Math Problem Statement
The mass of planet Mars is 6.42 × 1023 kg and mean radius of 3390 km. The duration of a day on Mars is 24 hours 40 minutes. Perhaps in the future, a geostationary satellite may be planned for Mars. Determine the orbital radius (in km) of such a satellite (measured from center of Mars)
Solution
To determine the orbital radius for a geostationary satellite around Mars, we need to find the radius at which the satellite's orbital period matches the rotational period of Mars. We can apply Kepler's third law here, which connects the period of orbit with the orbital radius as follows:
where:
- is the orbital period (equal to the rotational period of Mars for a geostationary satellite),
- is the gravitational constant ,
- is the mass of Mars ,
- is the orbital radius from the center of Mars.
Step 1: Convert the period to seconds
The day on Mars is 24 hours 40 minutes, so:
Step 2: Rearrange Kepler's Third Law to solve for
Rearrange the equation for :
and then take the cube root:
Step 3: Substitute values and solve
Substitute , , and :
Carrying out this calculation yields an approximate value for .
Result
Upon calculating, we find:
Thus, the orbital radius required for a geostationary satellite around Mars is approximately 20427 km from the center of Mars.
Would you like further explanation or a breakdown of any part?
Related Questions
- How would the orbital radius change if the rotational period of Mars were different?
- What other factors affect a satellite’s stability in geostationary orbit?
- How does Mars’ lower gravity impact the satellite’s orbital characteristics compared to Earth?
- Could the geostationary satellite experience tidal effects from Mars' moons?
- What would be the energy requirements to place a satellite in this orbit?
Tip:
Remember that Kepler’s laws apply universally to any orbiting body, making them useful for analyzing celestial mechanics across various planets.
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Math Problem Analysis
Mathematical Concepts
Orbital Mechanics
Kepler's Third Law
Formulas
T^2 = (4 * π^2 * r^3) / (G * M)
r = (G * M * T^2 / (4 * π^2))^(1/3)
Theorems
Kepler's Third Law of Planetary Motion
Suitable Grade Level
Undergraduate Physics/Astronomy