MadSci Network: Astronomy Query:

### Re: Why does Phobos orbit Mars in ~ 8 hours while Earth's moon takes 24 hours?

Date: Thu Oct 15 00:07:18 1998
Posted By: Justin Miller, Undergraduate, Computer Science, Geneva College
Area of science: Astronomy
ID: 908281767.As
Message:
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Don--

Well, the masses and diameters (actually, radii) do play a role in the
length of time a moon takes to revolve around a planet.  The only problem
with your question is that it takes the moon closer to 27 days to orbit the
earth.  This is where we get our months from.

Let's see if I can explain a little more in-depth than that now.  First,
I'll push some numbers up here for you.

mass of Mars:    6.42 x 10^23 kg
mass of Phobos:  1.08 x 10^16 kg
mass of earth:   5.97 x 10^24 kg
mass of Moon:    7.35 x 10^22 kg

radius of Mars:    3397 km
radius of Phobos:    11 km
radius of earth:   6378 km
radius of Moon:    1738 km

Time Phobos takes to orbit Mars:   0.318910 days
Time Moon takes to orbit earth:   27.321582 days

The average distance from the center of Mars to the (approximate, as it is
not spherical) center of Phobos is 9378 km.  The distance from the center
of earth to the center of the Moon is 384400 km.

Now that that's out of the way, the fun part comes in...the equations.  You
probably saw this coming. =)

Fg = G (m1 x m2)/d^2  where Fg is the force of gravity, G is the
gravitational constant, m1 and m2 are the masses of the two objects, and r
is the distance between the centers of the objects.

Using this equation (I'll spare you the number crunching) the force of
attraction between Mars and Phobos is 5.26 x 10^15 N, and between earth and
the Moon is 1.98 x 10^20 N.

Next equation:  v = sqrt(rg)  where v is the velocity of the satellite, r
is the radius of its orbit (or distance between the centers) and g is the
acceleration of gravity.  The velocity of Phobos is 2137 m/s and the
velocity of the Moon is 1017 m/s.  Now we're beginning to see why Phobos
orbits in less time...

The last value we'd need to find the time of orbit is the length of the
orbit.  This one's C = 2(pi)r, where C is the circumference of the orbit
(true, the orbits aren't circular, but we're working with rough averages
here anyway).  Therefore, the length of Phobos's orbit is 58900 km, and of
the Moon's orbit, 2420000 km.

Using these (this has been long enough, hasn't it?  heh), we can find the
time it takes each body to orbit.  Note that the numbers won't be perfect,
because there's been quite a bit of rounding involved, and these numbers
can't necessarily be trusted with as many decimals as they've been taken
out to.

t = d/v, where t is time, d is distance, and v is velocity.  Plug the
numbers into this equation, and the time it takes Phobos to orbit Mars is
7.66 h (0.3190046622 days...looks pretty close to the accepted value), and
the time it takes the Moon to orbit the earth is 661 h (27.54106122 days,
again close enough to be within an acceptable margin of error).

I got my information here from
http://seds.lpl.arizona.edu/nineplanets/nineplanets/data1.html and from
telnet://ssd.jpl.nasa.gov:6775.  Your browser may or may not be able to
view the second address.  You may instead need a telnet client to get the
information there.

Justin Miller
gemiller@bellatlantic.net

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