> > probably calculate how fast they were decelerating if we knew the altitude > > of the gravity block. > Indeed the equation is pretty simple. But honestly I don't feel like > working it out now, it's getting a little too technical for the list.
Ok can't stop myself :P
E^2 ( 1 - r^3/S^3)
a = g --------- ------------------------
r^2 ( 1 - E^3/S^3)
Where a is the gravitational acceleration anywhere along the space
elevator at a distance r from center of Earth. Remember altitude = (r
- E)
g = gravitational acceleration on Earth surface, = 9.8 m/s^2
S = radius of geostationary orbit = ~42,000,000 m
E = radius of Earth = ~ 6,300,000 m
So for r = E, a = g, the normal gravity on Earth surface.
For r = S, a = 0, zero gravity
For r > S, a < 0, meaning you "fall" away from Earth
Hope the list doesn't mind a small GIF. It's a graph of gravity pull
as percent of normal gravity versus altitude, along the space
elevator. At zero altitude, gravity is 100%, at 35,700 km, it's zero.
BTW, a minor point I didn't cover yet: you do fall straight down
(below geosyn. altitude) or straight up (above geosyn altitude).
If my eyes were not too mistaken, Louise was falling at about 0.1g
(see 10% on the graph). That means the lower ring's altitude was
under 12,750 km, almost 23,000 km below Geostationary orbit. That's
only 36% of the distance from Earth surface to Geostationary orbit.
Frankly even I am shocked how low that is.
Ok let's say my eyes were deceived, and she was only falling at 0.02g
(see 2% on the graph). Then the lower ring's altitude was under 26,400
km, still more than 9300 km below the geosyn orbit. At this altitude
it takes more than 3.9 seconds to fall 1.5m. Even if Louise were very
short, she has almost 4 seconds after she steps off the 'cliff', to
reach out a hand and grab on to the edge. I seem to remember the
scene from when Louise fell off to the time when Saji fell below the
edge to last less than one second. Even then... we are still talking
about using over 9000 km of tethers to hold the lower ring up.
>From 26,400 km altitude, after 14 minutes (going to the gravity block
scene), the block would fall by almost 70 km and attain an extra
velocity of almost 600 km/hr. Meaning in 14 minutes it's altitude
fell from 26,400 km to 26,330 km, and it's original velocity of over
8500 km/hr suffered a maximal 7% perturbation. By comparison, ISS's
altitude is under 350 km. Hardly a crisis situation.
Ok, go back to 12,750km altitude scenario (0.1g). After 14 minutes,
the gravity block would have fell over 345 km and acquired an extra
velocity of about 3000 km/hr. So it's altitude is still over 12,400
km, but its velocity is now an uncomfortably slow 5830 km/hr. But the
velocity problem has much more to do with the ridiculous dangerous
position they started off with, not from 14 minutes of free-fall.
Nevertheless, they probably still have an hour of free-falling before
reaching the top of the atmosphere (120 km).
Either way, the amount of tether material need to hold up the lower
ring runs in the ridiculous range, more than 1 tons of tether per 375
tons of lower ring mass (for high altitude case, for the low altitude
case, it is many many times more). And the time limit of a rescue is
nowhere near 7-14 minutes, but would in the range of one to many
hours.
--
Dr. Core
<<attachment: gravity_space_elevator.gif>>
