The crew was weightless at nine tenths of the gravity

# Self-authored for the EML case corpus (no external origin). A visitor watching
# the crew of a space station float across the cabin reasons that the station
# is high enough to be out of the Earth's gravity. How much of the gravity is
# still there at 400 kilometres, and why the crew floats anyway, is computed
# below.
#
# The reasoning is careful. Gravity really does weaken with distance from the
# Earth; the station really is 400 km up; a scale aboard really does read zero;
# and the intent is exactly 'is there gravity up there'.
#
# Gravity falls with the square of the distance from the Earth's centre, and 400
# km on top of a 6371 km radius is only six percent farther out - 885 per mille
# of the surface pull remains. The crew floats because the station, the crew and
# the scale are all falling together: in each second they drop 434 cm toward the
# Earth, and at 7673 m/s the surface curves away 434 cm beneath them.

6371 => earth_radius_km
400 => station_altitude_km
982 => surface_gravity_cm_per_s2
7673 => station_speed_m_per_s
70 => astronaut_mass_kg
31416 => pi_times_ten_thousand

earth_radius_km + station_altitude_km => station_distance_from_the_centre_km
int(earth_radius_km * earth_radius_km * 1000 / (station_distance_from_the_centre_km * station_distance_from_the_centre_km)) => gravity_at_the_station_per_mille
int(surface_gravity_cm_per_s2 * earth_radius_km * earth_radius_km / (station_distance_from_the_centre_km * station_distance_from_the_centre_km)) => gravity_at_the_station_cm_per_s2
int(astronaut_mass_kg * gravity_at_the_station_per_mille / 1000) => pull_on_the_astronaut_kg_force
0 => scale_reading_kg
int(gravity_at_the_station_cm_per_s2 / 2) => fall_in_one_second_cm
int(station_speed_m_per_s * station_speed_m_per_s * 100 / (2 * station_distance_from_the_centre_km * 1000)) => surface_curves_away_in_one_second_cm
int(2 * pi_times_ten_thousand * station_distance_from_the_centre_km * 1000 / (station_speed_m_per_s * 10000)) => one_orbit_s
int(one_orbit_s / 60) => one_orbit_minutes

"earth's radius                  : " + str(earth_radius_km) + " km" ^0
"station altitude                : " + str(station_altitude_km) + " km, " + str(station_distance_from_the_centre_km) + " km from the centre" ^0
"gravity at the station          : " + str(gravity_at_the_station_per_mille) + " per mille of the surface's, " + str(gravity_at_the_station_cm_per_s2) + " cm/s2" ^0
"pull on a " + str(astronaut_mass_kg) + " kg astronaut       : " + str(pull_on_the_astronaut_kg_force) + " kg of force" ^0
"a scale aboard reads            : " + str(scale_reading_kg) + " kg" ^0
"" ^0
"station speed                   : " + str(station_speed_m_per_s) + " m/s, one orbit in " + str(one_orbit_s) + " s, " + str(one_orbit_minutes) + " minutes" ^0
"fall toward the earth in 1 s    : " + str(fall_in_one_second_cm) + " cm" ^0
"surface curves away in that 1 s : " + str(surface_curves_away_in_one_second_cm) + " cm, under " + str(station_speed_m_per_s) + " m of travel" ^0
"" ^0

# ---- what the visitor verified ----

"the out-of-gravity reasoning" ^0
"  gravity : weakens with distance from the earth" ^0
"  altitude : " + str(station_altitude_km) + " km, measured" ^0
"  scale aboard : reads " + str(scale_reading_kg) + " kg" ^0
"  intent : is there gravity up there" ^0
"  facts wrong : 0" ^0
"  verdict : UP THERE, GRAVITY IS GONE" ^0
"" ^0
"  reading the scale aboard is the part done right here, and" ^0
"  it is why " + str(scale_reading_kg) + " kg is exactly what the crew presses on the floor" ^0
"" ^0

# ---- how much gravity is left, and why they float ----

"falling around the earth" ^0
"  how gravity weakens : with the square of the distance from the" ^0
"    centre, and " + str(station_distance_from_the_centre_km) + " km is not much more than " + str(earth_radius_km) ^0
"  what is left : " + str(gravity_at_the_station_per_mille) + " per mille of the surface pull" ^0
"  what the station does with it : falls, " + str(fall_in_one_second_cm) + " cm in each second," ^0
"    like anything dropped" ^0
"  why it never lands : at " + str(station_speed_m_per_s) + " m/s sideways the ground curves" ^0
"    away " + str(surface_curves_away_in_one_second_cm) + " cm in that same second" ^0
"  why the scale reads zero : the scale falls with the crew, and" ^0
"    nothing can press on a floor that falls just as fast" ^0
"" ^0

# ---- what the visitor got ----

"the explanation" ^0
"  believed : the station is above the gravity" ^0
"  actual : " + str(gravity_at_the_station_per_mille) + " per mille of it, pulling a " + str(astronaut_mass_kg) + " kg astronaut with " + str(pull_on_the_astronaut_kg_force) + " kg" ^0
"    of force" ^0
"  is the scale wrong : no; nothing presses on it" ^0
"  is weightlessness the absence of gravity : no; it is falling" ^0
"    freely in it" ^0
"" ^0

# ---- null control ----

# The same crew measured by the pull of gravity on them instead of by a scale
# that falls with them.
0 => nc_weight_read_from_the_scale_kg
61 => nc_weight_read_from_the_pull_of_gravity_kg
61 => nc_kg_the_falling_scale_hides

"null control - measure the pull, not the scale" ^0
"  weight, read from the scale : " + str(nc_weight_read_from_the_scale_kg) + " kg" ^0
"  weight, read from the pull of gravity : " + str(nc_weight_read_from_the_pull_of_gravity_kg) + " kg" ^0
"  kg the falling scale hides : " + str(nc_kg_the_falling_scale_hides) ^0
"  no astronaut and no orbit changed; the force was measured" ^0
"  instead of what a falling floor pushes back with" ^0
"" ^0

# ---- the rule ----

"what a scale reading zero aboard the station guarantees" ^0
"  nothing presses on the floor : exactly" ^0
"  there is no gravity up there : not addressed; at " + str(station_distance_from_the_centre_km) + " km" ^0
"    from the centre gravity is " + str(gravity_at_the_station_per_mille) + " per mille of the surface's, and" ^0
"    the crew floats because crew, scale and station fall" ^0
"    together, " + str(fall_in_one_second_cm) + " cm a second, around a curve of the same size" ^0
"" ^0

"weightless is not the same as without weight; it is what gravity feels like" ^0
"when nothing stands in its way, and an orbit is a fall that keeps missing" ^0
"the ground" ^0
"" ^0

"The scale reads " + str(scale_reading_kg) + " - that is exact. But gravity at " + str(station_altitude_km) + " km is " + str(gravity_at_the_station_per_mille) + " per" ^0
"mille of the surface's: the station falls " + str(fall_in_one_second_cm) + " cm in each second while the" ^0
"ground curves away " + str(surface_curves_away_in_one_second_cm) + " cm under its " + str(station_speed_m_per_s) + " m/s, and a scale falling with the" ^0
"crew reads nothing, until the pull is measured instead of the floor." ^0

earth_radius_km = 6371
station_altitude_km = 400
surface_gravity_cm_per_s2 = 982
station_speed_m_per_s = 7673
astronaut_mass_kg = 70
pi_times_ten_thousand = 31416
station_distance_from_the_centre_km = earth_radius_km + station_altitude_km
gravity_at_the_station_per_mille = int(earth_radius_km * earth_radius_km * 1000 / (station_distance_from_the_centre_km * station_distance_from_the_centre_km))
gravity_at_the_station_cm_per_s2 = int(surface_gravity_cm_per_s2 * earth_radius_km * earth_radius_km / (station_distance_from_the_centre_km * station_distance_from_the_centre_km))
pull_on_the_astronaut_kg_force = int(astronaut_mass_kg * gravity_at_the_station_per_mille / 1000)
scale_reading_kg = 0
fall_in_one_second_cm = int(gravity_at_the_station_cm_per_s2 / 2)
surface_curves_away_in_one_second_cm = int(station_speed_m_per_s * station_speed_m_per_s * 100 / (2 * station_distance_from_the_centre_km * 1000))
one_orbit_s = int(2 * pi_times_ten_thousand * station_distance_from_the_centre_km * 1000 / (station_speed_m_per_s * 10000))
one_orbit_minutes = int(one_orbit_s / 60)
print("earth's radius                  : " + str(earth_radius_km) + " km")
print("station altitude                : " + str(station_altitude_km) + " km, " + str(station_distance_from_the_centre_km) + " km from the centre")
print("gravity at the station          : " + str(gravity_at_the_station_per_mille) + " per mille of the surface's, " + str(gravity_at_the_station_cm_per_s2) + " cm/s2")
print("pull on a " + str(astronaut_mass_kg) + " kg astronaut       : " + str(pull_on_the_astronaut_kg_force) + " kg of force")
print("a scale aboard reads            : " + str(scale_reading_kg) + " kg")
print("")
print("station speed                   : " + str(station_speed_m_per_s) + " m/s, one orbit in " + str(one_orbit_s) + " s, " + str(one_orbit_minutes) + " minutes")
print("fall toward the earth in 1 s    : " + str(fall_in_one_second_cm) + " cm")
print("surface curves away in that 1 s : " + str(surface_curves_away_in_one_second_cm) + " cm, under " + str(station_speed_m_per_s) + " m of travel")
print("")
print("the out-of-gravity reasoning")
print("  gravity : weakens with distance from the earth")
print("  altitude : " + str(station_altitude_km) + " km, measured")
print("  scale aboard : reads " + str(scale_reading_kg) + " kg")
print("  intent : is there gravity up there")
print("  facts wrong : 0")
print("  verdict : UP THERE, GRAVITY IS GONE")
print("")
print("  reading the scale aboard is the part done right here, and")
print("  it is why " + str(scale_reading_kg) + " kg is exactly what the crew presses on the floor")
print("")
print("falling around the earth")
print("  how gravity weakens : with the square of the distance from the")
print("    centre, and " + str(station_distance_from_the_centre_km) + " km is not much more than " + str(earth_radius_km))
print("  what is left : " + str(gravity_at_the_station_per_mille) + " per mille of the surface pull")
print("  what the station does with it : falls, " + str(fall_in_one_second_cm) + " cm in each second,")
print("    like anything dropped")
print("  why it never lands : at " + str(station_speed_m_per_s) + " m/s sideways the ground curves")
print("    away " + str(surface_curves_away_in_one_second_cm) + " cm in that same second")
print("  why the scale reads zero : the scale falls with the crew, and")
print("    nothing can press on a floor that falls just as fast")
print("")
print("the explanation")
print("  believed : the station is above the gravity")
print("  actual : " + str(gravity_at_the_station_per_mille) + " per mille of it, pulling a " + str(astronaut_mass_kg) + " kg astronaut with " + str(pull_on_the_astronaut_kg_force) + " kg")
print("    of force")
print("  is the scale wrong : no; nothing presses on it")
print("  is weightlessness the absence of gravity : no; it is falling")
print("    freely in it")
print("")
nc_weight_read_from_the_scale_kg = 0
nc_weight_read_from_the_pull_of_gravity_kg = 61
nc_kg_the_falling_scale_hides = 61
print("null control - measure the pull, not the scale")
print("  weight, read from the scale : " + str(nc_weight_read_from_the_scale_kg) + " kg")
print("  weight, read from the pull of gravity : " + str(nc_weight_read_from_the_pull_of_gravity_kg) + " kg")
print("  kg the falling scale hides : " + str(nc_kg_the_falling_scale_hides))
print("  no astronaut and no orbit changed; the force was measured")
print("  instead of what a falling floor pushes back with")
print("")
print("what a scale reading zero aboard the station guarantees")
print("  nothing presses on the floor : exactly")
print("  there is no gravity up there : not addressed; at " + str(station_distance_from_the_centre_km) + " km")
print("    from the centre gravity is " + str(gravity_at_the_station_per_mille) + " per mille of the surface's, and")
print("    the crew floats because crew, scale and station fall")
print("    together, " + str(fall_in_one_second_cm) + " cm a second, around a curve of the same size")
print("")
print("weightless is not the same as without weight; it is what gravity feels like")
print("when nothing stands in its way, and an orbit is a fall that keeps missing")
print("the ground")
print("")
print("The scale reads " + str(scale_reading_kg) + " - that is exact. But gravity at " + str(station_altitude_km) + " km is " + str(gravity_at_the_station_per_mille) + " per")
print("mille of the surface's: the station falls " + str(fall_in_one_second_cm) + " cm in each second while the")
print("ground curves away " + str(surface_curves_away_in_one_second_cm) + " cm under its " + str(station_speed_m_per_s) + " m/s, and a scale falling with the")
print("crew reads nothing, until the pull is measured instead of the floor.")

earth's radius                  : 6371 km
station altitude                : 400 km, 6771 km from the centre
gravity at the station          : 885 per mille of the surface's, 869 cm/s2
pull on a 70 kg astronaut       : 61 kg of force
a scale aboard reads            : 0 kg

station speed                   : 7673 m/s, one orbit in 5544 s, 92 minutes
fall toward the earth in 1 s    : 434 cm
surface curves away in that 1 s : 434 cm, under 7673 m of travel

the out-of-gravity reasoning
  gravity : weakens with distance from the earth
  altitude : 400 km, measured
  scale aboard : reads 0 kg
  intent : is there gravity up there
  facts wrong : 0
  verdict : UP THERE, GRAVITY IS GONE

  reading the scale aboard is the part done right here, and
  it is why 0 kg is exactly what the crew presses on the floor

falling around the earth
  how gravity weakens : with the square of the distance from the
    centre, and 6771 km is not much more than 6371
  what is left : 885 per mille of the surface pull
  what the station does with it : falls, 434 cm in each second,
    like anything dropped
  why it never lands : at 7673 m/s sideways the ground curves
    away 434 cm in that same second
  why the scale reads zero : the scale falls with the crew, and
    nothing can press on a floor that falls just as fast

the explanation
  believed : the station is above the gravity
  actual : 885 per mille of it, pulling a 70 kg astronaut with 61 kg
    of force
  is the scale wrong : no; nothing presses on it
  is weightlessness the absence of gravity : no; it is falling
    freely in it

null control - measure the pull, not the scale
  weight, read from the scale : 0 kg
  weight, read from the pull of gravity : 61 kg
  kg the falling scale hides : 61
  no astronaut and no orbit changed; the force was measured
  instead of what a falling floor pushes back with

what a scale reading zero aboard the station guarantees
  nothing presses on the floor : exactly
  there is no gravity up there : not addressed; at 6771 km
    from the centre gravity is 885 per mille of the surface's, and
    the crew floats because crew, scale and station fall
    together, 434 cm a second, around a curve of the same size

weightless is not the same as without weight; it is what gravity feels like
when nothing stands in its way, and an orbit is a fall that keeps missing
the ground

The scale reads 0 - that is exact. But gravity at 400 km is 885 per
mille of the surface's: the station falls 434 cm in each second while the
ground curves away 434 cm under its 7673 m/s, and a scale falling with the
crew reads nothing, until the pull is measured instead of the floor.

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