I was bored:
We can do a massively simplified thought experiment in which the rotation of the Earth is abstracted away - imagine the Earth is the only thing in the universe, is not rotating, and that we’re gently prodding a pillow towards Earth from an initial distance of infinity. Let’s make it a kingsize, soft filled, hypoallergenic pillow; there’s going to be a lot of napping to be done before we reach Earth, and with infinity to travel over, the probability of anything occurring during our journey approaches one - and that includes hyperallergenic Space Cats.
Anyway; that’s an 18oz pillow, or 0.5kg (for simplicity’s sake). At infinity, our pillow’s gravitational potential energy is zero. Let’s assume the atmosphere starts at 100km from the surface (this includes 99.99997% of the atmosphere, which seems sufficient). The gravitational potential energy at this point is:
U = -GMm/r = ( 6.67*10[sup]-11[/sup] * 5.9742 × 10[sup]24[/sup] * 0.5 ) / 6,378,260
where G is the gravitational constant, M is the mass of the Earth, m that of the pillow, and r the distance from the centre of the earth. Thus our pillow’s change in potential energy is:
delta_U = -31,237,292 joules
Assuming that all of this energy has been converted into kinetic energy, and not waylaid by Space Cats, this gives us
0.5mv[sup]2[/sup] = 31,237,292
=> v = 2 * sqrt( 31.2*10[sup]6[/sup] )
= 11.2 * 10[sup]3[/sup]
So our pillow is travelling at about 11km/s as it enters the lower thermosphere. More significantly, perhaps, it’s carrying some 31 megajoules of energy that must be dissipated in a non-conflagratory manner before it nestles to earth in a manner that supports crucial vertebrae while suppressing the causes of snoring. It only picks up another 500 or so kilojoules from here to the ground, so let’s ignore that.
Now, pretty much all of this energy has to be dissipated within a distance of 100km (assuming our pillow isn’t coming in obliquely), a distance which at its present velocity it will cover in a little less than 10 seconds (obviously the atmosphere will have something to say about this). Let’s assume that this energy is dissipated at a constant rate (an outlandish assumption, but generous to the pillow in that it’s the maximum energy dissipation rate that will determine its fate, so linear dissipation gives the best chance of survival).
Let’s further assume that the pillow lands at its terminal velocity for sea level; assuming a cross section of 20"x5" (from the pillow website), a C[sub]D[/sub] of 2.1 (for the sake of argument) and density 1.225kg/m[sup]3[/sup], I get a V[sub]t[/sub] of 7.6m/s, giving a kinetic energy of 14 joules, which I will completely ignore because it’s so tiny relative to our starting point.
Now, how long will it take to reach the ground with these assumptions? Well, our kinetic energy is decreasing linearly, so our velocity is decreasing as the square root of time.
v(t) = 11.2 * 10[sup]3[/sup] + sqrt( 2 * delta_E(t) / m )
where delta_E is the decrease in kinetic energy at time t. But we “know” (read: have assumed) that the rate of energy decrease is linear:
delta_E(t) = - ( 31.2 * 10[sup]6[/sup] / t[sub]L[/sub] ) * t
where t[sub]L[/sub] is the time of landing. So now we have:
v(t) = 11.2 * 10[sup]3[/sup] - sqrt( 2 * 31.2 * 10[sup]6[/sup] * t / ( m * t[sub]L[/sub] )
Integrating this with respect to time from zero to t[sub]L[/sub] will give us the distance travelled by time t[sub]L[/sub], which is 100km. Then simply solve for t[sub]L[/sub]. I get 27s (this post is already long enough without me showing all those workings).
So our approximately 30MJ have been dissipated in approximately 30s. That means our pillow is shedding kinetic energy at a rate of about 1MW (in practice, the dissipation will not have been linear, meaning the actual maximum figure will be higher). Obviously a lot of that will end up heading into the atmosphere rather than the pillow itself, but it still sounds like an awful lot for a humble pillow to absorb in the form of heat; even a pillow that has survived the lasers of intergalactic cats. If our pillow were made of water, 1MW would be sufficient to raise its temperature by nearly 500K every second. Here we also have to take into account the insulating properties of down - the heat won’t be able to escape from the pillow’s surface very quickly at all, leading to a very rapid increase in surface temperature. While our pillow promises to be flame-retardant, I’m not sure that the Allergy Buyers’ Club have subjected their pillows to heat flows in the megawatt region.
In summary, our pillow is doomed.
All rather silly, I realise, and in particular ignoring the angle of re-entry renders the whole exercise rather academic, but still, it saved me doing some ridiculously involved integrations. 