Planet Skimming Satellites Orbit in about an Hour
OK, OK, so it's closer to 1.5 hours for a satellite skimming the earth's surface, but we*
came across this little gem (that all planet-skimming satellites orbit in about an hour)
earlier this summer and I couldn't resist sharing it with an audience who would
appreciate it (I hope so anyway!).
"Discovering" this kind of thing is one of the reasons why I love physics! If you want
more details, keep reading …
*Carl Campbell, one of our 2010 SPS summer interns, and I were discussing the Lunar
Reconnaissance Orbiter (see http://lunar.gsfc.nasa.gov/ for "Ten Cool Things Seen in
the First Year of LRO" and see http://www.spsnational.org/programs/internships/ for
more about SPS interns), and it was noted that one of the previous interns (Jesus
Cantu) had indicated that the LRO orbited the moon in about an hour-and-a-half. This
caused me to perk up my ears because I recalled that the space shuttle orbits the earth
in about the same time....probably not a coincidence, right?
So, we applied the usual Newtonian prescription to a satellite of mass m orbiting very
near the surface of a spherical mass M with density d; specifically, in the direction
toward the center of the planet we have:
F = ma
GMm/R2 = mV2/R,
V = 2R/T
GM/R2 = 42R/T2
or
but since
where T is the period, so
or
T2/R3 = 42/(GM),
which is the usual Newtonian form of Kepler’s third law. For a planet-skimming satellite
though, M ~ d(4/3)R3, so we find the happy result
T2 = 3/(G*d)
which shows that the period of a planet-skimming satellite depends only on the density
of the planet (G is Newton's universal gravitational constant, 6.67E-11 N*m^2/kg^2).
Wow! This is why I love physics! It is very cool (to me anyway) that the time of orbit
for near-surface satellites is about the same no matter the size or mass of the planet,
and that this kind of knowledge is "discoverable".
Thus, the LRO has about the same orbital time as the space shuttle, the terrestrial
planets all have near-satellite periods between 1 and 2 hours, and the Jovian planets'
are not much more than that.** Incidentally, I remember an exam once where I was
asked for the period of oscillation for a rock dropped into a hole that went all the way
through the earth---it takes a similar amount of time, as I recall, but in that case, I
guess you'd have to assume uniform density to get that result. Cool beans, as they
say...GDW
**Carl Rutledge (Eastern Central University) wrote in to say that Saturn, with its very low
density (considerably less than water, so it would float) has planet-skimming satellites with
periods of nearly 4 hours, the extreme case for the solar system…