Circular orbits

1. Mar 23, 2005

dearborne

back story . i bombed my midterm . and i get a chance to make up some marks by redoing it at home . but i'm stuck on one question . so here goes .

the planet mars has two satellites, Phobos and Deimos, that orbit mars along almost circular paths .
a) Phobos average orbital radius [R] is 9.378 * 10^6m , and it's orbital period [T] is 2.754 * 10^4s. Calculater Mars's mass .

b) Deimos's orbital period is 1.08 * 10^5s . Calculate the ratio of the satellites orbital speeds (Vp/Vd) .

----

so the first thing i did was get the acceleration with a=(v^2/r) .
so a=4.95*10^-3 .
my thought was that i was going to use F=(G*M1*M2)/R^2 to get the F , and then use that to get the mass . but obviously i forgot that i don't have one of the masses . so that won't work .
and i know that for b) the velocity[v] = 2?r/t . but then i only have the orbital radius for Phobos .

so i have:
Rp = 9.378*10^6
Tp = 2.754*10^4
Td = 1.08*10^5
the centripetal acceleration [a] of mars and phobos = 4.95*10^3
Vp = 2.139*10^3

now from here i'm stumped . i have all sorts of crap , and i'm still missing the mass of mars and one of the velocities . so if anybody out there had a hint or two they could give me , i would be forever gratefull . maybe i'll even shovel your walk in the winter , or rake your leaves in the fall .

2. Mar 23, 2005

tony873004

You have a cool teacher! That or too many people did poorly and the teacher needs to do something to raise the overall curve.

This is the formula you need. Keep in mind that whichever m you use for Phobos is insignificant compared to the mass for Mars, and you can just eliminate it. Also, a stands for semi-major axis, which is "average orbital radius".

Now just use your algegra skills to isolate the m you didn't eliminate.
$$p=\sqrt{\frac{a^3}{m_1+m_2}}$$

This formula gives P (period) in Earth years, and wants a (semi-major axis) in AU, and m in solar masses. So you have to do some unit conversion too!
Now that you have Mars's mass from part 1, you can use the formula I gave you "as-is" to compute the period of Deimos. Remember that the mass of Diemos is insignificant compared to the mass of Mars, so again, you can simply ignore one of your m's

As a double check, do a Google for "Mass of Mars" and "Mass of Diemos" to see if your answers agree with those published on various web sites.
That would be cool!

Last edited: Mar 23, 2005
3. Mar 23, 2005

Andrew Mason

Use:

$$a = GM/R^2 = \omega^2r = 4\pi^2 r/T^2$$

Use the result for M from a) and find the radius of Deimo's orbit. Or you could use Kepler's Third law:

$$R^2/T^3 = Constant$$ so:

$$R_p^2/R_b^2 = T_p^3/T_b^3$$

and: $v = \omega r = 2\pi r/T$

AM