We hypothesize that in the feathered
proto-birds, flutter-gliding was functional and highly
adaptive during descent from high perches, such as trees
or cliffs, and preceded gliding with set wings and
eventually flapping flight. Physical theory based on
Newton’s principles supports this synthesis, and rules
out the ground-up theory.
Acknowledgements
We thank Walter J. Bock, Professor of Evolutionary
Biology at Columbia University for reading and
commenting on the manuscript. We are also grateful
to Prof. Arthur J. Pejsa for much advice and for
pointing out various relationships between the drag and
the velocity for metal aircraft, as discussed in his book
Modern practical ballistics, 1989. We would like to thank
Prof. David Hillier, Biology, University of WisconsinStevens Point, for assistance with computers and other
advice. Prof. Christopher Yahnke, Biology, University
of Wisconsin-Stevens Point, assisted us with the development of character matrices and dendrograms.
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