# Airplane Orbits and Satellite Orbits and Orbitfall: Physics Hidden in Plain
Sight

John P. Boyd
Department of Climate & Space Sciences and Engineering
University of Michigan, 2455 Hayward Avenue, Ann Arbor MI 48109
<EMAIL_ADDRESS>
###### Abstract

An airplane flying at constant speed and altitude is an example of physics
invisible to the pilots and passengers, but visible to remote observers and
manifest in the mathematics. The optimum flight path is an arc of a Great
Circle, specifically that circle which is the result of rotating the equator
to intersect the origin and destination airports. In order that the velocity
vector remain parallel to the surface of the spherical earth, a centripetal
force is required to rotate the velocity without altering its magnitude. This
force must be radially inward and thus parallel to the local vertical. The
assertion that “lift balances gravity” is only an approximation. To follow the
curve of the earth, the vertical component of aerodynamic lift must be
_slightly weaker_ than gravity so that the plane can be in “orbitfall”. That
is, to follow the curvature of the earth, maintaining a constant distance $R$
from the center of the earth while flying at a constant speed $V$ and
generating a vertical lift per unit mass $L$, the plane and all inside it must
fall towards the center of the earth with an acceleration, the “orbitfall
acceleration”, $g_{orbitfall}\equiv g-L=V^{2}/R$ where $g$ is the usual
gravitational acceleration constant. If the plane travels a fraction
$\digamma$ of the earth’s circumference , then it must execute $\digamma$ of
an outside loop. This requires that the pitch angle $\xi$ must rotate nose-
down at a rate of $d\xi/dt=V/R$. The dynamic stability of a non-military
aircraft makes this pitch change continuously without pilot intervention: the
horizontal stabilizers act as weathervanes, suppressing small perturbations so
that the longitudinal axis remains at a constant angle of attack. Neither
pilot nor passenger is aware of the orbitfall or the automatic pitch changes —
physics invisible but essential.

To be submitted to _American Journal of Physics_

> “All that’s needed for a plane to fly around a spherical earth is a
> miniscule difference betweeen lift and weight.”

— “roohif” in [1]

##  1 Introduction

If an aircraft is flying parallel to the earth’s surface, maintaining a
constant altitude and thus flying an arc of a Great Circle , then constant
speed $V$ requires that the net horizontal force (thrust of engines minus
aerodynamic drag) must be zero. However, as shown in Fig. 1, following the
curve of the earth requires that the the pitch angle $xi$, the vertical and
horizontal coordinates axes, and the orientation of the velocity vector must
_rotate_ in synchronization. For a flight that covers a fraction $\digamma$ of
the circumference of the earth, all these angles and vectors must rotate
through a total of $2\pi\digamma$ radians.

The velocity vector will rotate if and only if there is a centripetal
acceleration, a force acting towards the center of the Great Circle. For
movement at constant speed in a circle, the centripetal force is always a
_normal_ force, that is, a force that acts perpendicular to the velocity
vector. Normal forces _turn_ the velocity vector without altering the speed
$V$. But what is the source of the centripetal force? And how do we calculate
it as a function of $g$, $R$ and $V$ where $g$ is the gravitational constant
at distance $R$ from the center of the planet,

$~{}g=g_{E}\dfrac{R_{E}^{2}}{R^{2}},\qquad g_{E}$ (1)

where the gravitational constant $g_{E}$ at the earth’s surface is about
$9.81$ meters/second-squared.

Figure 1: A schematic diagram of the flight of an airplane on an arc of a
Great Circle. The shaded disc is a side view of the sphere, which has a radius
of roughly $R_{E}=6.366\times 10^{6}$ m. The dashed curve is the Great Circle
which is the route of the aircraft; its radius is $R=R_{E}+H$ where $H$ is the
altitude of the jet, typically $H\approx 12$ km. The aircraft icons show the
orientation of the aircraft at different points on the Great Circle. To fly a
path of 1/8 or 1/4 or 1/2 of the Great Circle, the plane must pitch nose-down
by 45 degrees, 90 degrees or 180 degees, respectively. These changes in the
pitch angle $\xi$ are changes in this angle as seen by an observer off the
globe, such as in a spaceship; this is the viewpoint of this schematic. In
contrast, the pilots and passengers in the plane perceive no change in pitch
whatsoever. A centripetal acceleration is needed to rotate the velocity vector
through an angle which is equal to the change in the pitch angle $\xi$; this
downward acceleration has the magnitude $a_{orbitfall}=g-L$. The black arrows,
attached to each aircraft icon, show the direction of the orbitfall
acceleration. This is always normal to the plane’s velocity vector, that is,
radially inward (in spherical coordinates) towards the center of the earth;
this direction is downward in the local vertical direction as perceived by
those on the airplane.

Although space is three-dimensional, the Great Circle lies entirely within a
plane that intersects the center of the planet. This allows us to use a two-
dimensional polar coordinate system with the origin at the center of the
earth, $r$ as the radial coordinate and $\theta$ as the polar angle. Newton’s
equations in an inertial reference frame are, with the forces per unit mass on
the right side (Eq. 2.14) of [2])

$\displaystyle\dfrac{d^{2}r}{dt^{2}}-r\left\\{\dfrac{d\theta}{dt}\right\\}^{2}$
$\displaystyle=$ $\displaystyle-g\,+L$ (2)
$\displaystyle\dfrac{d^{2}\theta}{dt^{2}}+2r\dfrac{dr}{dt}\dfrac{d\theta}{dt}$
$\displaystyle=$ $\displaystyle 0$ (3)

where the speed $V=r\,d\theta/dt$.footnoteAirplanes fly slightly nose up so as
to generate a positive angle-of-attack, as explained in aeronautics texts; the
lift vector is not purely vertical, but includes a small horizontal component
that is cancelled by the thrust generated by the propeller or jet engine. Only
the vertical component of lift is relevant to the vertical balance of forces.

Since for a constant altitude, constant velocity orbit, $r(t)=R$, $dr/dt=0$
and $r\,dtheta/dt=V$, the equations simplify to

$\displaystyle\dfrac{V^{2}}{R}$ $\displaystyle=$ $\displaystyle g\,-\,L$ (4)
$\displaystyle 0$ $\displaystyle=$ $\displaystyle 0$ (5)

A space station or satellite is the special case that the lift is zero. The
speed $V_{freefall}$ required for a space station to orbit without lift is

$\displaystyle V_{freefall}$ $\displaystyle=$ $\displaystyle\sqrt{g\,R}$ (6)

Every part of the space station including the astronauts is accelerating
downward at an acceleration $g$ where $g$ is defined by Eq. 1. All these
downward accelerating objects are said to be in “freefall”. The astronauts and
everything loose appear to be experiencing no gravity. This is an illusion, a
spectacular counter-example to the maxim “trust your senses”; in reality,
everything in the space station is accelerating towards the center of the
earth at an acceleration $g$. The space station or satellite does not hit the
surface of the earth, or even change distance from the surface because (i) the
spacecraft has a high horizontal velocity and (ii) the surface of the earth is
curving downward, away from the falling spacecraft.

An aircraft circling the earth at constant altitude is doing a perfect
imitation of the space station except that the downward acceleration, the
“orbitallfall”, is

$\displaystyle~{}a_{orbitfall}=g-L$ (7) $\displaystyle=$ $\displaystyle
V^{2}/R$ (8)

When the aerodynamic lift is not zero, it is possible to maintain constant
altitude with much smaller speed $V$,

$\displaystyle V=V_{freefall}\,\sqrt{1-L/g}$ (9)

A numerical example is instructive. If the jet is traveling at 255 m/s, then
$R=6.378\times 10^{7}$ meters gives

$\displaystyle V/V_{freefall}\approx 0.032\qquad\Leftrightarrow\qquad$ (10)

or in other words the orbital speed of a space station is about thirty-one
times larger Eq. 7 shows that the vertical lift is smaller than $g$ by
slightly more than 1 part in a thousand or about 1 centimeter per second-
squared.

Airplanes do not have instruments to measure lift directly; a pilot climbs to
the target height and then adjusts the trim, pitch and thrust until the
vertical velocity is zero. Physics then requires that the lift be slightly
smaller than gravity.

##  2 The Ever-Changing Pitch of the Airplane and Inherent Stability

> “A wing with a conventional aerofoil profile makes a negative contribution
> to longitudinal stability. This means that any disturbance (such as a gust)
> which raises the nose produces a nose-up pitching moment which tends to
> raise the nose further. With the same disturbance, the presence of a
> tailplane produces a restoring nose-down pitching moment, which may
> counteract the natural instability of the wing and make the aircraft
> longitudinally stable (in much the same way a weather vane always points
> into the wind). ”

— Wikipedia article _Tailplane_

A bicycle is dynamically unstable if the wheels are not turning; stop
pedaling, and it will slow and then fall over in seconds, which is why a
kickstand is mandatory. When moving at sufficient speed, however, the wheels
acts as gyroscopes, resisting perturbations, so a child can shout, “Look, Mom!
No hands!”

Airplanes in flight are also dynamically stable. Without pilot intervention,
aircraft automatically correct for small perturbations, such as internal
gravity waves, wind shear, convective plumes and the constant buffeting by
convective bubbles.

The horizontal parts of the tail fin are the “horizontal stabilizers”. The
elevators, which allow manual control of pitch, are moveable panels embedded
in the horizontal stabilizer.

The horizontal stabilizers provide stability in pitch. As the aircraft orbits
around the earth at a constant altitude, the horizontal fins and the fuselage
act like a weather vane; the flow deflecting off the the horizontal
stabillizers provides a torque that rotates the airplane about the pitch axis.
This axis is parallel to the line from one wingtip to the other wingtip. This
dynamical stability, this weathervane effect, keeps the aircraft always at the
proper pitch parallel to the sphere as illustrated in Fig. 2.

Figure 2: If the aircraft is pitched nose-high as shown on the left while the
center-of-mass is moving horizontally, the airflow [black arrow] deflecting
off the horizontal stabilizer fin [red] will create a torque that will rotate
the airplane nose-down. This “wind vane effect” realigns pitch angle as shown
on the right.

.

_Large_ perturbations, of course, require manual adjustments by the pilot.
Only _small_ perturbations are damped out by the horizontal fins in the tail.

This is not a problem for constant altitude flight. As the airplane flies an
arc of a circle, the pitch needs to be adjusted _smoothly_ and _continuously_
in what, from a calculus viewpoint, can be interpreted as an infinite number
of infinitesimal corrections.

##  3 Summary

Proponents of the “Flat Earth” have invoked airplane orbits as important
evidence for their cause. Their argument notes, correctly, that (i) the
aircraft must fall around the curve of the sphere and (iii) the pitch of the
jet must rotate through a fraction $\digamma$ of a full circle when flying an
arc that is a fraction $\digamma$ of the circumference of the earth.

Their false part of their argument is the assertion that neither can happen
without constant intervention of the pilots and awareness of pitch-down-and-
fall by the passengers; therefore, the absence of these perceptions implies
that the surface of the planet is flat. We have shown that this not true;
orbitalfall (downward acceleration) and pitch adjustments are automatic,
unperceived by the passengers, requiring no action by the pilots.

The other insight is that satellite orbits are just a special case, namely the
special case of _zero lift_ , of the larger family of objects flying at
constant speed and altitude above a spherical planet.

## References

  * [1] roohif, Quantum Eraser Is Not a Pilot.  video, October 22 2018.  Channel= roohif , duration=7:26.
  * [2] H. M. Stommel and D. W. Moore, An Introduction to the Coriolis Force, Cambridge University Press, Cambridge, 1989.  297 pp.