The moon at 2150 km – back of the envelope

I’ve been seeing a lot of this video recently:

We can probably all agree that it looks pretty cool, particulary around the time when Luna occludes Sol at 0:53. But the imagery flipped some interrupt flags and, per my programming, I had to head to the back of the envelope to get an idea of just what would happen if Luna were to switch places with the ISS. First of all, it would be silly to take the statement “at the same distance” as literal. The moon’s radius is almost 4 times the height above ground of the ISS at apogee, so placing the lunar center of gravity within the ISS orbital range of 415-419 km [1, Oct. 18, 2013] would lead to a bit less idyllic scene than seen in the video.

earthLunaSuperposition

I’m not sure what proportion of matter would have a significant probabability of fusing, or what the average proportion of each nuclei pair would be converted to energy, but the density of the Earth/Luna overlapping volume would immediately increase to 160% of normal Earth density, increasing in temperature by about the same proportion.

Instead, let’s consider transporting the moon from its current orbit, with an apogee of about 407,000 km, to an orbit where the edge of the moon is the same distance from the earth’s surface as the ISS. That would place the approximate center of gravity of the moon at 2150 km.

earthLunaJustFine

This puts our only natural satellite well within the Roche limit[3], the minimum distance at which a satellite can remain intact without tidal forces tearing it apart, leaving a debris field in its place.

Roche

But we would run into trouble long before the debris field began to give us problems. Luna’s tangent velocity is about 1.022 km/s on average[4]. Remember the cartoon you saw in physics class where Newton shoots a cannon over the hill fast enough to enter orbit? This would be the one that didn’t make it. To maintain an orbit at this altitude, an object would have to be travelling at 6.844 km/s [5] to avoid spiralling down to the earth’s surface.

earthLunaVectors

The energy imparted throughout the impact would pretty much defy any metric we have for intuitively thinking about energy. At 2150 km above ground acceleration due to gravity is about 5.5 m/s2[7], and Lunar mass is about 7.34 X1022 kg[4], for a combined energy due to velocity and gravity potential of 3.5X1030 Joules. That’s a lot (5.2X1016, or over 50 quadrillion) Little Boy equivalents.

Even if we incresed lunar orbital velocity so that the the moon doesn’t immediately fall to ground, orbital decay would still come into play fairly quickly. The ISS loses about 2 km a month, making it dependent on expensive station keeping maneuvers. Even at the 6.844 km/s required to maintain orbit at 2150 km, a great deal of Luna would be dragging through the upper atmosphere (The thermosphere tops out between 500-1000 km [6]). So we would have a continuous rain of meteoric debris to look forward to as tidal forces ripped our moon apart, the majority of the mass of the moon following a decaying orbit until eventually (my guess is well within 20 years) the largest chunks and the majority of the total mass had dissipated their kinetic and potential energy relative to the Earth in a concussive fashion.

Free Luna, comrades.

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