[QUOTE=Critical1]
ok this one is confusing? how is the radius not relevant? if it werent then wouldnt we already be flying off into space? and how do you arrive at the orbital distance of 1.85 million km? " The diameter of the earth at the equator is 7,926.41 miles (12,756.32 kilometers)."
my guess is that we would end up in an orbit that is smaller than the current diameter not larger? or am I reading that wrong?
(and I never really considered the total population of earth in this orbit, we would end up as Rings probably all pretty and red)
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I realise my answer was a little misleading - what I calculated is the radius of an orbit where the current rotational velocity of the earth is enough to keep you there. As others have noted, since you don’t have that velocity, you will go crashing into the surface unless you have some means of lifting yourself up to the required orbit of 1.85 million km.
And no, the orbit would be much larger than the current radius, for the simple reason that orbital velocity is inversely proportional to the radius of the orbit. The Space Shuttle orbits at an altitude of about 380 km above the earth’s surface, and
So to keep yourself in an orbit of radius 6758km (6378 + 380) already needs a velocity of 27,650 km/h. To come closer to the earth without falling, the Space Shuttle would have to go faster still. If it moved further out, the required velocity would drop.
[QUOTE=Critical1]
lets say the planet suddenly shrinks down to a golf ball in size leaving the atmosphere intact. how far above the golf ball sized earth would your average human orbit the new tiny planet? (orbit because in this case the spin remains so when the planet suddenly changes size everyone is still hurtling along at the current spin speed of whatever surface they were on)
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So, just to be clear, you’re asking: the Earth suddenly drops down to golf-ball-size under our feet. We keep moving at the same velocity we had before due to the rotation of the Earth. What is our orbit?
Well, the orbit you have is going to depend pretty critically on the latitude you’re at, since that’s going to determine your velocity and your angular momentum. Everyone’s new orbit will be highly elliptical— the velocity we have due to rotation is significantly smaller than the velocity required to maintain a circular orbit at Earth’s radius. Someone at the poles, whose velocity is zero to start with, will drop straight down to the new surface, regardless of how small the new Earth is; someone at the equator, with a lot more angular momentum, has the best chance of actually completing an orbit; if their perigee would need to be below the surface of the golf ball, then they’re hosed.
Going through the orbital calculations, I find that the new perigee of someone standing at latitude L when this happens will be approximately
r[sub]p[/sub] = 4 pi[sup]2[/sup] R[sup]4[/sup] cos[sup]2[/sup] L / (G M T[sup]2[/sup])
where R is the radius of the Earth, M is its mass, and T is its rotational period (1 day.) At the equator, this is about 0.3% of Earth’s radius, or about 22 km; even someone standing at 89° N (or S) latitude will still have a perigee of about 6 m. In other words, unless you’re standing really close to the pole, you’ll be able to orbit the golf ball successfully.
[QUOTE=Dervorin]
So to keep yourself in an orbit of radius 6758km (6378 + 380) already needs a velocity of 27,650 km/h. To come closer to the earth without falling, the Space Shuttle would have to go faster still. If it moved further out, the required velocity would drop.
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But a higher orbit requires more energy.
MikeS is reading this question the same way I did.
On the earth-sized earth, calculate the energy and velocity vector of a person standing on the surface. (Both are dependent on latitude. Potential energy should be with respect to the center of the earth.)
The earth shrinks to the size of a golf ball. The person is now at the apogee of a highly eliptical orbit around the new, smaller earth. That orbit can be calculated from the initial position and velocity.
If the perigee of that orbit is less than the radius of a golf ball, you’re hosed.