On August 12, 2005 the Mars Reconnaissance Orbiter (MRO) was launched from Earth and reached Mars on March 10, 2006. The orbiter is in roughly circular orbit about 390 km above the martian surface. Mars has a radius of 3396 km and a mass of 0.642 x 1024 kg (6.42E23 kg).
b) What is the radius of the MRO orbit?
c) What is the linear speed (v) of the orbiter above Mars?
d) What is the period of the orbiter in hours
To find the radius of the MRO orbit, we know that the orbiter is in a circular orbit around Mars. The radius of the orbit is equal to the sum of the radius of Mars and the altitude of the orbiter above the Martian surface.
b) Radius of the MRO orbit:
Radius of Mars (r) = 3396 km
Altitude of MRO (a) = 390 km
Radius of the MRO orbit = Radius of Mars + Altitude of MRO
Radius of the MRO orbit = 3396 km + 390 km
Radius of the MRO orbit = 3786 km
So, the radius of the MRO orbit is 3786 km.
To calculate the linear speed (v) of the MRO orbiter, we can use the formula for orbital speed:
v = (G * M)^(1/2) / r
Where:
G is the gravitational constant (6.67430 x 10^(-11) m^3 kg^(-1) s^(-2)).
M is the mass of Mars (0.642 x 10^24 kg).
r is the radius of the MRO orbit (converted to meters).
c) Linear speed (v) of the MRO orbiter:
Radius of the MRO orbit (r) = 3786 km = 3,786,000 meters
v = (G * M)^(1/2) / r
v = (6.67430 x 10^(-11) m^3 kg^(-1) s^(-2) * 0.642 x 10^24 kg)^(1/2) / 3,786,000 meters
Calculating the value of v will provide the answer.
The period of an orbit is the time it takes for an object to complete one revolution around the central body. The formula for the period of an orbit is:
T = 2π * (r / v)
Where:
T is the period of the orbit.
r is the radius of the MRO orbit.
v is the linear speed of the MRO orbiter.
d) Period (T) of the MRO orbiter:
r = 3786 km = 3,786,000 meters
v = Calculated linear speed from previous step
T = 2π * (3,786,000 meters / v)
Calculating the value of T will provide the period of the MRO orbiter in hours.