Mars rotates slowly about its axis, the period being 687 days. The mass of mars is 6.39 x 10^23 kg. Determine the radius for a synchronous satellite in orbit around Mars. (assume circular orbit)
GMm*ms/r^2=mw^2 r
w=2pi rad/687days
=2PI/(687*24*3600) rad/sec
solve for r.
r^3=GMm/w^2
r = cubr of that
To determine the radius for a synchronous satellite in orbit around Mars, we need to use the concept of synchronous orbit. In a synchronous orbit, the satellite takes the same amount of time to orbit its planet as the planet takes to rotate on its axis.
We know that Mars has a rotation period of 687 days. Therefore, a satellite in synchronous orbit around Mars will also complete one orbit in 687 days.
We can use Kepler's Third Law, which relates the orbital period, radius, and mass of a planet, to find the radius of the synchronous orbit.
Kepler's Third Law states that the square of the orbital period (T) of a satellite is directly proportional to the cube of the radius (r) of its orbit. Mathematically, it can be written as:
T^2 = (4π^2/GM) * r^3
Where:
T = orbital period (in seconds)
r = radius of the orbit
G = gravitational constant (6.67430 × 10^-11 m^3 kg^-1 s^-2)
M = mass of Mars (6.39 x 10^23 kg)
First, let's convert the rotation period of Mars to seconds:
687 days = 687 * 24 * 60 * 60 seconds
Now, we have all the information to calculate the radius of the synchronous orbit.
T^2 = r^3 * (4π^2/GM)
r^3 = (T^2 * GM) / (4π^2)
r = (T^2 * GM / 4π^2)^(1/3)
Substituting the values:
r = ((687 * 24 * 60 * 60)^2 * (6.67430 × 10^-11) * (6.39 x 10^23)) / (4π^2)^(1/3)
Calculating this expression will give us the radius of the synchronous satellite orbit around Mars.