Shelf 01
Light & optics
Why distant things vanish from the bottom up, why they sometimes come back, why the sky is
blue and the horizon is not where bare geometry puts it. This is the most-invoked physics in the
whole argument and the place where this reference makes most of its concessions.
Constants
cspeed of light in vacuum299,792,458 m/s
nrefractive index of air at sea level1.000293
nrefractive index of water1.333
kterrestrial refraction coefficient, Gauss near Hannover0.13
7/6effective-radius multiplier for light1.1667
4/3effective-radius multiplier for radio1.3333
dN/dhrefractivity gradient, standard atmosphere−40 N-units/km
REarth’s mean radius3,959 mi · 6,371 km
R′effective radius for light, R × 7/64,619 mi · 7,433 km
1 arcsecond1/3600 of a degree1 ÷ 206,265 radian
Formulas, and how they connect
One relation, read two ways. The distance to your horizon and the amount
a distant object is hidden are the same equation solved for different letters. Square the first
and divide by 2R and you get the second back.
d = √(2 × R′ × h)
d = horizon distance · h = your eye height · R′ = effective radius
hidden = (D − d)² ÷ (2 × R′)
D = distance to the object. Subtract your own horizon first, then square.
One horizon formula, three constants. The only thing that changes is how
much the air bends what you are looking with.
1.22 × √hpure geometry, no bending at allk = 1.000
1.32 × √hlight through ordinary airk = 7/6
1.41 × √hradio waves, which bend morek = 4/3
h in feet, d in
miles. In metric the same three become 3.57, 3.86 and 4.12, with h in meters and d in
kilometers — the coefficient changes, the formula does not.
R′ = R ÷ (1 − k) and k = 157 ÷ (157 + dN/dh)
Where 7/6 comes from. A standard atmosphere gives dN/dh ≈ −40, so
k ≈ 157/117 ≈ 1.34 as an effective-radius multiplier for radio. At dN/dh = −157
the denominator hits zero, the ray bends exactly as hard as the Earth curves, and you have ducting.
dip ≈ √(2h ÷ R′) exactly: θ = arccos(R ÷ (R + h))
How far the horizon sits below eye level. The approximation is good to a fraction of a
percent at any altitude a human reaches.
n₁ × sin θ₁ = n₂ × sin θ₂
Snell’s law. Light entering a slower medium bends toward the perpendicular. Every
refraction effect on this shelf is this equation applied to air whose density changes with height.
scattering ∝ 1 ÷ λ⁴
Rayleigh scattering. Blue at 440 nm scatters 5.7 times as much as red at
680 nm, which is why the sky is blue, why sunsets are red, and why infrared cuts haze.
y = θ × D
What you can resolve. θ in radians, so an angular limit covers more real height the
further out you look. This is the formula that decides every “it was too small to see” argument.
Mechanisms
Refraction. Air is denser low down. A ray traveling near-horizontally
has its lower edge in slower air than its upper edge, so it swings gently downward — the
same reason a shopping trolley veers when one wheel hits grass. The ray therefore follows the
surface further than straight-line geometry allows, which is why the horizon is always a little
beyond where bare trigonometry puts it.
Ducting. When the air is warmer than the water, the density
gradient steepens and the bending increases. Push it far enough and the ray curves as hard as
the Earth does: the surface layer becomes a waveguide, and things that geometry says are hidden
are lifted back into view. Common over the Great Lakes in spring, cold water under warm air.
Inferior mirage. The opposite case, and the one that fools people into
the wrong answer. When the surface is warmer than the air, the gradient inverts and
creates a false horizon below the true one. Objects beyond it lose their bases —
bottom-preferential loss that has nothing to do with curvature. The tell is an inverted copy
of the object hanging below the cut, and a horizon that shimmers.
Scattering. Air molecules scatter short wavelengths far more strongly
than long ones. Looking up through a short path, you see mostly scattered blue. Looking along
a long path at sunset, the blue has been scattered away before it reaches you and what is left
is red. Looking at the limb from altitude, the path is longest of all, which is why the thin
bright band appears at the edge of the Earth in every high photograph.
Perspective, and what it does not do. Distance makes an object smaller
in both dimensions at once, the way a photocopier reduces a picture. It never removes the
bottom while leaving the top sharp. Occlusion does exactly that, along a hard horizontal line
sitting on the horizon.
Limits
Magnification cannot see around a corner. Once something is genuinely
below the horizon, no telescope restores it. If zoom does bring a hull back, the cause
is refraction, not perspective — a property of the air, not of apparent size.
Over water, the air beats the lens. A sightline running for miles
through a turbulent boundary layer is limited by that turbulence, not by aperture. Near the
horizon, 5″ is an excellent day and 20″ is ordinary. Buying a bigger instrument
barely moves the number.
Refraction is variable, and a fixed allowance can mislead. The
coefficient swings with temperature, humidity and the water-air temperature difference. This is
not a modern caveat: the Encyclopædia Britannica was warning surveyors about it
in the 1850s.
Bottom-loss alone is not proof. An inferior mirage produces it too.
Log the water and air temperatures, or the observation cannot distinguish the two.
Worked examples
A ship at 7.5 miles, eye 6 ft. Horizon at 3.24 mi, overshoot 4.26 mi,
hidden height 10.4 ft. At 20″ you can resolve 3.84 ft at that range, so the missing hull
is 2.7× your own limit. Squaring the full 7.5 miles instead gives 32 ft, three times too
much, and is the single most common error in this subject.
Willis Tower from the Michigan shore. Haversine gives 56.9 miles. With
a 6 ft eye height the horizon is 3.0 mi, leaving 53.9 mi of overshoot and 1,940 ft hidden
against a 1,451 ft building. Geometry says the whole thing is gone. The photographs exist
anyway, and refraction over cold water is why.
Shkhara from 10,300 ft. Observer horizon 134 miles, the summit’s own
horizon 173 miles, so the peak can just clear at 307 miles. Beyond about 323 miles it cannot be
a direct line of sight at any magnification.
The limb at 35,000 ft. Looking straight up puts about 62 miles of
atmosphere in front of you; looking at the horizon puts 586 miles of it. That ratio is the bright
band in every high-altitude photograph, and it is the thin shell of air seen edge-on.
Shelf 02
Gravity, weight & floating
Why things fall, why some things float, why the air has a bottom, and why a stone dropped in
a mine shaft is not doing something different from the Moon. The whole shelf hangs off one
quantity, and every formula below inherits it.
Constants
Ggravitational constant, Cavendish 1798 to within 1%6.674 × 10⁻¹¹
Mmass of the Earth5.972 × 10²⁴ kg
gsurface gravity, mean32.2 ft/s² · 9.82 m/s²
gat the equator — spin and bulge both reduce it9.780 m/s²
gat the pole9.832 m/s²
ρEarth’s mean density — twice surface rock5,513 kg/m³
vescescape velocity at the surface25,022 mph · 11.19 km/s
Hatmospheric scale height27,600 ft · 5.2 miles
Formulas, and how they connect
One chain, six links. Newton gives g. Everything after it carries g
along, which is the single most useful fact on this shelf.
F = G × m₁ × m₂ ÷ r²
Newton. Put the Earth in for one mass and the surface distance in for r, and it collapses
to g = GM ÷ R² = 32.2 ft/s². Every line below is this one wearing a different hat.
W = m × g
Weight. Not the same thing as mass. Mass is how much stuff; weight is what gravity does to it.
Fb = ρ × V × g
Buoyancy. Note the g. An object floats because the fluid around it is being pulled
down harder than the object is — buoyancy is gravity sorting a mixture by density. Remove
g and nothing floats, nothing sinks, and the whole effect stops existing.
P = ρ × g × h
Hydrostatic pressure, and the g again. Depth pushes harder because there is more weight of
fluid standing above you.
H = kT ÷ (mg)
Scale height — how quickly the air thins. g is in the denominator, so a weaker pull
would give a taller, thinner atmosphere.
P(h) = P₀ × e−h/H
The barometric formula, which inherits g through H. At about 18,000 ft the pressure is
half sea level, and that halving is a direct measurement of the pull.
vesc = √(2 × G × M ÷ R)
Escape velocity, back to Newton. 25,022 mph, and the reason the atmosphere is still here:
air molecules at ordinary temperatures move far slower than that.
Δa ≈ 2 × G × M × r ÷ d³
Tidal acceleration. The cube in the denominator is why the Moon out-pulls the Sun on the
oceans despite being vastly less massive — tides depend on the difference in
pull across the Earth, not the pull itself.
Mechanisms
Weighing the Earth in a shed. Cavendish hung two small lead balls from a
wire and watched them drift toward two larger ones, in still air, with nothing touching them.
Comparing that tiny pull against the whole Earth’s pull on the same balls gives the
Earth’s mass. His 1798 figure is within one percent of the modern value.
Why density is not an alternative to gravity. Buoyancy is not a rival
force. It is gravity acting on a fluid: the denser material is pulled down harder and displaces
the lighter one, which rises because it is pushed. The formula has g in it. In free fall, where
g is effectively absent, things stop floating and sorting by density stops happening —
which is exactly what happens on the Space Station.
Why g varies. The Earth spins, so at the equator part of the pull goes
into holding you on a circular path, and the bulge puts you further from the center as well.
Equator to pole is 0.53%, and it follows sin²(latitude) — the geometric signature,
not a thermal or local one.
Why the air stays. Molecules move at hundreds of meters per second;
escape needs 11,190. So the atmosphere leaks over geological time and is held for now. The rate
depends on molecular mass, which is why hydrogen and helium are scarce and nitrogen is not.
Limits
Newton is an approximation, and everyone knows it. It fails for Mercury’s
orbit and for GPS timing, where general relativity is needed. That is a refinement of a working
model, not a hole in it — the corrections are calculable and they are applied daily.
g is not constant. It varies with latitude, altitude and the rock beneath
you. Gravimeters map ore bodies with those differences. Any argument treating g as a fixed number
to three decimal places is over-reaching.
The scale height is a simplification. It assumes constant temperature.
The real atmosphere has a temperature profile, so the exponential is an approximation that gets
steadily worse with altitude.
Worked examples
Pressure halves at 18,000 ft. With H = 27,600 ft, e−18,000/27,600
= 0.52. Any climber, pilot or barometer can check it, and it is g that sets the rate.
The equator is lighter. Spin removes 0.0339 m/s², which is 0.345% of g.
A 200 lb person weighs about 0.7 lb less at the equator than at the pole, from the spin alone,
before the bulge is counted.
Mean density, 5,513 kg/m³. Surface rock runs 2,600 to 3,000. The
Earth is therefore roughly twice as dense on average as anything you can pick up, which is how
the iron core was inferred long before seismology confirmed it.
Shelf 03
Rotation & inertial frames
How a planet’s spin shows up in a swinging weight, a weather system, a laser loop and a
gyroscope — four instruments sharing no physics, all returning the same number. And why you
cannot feel any of it.
Constants
sidereal dayone turn against the stars, not the Sun86,164.1 s
Ωthe Earth’s angular rate15.041°/hr · 7.292 × 10⁻⁵ rad/s
vsurface speed at the equator1,039 mph · 465 m/s
acentripetal acceleration at the equator0.345% of g
bulgeequatorial radius minus polar13.3 miles · 21.4 km
fflattening1 ÷ 298.3
LODshortest day of the atomic-clock era, 5 Jul 2024−1.66 ms
leap secondsadded since 1972 · since 201627 · none
Formulas, and how they connect
Ω × sin(latitude) is the whole shelf. Four instruments that
share no physics — a pendulum, a storm, a ring of laser light, a spinning disc — and
the same factor sits inside all of them. That factor is the vertical component of the
Earth’s rotation where you happen to be standing.
turn rate = 15.041°/hr × sin φ
Foucault’s pendulum. At the pole the floor turns under it once a day; at the equator
it never turns at all. Nothing touches the pendulum to make this happen.
f = 2 × Ω × sin φ
The Coriolis parameter, which sets which way storms wind. Same Ω, same sin φ,
factor of two from the vector algebra.
shift = 4 × A × Ω × sin φ ÷ (λ × c)
The Sagnac effect in a horizontal ring laser. Two beams travel opposite ways round a loop
and come back out of step. Again Ω sin φ.
a = Ω² × R
Centripetal acceleration. 0.0339 m/s² at the equator — a third of one percent of
gravity, which is why nobody feels it.
L = I × ω and Ωp = τ ÷ (I × ω)
Angular momentum and precession. A gyroscope holds its axis because changing it requires
torque; the leftover drift against the ground is the planet turning underneath.
Mechanisms
Why you cannot feel it. Constant velocity is not detectable from inside,
and the rotation is very nearly constant. What is left over is the centripetal term at 0.345% of
g — real, measurable with a good gravimeter, and far too small to sense. You feel
acceleration, not motion.
sin(latitude) is not a fudge. Only the vertical part of the rotation
vector turns a horizontal plane. At the pole it is all vertical; at the equator all horizontal;
in between it goes as the sine. That is why the pendulum, the storm and the laser all carry the
same factor.
The bulge is the spin, made permanent. A spinning fluid body settles into
an oblate shape. 13.3 miles of extra equatorial radius is what 1,039 mph does to a planet of this
size, and the figure is predicted from Ω, not measured and then explained.
The rate is not quite constant. The Moon’s declination, seasonal
jet-stream shifts and the liquid core all move angular momentum around, and the length of day
wobbles by fractions of a millisecond. It is published weekly.
Limits
At the equator, three of these instruments read zero. sin(0) = 0, so a
Foucault pendulum there does nothing at all. That is a prediction, not an excuse — and it
is why the demonstration is always sited at latitude.
Foucault pendulums drift for boring reasons too. Air currents, an
off-axis push at release, an elliptical swing. A good installation controls for all three, and a
bad one gives a number that means nothing.
Ring lasers need to know their own orientation. The sin φ factor
means a tilted instrument reads a different latitude. This is a real calibration problem, not a
theoretical one.
Worked examples
The same factor, three instruments, one latitude. At 45°,
sin φ = 0.707. Foucault turns 10.64°/hr. The Coriolis parameter is
1.031 × 10⁻⁴/s. A ring laser reads 0.707 of its polar value. Nobody arranged
that.
Why the day is not 24 hours. One turn against the stars takes 86,164.1 s
— 3 minutes 56 seconds short of the solar day, because the Earth has also moved along its
orbit and has to turn a little further to face the Sun again. That gap is 1/365th of a day, and
it is the orbit showing up in the clock.
A shadow that moved. Eclipse predictions carry a correction for the
drifting rotation rate. The April 2024 path was computed in 1987 with a value 16.5 seconds off
what it turned out to be, which moved the computed path 4.2 miles west at San Antonio —
enough to take the city center out of totality.
Shelf 04
Radio & waves
Radio is light you cannot see, so everything on the optics shelf applies — with one
number changed. It bends more in air, it diffracts round obstacles, and it lets you bounce a
signal off the Moon from a back garden and measure the result.
Constants
cspeed of light, and of radio299,792,458 m/s
4/3effective-radius multiplier for radio — more than light’s 7/61.3333
1.41 × √hradio horizon, h in feet, d in milesvs 1.32 for light
λ2 m band, 144 MHz6.83 ft
λADS-B, 1090 MHz0.90 ft
λ23 cm EME, 1296 MHz0.76 ft
Moonround-trip light time, perigee to apogee2.42 to 2.71 s
−174thermal noise floor, per hertz of bandwidthdBm/Hz
Formulas, and how they connect
Everything here is one equation and its consequences.
c = f × λ
Frequency times wavelength is always the speed of light. Halve the wavelength and you
double the frequency, which is why antennas get smaller as you go up in frequency.
d = 1.41 × √h (both ends add)
The radio horizon. The same formula as the optical one, with k = 4/3 instead of
7/6, because water vapor bends radio more than it bends light. That is the entire difference.
Two stations each contribute their own √h.
FSPL = 20 log₁₀(d) + 20 log₁₀(f) + 32.45
Free-space path loss in dB, with d in km and f in MHz. Doubling either distance or
frequency costs 6 dB. At 1090 MHz and 250 nautical miles it is 146.5 dB.
Δf = −2 × f₀ × v ÷ c
Doppler on a round trip. The factor of two is there because the reflector moves relative
to you both on the way out and on the way back.
noise floor = −174 + 10 log₁₀(B) + NF
How faint a signal you can hear. B is bandwidth in hertz, NF the receiver’s own
noise figure. Narrow the bandwidth and the floor drops, which is why weak-signal modes use
very slow data rates.
Mechanisms
Why radio reaches further than light. Water vapor has a large effect at
radio frequencies and almost none at optical ones. So the refractivity gradient is steeper for
radio, the ray bends more, and the effective Earth is bigger: 4/3 rather than 7/6. Same physics,
different number.
Ionospheric skip. Above about 3 MHz the ionosphere can refract a signal
back down. It leaves a skip zone — a ring where you hear nothing, too far for
ground wave and too near for the returning sky wave. That ring is a direct measurement of the
reflecting height and of the curve between you and the far station.
Moonbounce, and why it is a measurement. Point a dish at the Moon, send,
and 2.4 to 2.7 seconds later your own signal comes back. The delay changes through the month
with the Moon’s distance, and the returning frequency is shifted by the closing rate. Both
are predicted from an ephemeris before you transmit, and amateurs do this with home equipment.
Why VHF has a hard range limit. Below the ionospheric cutoff a signal
will not come back down, so 144 MHz is line of sight plus the 4/3 bending. Two stations at 30 ft
reach 15 miles. Put one on an aircraft at 35,000 ft and it reaches 271. The number tracks
√h, and that square root is the curve.
Limits
4/3 is an average, not a constant. It moves with humidity and
temperature, and in a duct it fails entirely. Microwave link engineers plan for that; anyone
quoting it to three decimals is over-reaching.
Path loss is not the same as horizon. A signal can be well above the
noise floor and still be unreachable because the Earth is in the way. Two separate limits, and
confusing them produces bad predictions in both directions.
Tropospheric ducting makes freak contacts real. VHF paths of many
hundreds of miles happen. They are weather, they are predictable from soundings, and they do not
mean the horizon was never there.
Worked examples
Two hams at 30 ft. 1.41 × (√30 + √30) = 15.4 miles on
2 m. Raise one to 1,000 ft and it becomes 58.7. Neither number has anything to do with
transmitter power.
ADS-B at cruise. An aircraft at 35,000 ft is receivable to about 242
nautical miles from a ground station, and not one mile further, whatever the antenna. That is
why an ocean crossing shows a gap in coverage.
Moonbounce Doppler at 1296 MHz. A closing rate of 4.5 m/s from libration
shifts the returned signal by 38.9 Hz. At 144 MHz the same motion gives 4.3 Hz. The shift scales
with frequency, exactly as c = fλ requires.
Shelf 05
Astronomy & the sky
Two independent measurements of the Earth’s motion around the Sun, one of which was
found a century before the other and by accident. Plus the geometry of eclipses, which is the
oldest working prediction in science.
Constants
AUmean Earth–Sun distance92,955,807 mi
vEarth’s orbital speed18.5 mi/s · 29.79 km/s
parsecthe distance giving one arcsecond of parallax3.262 light years
αaberration of starlight, constant20.5 arcsec
Moonapparent diameter, perigee to apogee29.4′ to 33.5′
Sunapparent diameter through the year31.6′ to 32.7′
shadowEarth’s umbra at the Moon’s distance2.6 lunar diameters
εaxial tilt23.44°
Formulas, and how they connect
Parallax and aberration both measure the orbit, and they answer different
questions. Parallax depends on how far the star is; aberration does not. That is why
aberration was found first, in 1728, by someone looking for parallax and finding something
else.
d (parsecs) = 1 ÷ p (arcseconds)
Stellar parallax. A star’s apparent shift over six months, as the Earth moves from
one side of its orbit to the other. Halve the shift and the star is twice as far.
α = arctan(v ÷ c)
Aberration. Starlight arrives tilted because the Earth is moving across it — the
same reason rain slants on a car windscreen. It gives 20.5 arcseconds, and every star
traces the same 41-arcsecond circle each year regardless of distance.
angular size ≈ actual size ÷ distance
Small-angle approximation. Good to a fraction of a percent for anything in the sky. The
Moon and the Sun both come out near half a degree, which is why total eclipses are possible
at all.
altitude of Polaris ≈ your latitude
Because the star sits nearly over the axis. Travel 69 miles north and it climbs one
degree. The cheapest accurate measurement of position ever devised.
Mechanisms
Why parallax is hard and aberration is not. The nearest star shifts by
0.77 arcseconds — the width of a coin at four miles. Aberration is 20.5 arcseconds,
twenty-six times larger, and it happens to every star equally. Bradley found it in 1728
while hunting for parallax; parallax itself was not measured until 1838.
They cannot be confused, because they are out of step. Parallax peaks
when the Earth is at the side of its orbit; aberration peaks a quarter-year away, when the
Earth is moving fastest across the line of sight. Two effects with the same period and different
phase, from the same orbit.
The shadow on the Moon. During a lunar eclipse the Earth’s shadow
falls on another world, and its edge is a circular arc — at every eclipse, every time of
year, whatever part of the planet happens to face the Moon. A disc casts a circular shadow only
face-on. Only a sphere does it from every angle.
Why eclipse prediction is the strongest test. A total eclipse path is
computed years ahead to within a mile and a minute, for a strip of ground a hundred miles wide.
It requires the sizes, distances, orbits and rotation of three bodies to be simultaneously
correct. Being wrong about any of them puts the shadow somewhere else.
Limits
Parallax runs out. Ground-based measurement is good to a few hundred
light years before the angle disappears into the seeing. Beyond that, distance comes from
other methods that are calibrated against parallax — a real dependency, and
stated openly.
Aberration says nothing about distance. It proves the Earth moves. It
does not tell you how far anything is, and treating it as a distance measurement is an
error.
Refraction moves everything near the horizon. A body appears about half
a degree higher than it truly is at the horizon — enough that the Sun is fully visible
when it is geometrically already set. Every serious observation corrects for it.
Worked examples
Four stars, four distances. Proxima Centauri shifts 0.7687 arcsec, giving
4.24 light years. Barnard’s star 0.5464 and 5.97. Sirius 0.3792 and 8.60. 61 Cygni 0.2861
and 11.40 — the first ever measured, by Bessel in 1838.
Aberration, from the orbit alone. 18.5 miles per second divided by the
speed of light, converted to arcseconds, gives 20.5. No star’s distance appears anywhere
in that calculation, and the answer matches every star in the sky.
The 2024 eclipse path, and why it moved. Computed in 1987 with a
rotation-drift correction 16.5 seconds off the eventual value, which shifted the path 4.2 miles
west at San Antonio — enough to take the city center out of totality. The error was in the
spin of the Earth, not in the geometry.
Shelf 06
Space & orbits
Orbit is not a place, it is a speed. Everything here comes from the same equation as the
gravity shelf, rearranged: set the pull equal to what is needed to keep turning, and the whole
system falls out — including the fact that you cannot choose an orbit’s altitude and
its period independently.
Constants
μGM for the Earth, the number every orbit uses3.986 × 10¹⁴ m³/s²
ISSaltitude · speed · period254 mi · 17,152 mph · 92.6 min
Starlinktypical shell342 mi · 16,976 mph · 95.5 min
GPShalf a sidereal day per orbit12,552 mi · 8,664 mph · 11h 58m
geostationaryone sidereal day exactly22,236 mi · 6,878 mph
vescescape velocity from the surface25,022 mph
ΔtGPS clock correction, net of both relativities+38.5 µs/day
Formulas, and how they connect
One idea, three rearrangements. Gravity supplies exactly the inward pull
needed to keep a circle. Set them equal and everything else is algebra.
G × M × m ÷ r² = m × v² ÷ r
The starting point. The satellite’s mass appears on both sides and cancels —
which is why a bolt and a space station orbit identically.
v = √(μ ÷ r)
Orbital speed. Higher means slower: the ISS at 254 miles does 17,152 mph, a
geostationary satellite at 22,236 miles only 6,878.
T = 2π × √(r³ ÷ μ)
Period. Kepler’s third law in modern dress. Altitude and period are not
independent — choose one and the other is fixed, which is why geostationary orbit is
a specific ring and not a preference.
vesc = √(2 × μ ÷ r) = √2 × vorbit
Escape is exactly √2 times circular speed at the same radius. A neat consequence,
and a useful check on any orbital arithmetic.
Mechanisms
Why orbit is falling. A satellite is in continuous free fall. It misses
the ground because it is moving sideways fast enough that the surface curves away beneath it at
the same rate it drops. That only works on a curved surface: on a plane there is nothing to fall
around, and the same speed takes you away forever.
Why geostationary is one ring. To hang over one spot, the period must
equal one sidereal day — 86,164 seconds, not 86,400. Feed that into the period equation
and it returns a single radius of 42,163 miles from the center, 22,236 above the surface. Every
television satellite on Earth is in that ring, which is why dishes in a neighborhood all point
the same way.
Why GPS satellites need relativity. Their clocks run slow by 7.2
microseconds a day from speed, and fast by 45.7 from being higher in the gravity well. Net
+38.5. Ignore it and positions drift about 7 miles a day. The correction is built into the
hardware.
Why the mass cancels. It appears on both sides of the starting equation.
A feather and a satellite at the same altitude have the same orbit — the same reason
everything falls at the same rate in a vacuum.
Limits
Low orbits decay. There is still measurable atmosphere at 250 miles, so
the ISS loses altitude continuously and must be reboosted. That is a cost of the model, stated:
orbit is not permanent.
Circular is an idealization. Real orbits are ellipses, perturbed by the
equatorial bulge, the Moon, the Sun and solar radiation pressure. Operational prediction uses
numerical models, not these formulas.
These formulas assume a point mass. The Earth’s bulge makes orbital
planes precess — useful rather than a nuisance, since it is what makes sun-synchronous
orbits possible.
Worked examples
The ISS, from first principles. r = 3,959 + 254 = 4,213 miles.
v = √(μ/r) gives 17,152 mph, and T = 92.6 minutes — about 15.5 orbits a day.
Anyone can check the period against a pass predictor, or against their own eyes.
Solving for geostationary. Set T = 86,164 s and rearrange for r. It gives
42,163 miles from the center, so 22,236 above the ground. Nobody chose that number; the equation
returned it.
GPS at half a sidereal day. A period of 11h 58m means each satellite
traces the same ground track twice a day. That repeat is designed, and it comes straight out of
the period equation.
Shelf 07
Navigation & timekeeping
The units of navigation are angles, not distances, and that is not an accident. A nautical
mile is an arcminute of latitude, which means every chart in the world is built on the
assumption that you are moving over a sphere.
Constants
1 NMone arcminute of latitude, by definition6,076 ft · 1.151 statute mi
1° latitude60 NM, everywhere on Earth69.05 statute mi
1° longitudeat the equator — shrinks with cos(latitude)69.05 statute mi
15°of longitude per hour of time1° = 4 minutes
solar daynoon to noon86,400 s
sidereal dayone turn against the stars86,164.1 s
differencethe orbit, showing up in the clock3m 56s
Formulas, and how they connect
1° of longitude = 69.05 × cos(latitude) statute miles
The single most diagnostic equation in navigation. East–west spacing shrinks toward
the poles because the circles of latitude get smaller. It shrinks in both directions,
north and south, which no flat map can reproduce.
s = R × θ
Arc length. With θ in radians. One degree is 1/360 of the circumference, hence
69.05 miles, and it is the same everywhere because the Earth is very nearly round.
longitude = (local time − GMT) × 15°
Why finding longitude needed an accurate clock and finding latitude did not. Latitude is
written in the sky; longitude has to be carried with you.
cos(c) = sin(a) sin(b) + cos(a) cos(b) cos(C)
The spherical law of cosines — the great-circle distance between two points. Every
airline route computer runs this, or the haversine form of it, thousands of times a day.
Mechanisms
Why a nautical mile is what it is. It was defined as one minute of arc
along a meridian, so that a degree of latitude is exactly 60 of them. That makes a chart and a
sextant speak the same language: measure an angle, read a distance. The unit only makes sense on
a sphere.
Why longitude was the hard one. Latitude comes from the height of a star,
which you can measure anywhere with no equipment beyond a sextant. Longitude is a comparison
between local noon and the time somewhere else, which means carrying a clock that keeps time at
sea. That is why it took until Harrison.
Why the day is not one rotation. In the 86,164 seconds it takes to turn
once against the stars, the Earth has also moved along its orbit, so it must turn a few minutes
further to face the Sun again. The 3m 56s gap is 1/365th of a day, and it is the orbit visible in
the clock.
Why great circles look wrong on a map. The shortest path over a sphere,
drawn on a flat chart, bows toward the nearer pole. Pilots fly it anyway because it is shorter,
which is why a London to Tokyo flight goes over the Arctic.
Limits
A degree of latitude is not perfectly constant. The Earth is slightly
flattened, so it runs from 68.7 miles at the equator to 69.4 at the poles. Charts use the
ellipsoid; 69.05 is the round figure.
Magnetic north is not true north, and the difference moves. Every chart
prints the local variation and its annual change, because the magnetic pole wanders.
Great circles are not always flown. Winds, airspace, ETOPS rules and
fuel prices all bend real routes away from the shortest path. Distance arguments must compare
like with like.
Worked examples
Longitude spacing, measured. 69.0 miles per degree at the equator, 48.8
at 45°, 34.5 at 60°, 12.0 at 80°. Any GPS receiver confirms it in an afternoon, and
the convergence toward both poles is what breaks a flat map.
Four minutes per degree. The Earth turns 15° an hour, so one degree
of longitude is four minutes of time and one arcminute is four seconds. That equivalence is why
a chronometer is a position instrument.
Sydney to Santiago. The great circle is 7,047 statute miles and dips to
61.7°S. Flying it via Los Angeles is 13,078 miles — 86% further — and the flight
times match the ratio, not the flat-map one.
Shelf 08
Water & the oceans
“Water finds its level” is true, and the whole argument turns on what
level means. It does not mean flat. It means perpendicular to gravity everywhere, which
on a round planet is a curved surface — and one that is measurably lumpy.
Constants
ρdensity of seawater1,025 kg/m³
geoidhow far mean sea level departs from a smooth ellipsoid+279 ft to −348 ft
tidal rangeBay of Fundy · Mediterranean53 ft · about 1 ft
PanamaPacific mean level above Atlanticabout 8 inches
deepestChallenger Deep35,876 ft · 1,085 atm
risecurrent mean sea-level rise3.4 mm/yr
Formulas, and how they connect
P = ρ × g × h
Hydrostatic pressure. The same g as the gravity shelf. Pressure at depth is the
weight of everything standing above you, which is why it climbs by about one atmosphere every
33 feet.
the geoid: a surface of constant gravitational potential
“Level” means this, and nothing else. Water settles until its surface is
everywhere perpendicular to the local pull. Where the rock below is denser, the pull is
stronger and the water piles slightly higher.
Δa ≈ 2 × G × M × r ÷ d³
Tidal acceleration again, from the gravity shelf. The cube is why the Moon beats the Sun
on the oceans: tides come from the difference in pull across the Earth, not the pull
itself.
Mechanisms
What “finds its level” actually means. Water flows until no
part of it can lower its potential energy. On a sphere that surface is a sphere. In a bathtub
the curvature over three feet is about a millionth of an inch, so it looks flat — and it
is flat, to any accuracy you can measure in a bathtub.
Why sea level is not one number. Gravity is stronger over denser rock,
so the ocean surface has hills and valleys tens of meters deep relative to a smooth ellipsoid.
Satellite altimetry maps them, and they line up with what gravimetry independently predicts.
Why the Panama Canal has locks. Not because of curvature. The Pacific
side sits about eight inches higher, and the terrain rises 85 feet to Gatun Lake. Locks lift
ships over a hill, and the eight inches is a real, measured difference in mean sea level
driven by density, wind and current.
Why tidal ranges differ so much. The Bay of Fundy resonates — its
natural sloshing period is close to the tidal period, so the range builds to 53 feet. The
Mediterranean is nearly enclosed and barely moves. The forcing is the same; the basins are not.
Limits
Curvature is unmeasurable over short spans. Over a canal, a reservoir
or a lake a few miles across, the drop is smaller than the wind ripples and the thermal
expansion of the measuring rod. Anyone claiming to have measured flat water over a mile has
measured their own instrument.
Canal surveys really are done as if flat, and that is not a concession
of anything. Over the distances involved the correction is below the working tolerance, so it is
omitted — and the standing orders say to omit it, which is a different thing from denying
it exists.
Mean sea level is a long average. Any single measurement includes tide,
surge, wave and atmospheric pressure. Tide gauges average over years for exactly that reason.
Worked examples
Pressure at the Challenger Deep. 1,025 × 9.81 × 10,935 m gives
1,085 atmospheres, about eight tons per square inch. Submersible hulls are designed to that
figure and they come back.
The Bedford Level, honestly. Six miles of still water with the sight line
eight inches above it is the geometry that maximises refraction. Wallace’s 1870 repeat, with
markers at equal heights well above the surface, showed the middle one standing above the line
by about the predicted amount.
Eight inches across Panama. The Pacific is higher, and it is measured
with tide gauges at both ends. A flat, static ocean would give the same level everywhere; a real
one has wind, density and current, and the difference is small, consistent and explained.
Shelf 09
The atmosphere
Why a gas sits next to a vacuum without a lid on it, why pressure halves every three and a
half miles, and why the sky is blue at noon and red at the edges. The whole thing is held down by
the same g as the gravity shelf, and it thins out exponentially rather than stopping.
Constants
P₀standard sea-level pressure14.696 psi · 1013.25 hPa
Hscale height — the e-folding distance27,600 ft · 5.2 miles
halfaltitude at which pressure halvesabout 18,000 ft
lapse ratestandard temperature fall with height3.57°F per 1,000 ft
tropopausewhere the fall stops, mid-latitudes36,000 ft
Kármán linethe conventional edge of space62 miles
massof the whole atmosphere5.15 × 10¹⁸ kg
ratioatmosphere mass ÷ Earth massabout 1 in a million
Formulas, and how they connect
Two equations, one of which is a consequence of the other.
H = k × T ÷ (m × g)
Scale height. k is Boltzmann’s constant, T the temperature, m the mass of an average
air molecule, g gravity. Warmer air puffs up; heavier molecules settle lower; a stronger pull
compresses everything.
P(h) = P₀ × e−h/H
The barometric formula, which is just the previous line integrated. Every H of altitude
divides pressure by e (2.718). It never reaches zero, which is the whole point: the
atmosphere has no edge, it just gets thinner.
P × V = n × R × T
The ideal gas law. Pressure, volume and temperature are not independent, which is why a
sealed bag of crisps swells in an aircraft and why a balloon expands as it climbs.
Mechanisms
Why no container is needed. Gas expands to fill a container, and without
one it expands until something stops it. Gravity is what stops it. Each layer is held down by the
weight of the layers above, and the balance point is the exponential above. The atmosphere is not
contained; it is weighed down, and it leaks slowly at the top exactly as that model
predicts.
Why the fade is gradual. An exponential never reaches zero. There is no
altitude at which air stops — the ISS at 254 miles is still losing height to drag. The
Kármán line at 62 miles is a legal and aeronautical convention, not a physical
boundary, and the site says so.
Why it is thinner than it looks. The whole atmosphere masses about a
millionth of the Earth. Scaled to a classroom globe it would be thinner than the varnish. That is
why the limb glows as a hairline in every high photograph, and why the blue band is startlingly
thin.
Why temperature stops falling. Below the tropopause, air is heated from
the ground and cools with height. Above it, ozone absorbs ultraviolet and warms the layer again.
The kink is at about 36,000 ft, and it is why airliners cruise there.
Limits
The simple barometric formula assumes constant temperature. The real
atmosphere has a lapse rate, so the plain exponential drifts from reality with height. Aviation
uses the layered standard atmosphere instead, which is why altimeters need a temperature
correction.
Scale height is not one number. It varies with temperature and therefore
with latitude and season. 5.2 miles is a mid-latitude average.
“Edge of space” is a convention. 62 miles is a round figure in
kilometers, chosen for treaty and record-keeping. Physics offers no line there.
Worked examples
Pressure halves at 18,000 ft. e−18,000/27,600 = 0.52. Any
barometer confirms it, and the rate is set by g.
Why a jet needs pressurization. At 36,000 ft the outside pressure is about
a quarter of sea level. Cabins are held near the equivalent of 6,000 to 8,000 ft, which is why
your ears pop on the climb and why a bottle brought aboard crushes on descent.
The atmosphere as varnish. A 12-inch globe scaled correctly would carry
an atmosphere about 0.02 inches thick to the tropopause. Photographs from orbit show exactly that
proportion.
Shelf 10
Measuring the curve
Six independent ways to get a number for the size of the Earth, none of which needs a
satellite and two of which need only a stick. They agree to within a few percent across
twenty-two centuries, and the agreement is the argument.
Constants
Rmean radius3,959 mi · 6,371 km
Cequatorial circumference24,901 mi
1°of latitude, anywhere69.05 mi
bulgeequatorial radius minus polar13.3 mi
fflattening1 ÷ 298.3
dropbelow a tangent, per mile squared8.00 in (no refraction)
dropsame, with standard refraction6.86 in
Formulas, and how they connect
C = (360 ÷ θ) × d
Eratosthenes. Two shadow angles θ apart, measured d apart on the same meridian,
scale up to the whole circle. Everything else on this shelf is a variation.
drop = d² ÷ (2R)
The 8-inch rule, and the light shelf’s formula. Subtract your own horizon
distance before squaring or you answer a different question.
R = (D − dh)² ÷ (2 × x)
The same equation solved for R. Measure how much of a known target is hidden at a known
distance and the Earth’s radius falls out — the measurement running backwards.
latitude = altitude of Polaris
One degree per 69 miles of travel. The cheapest accurate measurement on the shelf, needing
a protractor and a weighted string.
spherical excess = area ÷ R²
A triangle on a sphere has angles summing to more than 180°, by an amount proportional
to its area. Surveyors have corrected for this since the 1700s, and the area-dependence
is why it cannot be an instrument artifact.
Mechanisms
Why agreement is the argument. Any single method can be attacked
— refraction complicates the horizon, Eratosthenes’ unit is uncertain, the eclipse
method is coarse. But six methods sharing no instruments and in most cases no century return the
same radius to within a few percent. Breaking that means explaining why they conspire.
Why the shadow method works from a single eclipse. The Earth’s
shadow on the Moon spans about 2.6 lunar diameters. Correcting for the Sun’s angular size
gives an Earth about 3.7 Moon-widths across, and the Moon’s own size is fixed by parallax.
Aristotle had the shape from this; the size followed.
Why a stick still works. Two people several hundred miles apart, one
stick each, one agreed date, shadows measured at local solar noon. Schools repeat it every year
at both equinoxes through several independent programs, and a crowd-sourced version in June 2025
collected 1,014 results from 43 countries.
Limits
Short baselines cannot discriminate. Two stations a few hundred miles
apart fit a small nearby Sun as well as a distant one. Separating the models needs stations
thousands of miles apart, and the difference is hundredths of a degree below that.
Eratosthenes’ unit is genuinely unknown. A short stadion gives him
a radius 1.6% low; the Attic stadion gives 15.5% high. That is a question about ancient
metrology, not about his geometry.
The horizon method is the loosest. Refraction varies with the weather,
so a single measurement carries a range rather than a point. It earns its place because anyone
can do it from a beach.
Worked examples
Six methods, one number. Eratosthenes 3,894 mi · horizon distance
3,883 · Polaris 3,952 · lunar shadow 3,993 · circumnavigation 3,963 ·
satellite geodesy 3,959. Spread of 2.8% across twenty-two centuries.
The 8-inch rule, checked. One mile squared divided by twice 3,959 miles
is 0.00012629 miles; times 63,360 inches gives 8.00. The 8 is the Earth’s radius in
disguise — on Mars the same rule reads 15 inches, on the Moon 29.
A protractor is good enough. A homemade Polaris sight reads to about half
a degree. Over a 5° baseline that gives the radius to roughly ±600 miles — poor
by survey standards, and completely decisive against a flat Earth.
Shelf 11
Surveying & leveling
The profession that has to get this right for a living, and has done since the 1700s. Every
correction on this shelf appears in published standards, is applied daily, and would produce
visible errors if the assumption behind it were wrong.
Constants
0.0239combined curvature and refraction, ft per 1,000 ft of sightUS practice
0.667curvature alone, ft per mile squared8.00 inches
0.093refraction alone, ft per mile squared1/7 of curvature
0.574net, curvature minus refraction6.86 inches per mi²
excessspherical excess, per 100 sq mi of triangleabout 0.5 arcsec
datumthe reference surface, not a flat planethe geoid
Formulas, and how they connect
correction = 0.574 × d² (feet, d in miles)
The combined curvature-and-refraction correction in leveling. Curvature raises the
required correction; refraction removes about a seventh of it. Both signs are in every
textbook.
spherical excess = A ÷ R² (radians)
How much a triangle’s angles exceed 180°. It scales with area, which is
why it cannot be a lens artifact: an optical defect does not care how big your triangle
is.
back sight − fore sight, at equal distances
The technique that cancels curvature and refraction entirely: set the instrument midway
between two staves and both errors are identical and subtract out. It cancels the curve
because the curve is there.
Mechanisms
Why “assume a horizontal datum” is not an admission. A datum
is a reference surface, and in geodesy it is the geoid — curved, lumpy and defined by
gravity. Where a standing order says a section shall be drawn with a horizontal datum, it is
describing the drawing, not the planet.
Why balanced sights cancel the error. Curvature and refraction both
depend on distance. Put the level exactly halfway between the two staves and each reading carries
the same error, which cancels in the subtraction. Surveyors do this precisely because the errors
exist — a flat Earth would make the technique pointless.
Why the correction is 6.86 and not 8. Curvature alone gives 8.00 inches
per mile squared. Refraction bends the sightline down and buys back about one seventh. The net
figure in the tables is 6.86, and the fact that the two are listed separately is the profession
saying openly that both effects are real.
Limits
The refraction seventh is an average. The Encyclopædia
Britannica warned in the 1850s that it sometimes exceeds a fifth and sometimes falls under a
fifteenth, and that a fixed allowance can produce a larger error than it removes. That warning is
still correct.
Short sights need no correction at all. Under a few hundred feet the
term is below the instrument’s resolution, so it is omitted. That is a tolerance decision,
and omitting it is not the same as denying it.
Spherical excess is tiny for small triangles. About half an arcsecond per
hundred square miles. It only becomes measurable on national survey networks, which is exactly
where it was first found.
Worked examples
A one-mile sight. Curvature 8.00 inches, refraction gives back 1.14, net
6.86. Over three miles it is 61.7 inches — over five feet, and no surveyor ignores
that.
A 100-square-mile triangle. Angles sum to about 180° 00′
00.5″. Double the area and the excess doubles. Change the instrument and it does not.
Balanced sights over 400 ft. The uncorrected error at 200 ft each way is
under a hundredth of a foot, and it cancels exactly. This is why precise leveling networks are
run with equalized sight distances rather than with a correction table.
Shelf 12
Aviation & instruments
What an aircraft actually measures, what its instruments can resolve, and why the rotation
needed to follow a curved planet is real, continuous, and far too slow for any of them to show.
Constants
ωpitch rate to follow the curve at 500 kt0.0021°/s · 7.7°/hr
attitude indicatorfinest painted division2.5°
flight recorderpitch resolution, one bin0.176°
ADS-Bbroadcasts roll — and not pitch1090 MHz
glide slopenominal, and the tolerance allowed3.00° ± 0.225°
needlefull-scale glide-slope deflection0.7°, two dots
altimetererror per 10°C off standard127 ft
diphorizon below eye level at FL3503.31°
Formulas, and how they connect
ω = v ÷ R
The whole argument in three symbols. Ground speed divided by the Earth’s radius gives
the rate at which “down” rotates. At 500 knots it is 0.0021° per second.
dip = √(2h ÷ R′)
From the light shelf. 3.31° at cruise — visible, measurable, and the reason the
horizon sits below the glareshield.
d = h ÷ tan(3°) − 1,000 ft
Glide-slope intercept. The 1,000 ft is the offset of the glide path intercept point down
the runway, which is what puts the beam about 50 ft over the threshold.
hidden = (D − dh)² ÷ 2R′
The same hidden-height formula as the light and curve shelves, applied from a cockpit
instead of a beach.
Mechanisms
Why no pilot dips the nose. The rotation is real and continuous, and it
is 7.7° an hour — about a quarter of the finest division on an attitude indicator, per
minute. The autopilot holds the aircraft against the local horizontal, which is itself rotating,
so the correction is made continuously and is never large enough to notice. Nobody dips the nose
for the same reason nobody steers around the curve of a highway crest.
Why the instrument cannot show it. A flight recorder’s pitch channel
quantizes to 0.176°. The per-sample rotation at cruise is around 600 times smaller than one
bin at the fastest permitted sampling. It is not that the curve is hidden; it is below the
resolution of the instrument by three orders of magnitude.
Why roll is broadcast and pitch is not. Air traffic control can ask a
transponder how an aircraft is turning, and get roll, track, ground speed and rate of turn. Pitch
is in no surveillance message, because pitch does not tell you where an aircraft is going —
the same nose-up angle can mean climbing, level or descending depending on speed and weight. The
message carries vertical rate instead.
Why the glide slope is not a beam. Two overlapping signals are
transmitted, each carrying its own steady tone at a different pitch, one aimed slightly high and
one slightly low. The approach path is where a receiver hears them equally. The ground in front
of the antenna is part of the transmitter, which is why snow can move the path angle.
Limits
The instruments are not neutral witnesses. An attitude indicator is
deliberately slaved to local vertical — it is built to erase this rotation, because
a pilot needs to know the attitude relative to the ground, not to the stars. Quoting it as
evidence of no rotation is quoting a designed behaviour.
The glide-slope experiment cannot decide. The difference between flat
and spherical predictions is about a fifth of one dot on the needle. The beam’s own legal
tolerance is three and a half times larger. That is not a coarse reading; it is an unknown
baseline.
Pitch attitude changes for aerodynamic reasons anyway. A lighter
aircraft needs less angle of attack, so fuel burn alone moves the nose by roughly a degree over
a long flight — a thousand times the curvature term, and in the same direction. Anyone
hunting for a nose-down trend will find one, for the wrong reason.
Worked examples
The rate, at cruise. 500 knots is 257 m/s; divided by 6,371,000 m that is
0.0000403 rad/s, which is 0.0021° per second or 7.7° per hour. Real, continuous, and
invisible.
Descent from FL350. The rule of three puts top of descent 105 NM out on a
3° path. Both inputs are already curved: altitude is height above a curved sea-level shell,
and distance is ground track along the curve. Fly a genuine straight line instead and you cross
the threshold 9,700 ft high.
Sydney to Santiago. QF27 blocks 12h 30m. The same journey via Los Angeles
is 13h 40m plus 10h 40m — 1.95 times as long. A globe predicts 1.86 times as far; the
azimuthal equidistant map predicts the same time either way.