Question

The planet Mars has a satellite, Phobos, which travels in an orbit of radius \(9400 \mathrm{~km}\) with a period of 7 h 39 min. Calculate the mass of Mars from this information. (The mass of Phobos is negligible compared with that of Mars.)

Short answer

After simplifying and calculating the above expression, the mass of Mars is approximately \(6.427 \times 10^{23} \mathrm{kg}\).

Step-by-step solution

Write down the given parameters

Here are the given parameters: The radius \(r = 9400 \mathrm{~km} = 9400 \times 10^3 \mathrm{~m}\) (as we will work in SI units) and the period \(T = 7 \mathrm{~h} 39 \mathrm{~min} = 7.65 \mathrm{~h} = 7.65 \times 3600 \mathrm{~s}\).

Use the formula for the period of revolution

The formula for the period of revolution of a body around a celestial object is \(T = 2\pi \sqrt{\frac{r^3}{GM}}\), where \(T\) - period, \(r\) - radius, \(G\) - the gravitational constant (\(6.674 \times 10^{-11} \mathrm{Nm}^2/\mathrm{kg}^2\)) and \(M\) - mass of the celestial object (in this case, Mars).

Solve for the mass of Mars

You want to solve for \(M\), so rearrange the equation in Step 2 to solve for \(M\): \(M = \frac{r^3}{G(T/2\pi)^2}\). Plug in the known values and solve for \(M\). Mars' mass \(M = \frac{(9400 \times 10^3) ^3}{6.674 \times 10^{-11} \times (7.65 \times 3600/2\pi)^2}\).

Key concepts

Mass Calculation

When determining the mass of Mars using Phobos' orbital data, understanding mass calculation in celestial mechanics is essential. The mass of a celestial body like a planet can be calculated using the motion of objects orbiting it. In this case, the formula involves the radius of the orbit and the period of the satellite.
To successfully calculate mass (\(M\)), we rearrange the formula for the orbital period: \[T = 2\pi \sqrt{\frac{r^3}{GM}}\]Where \(T\) is the period, \(r\) is the radius, \(G\) is the gravitational constant, and \(M\) is the mass of the planet.
Using algebra, solve for \(M\): \[M = \frac{r^3}{G(T/2\pi)^2}\]Plug in the known values for radius and period to find the mass. This process demonstrates how celestial mechanics allow us to understand properties of large bodies using satellites.

Gravitational Constant

The gravitational constant (\(G\)) is a crucial factor in calculating celestial forces. Its universal value is approximately \(6.674 \times 10^{-11} \mathrm{Nm}^2/\mathrm{kg}^2\). This constant helps quantify the amount of force that gravity exerts between two objects with mass.
In orbital mechanics, \(G\) is used in formulas that relate the masses of two celestial bodies, the distance between them, and the orbital characteristics of their motions. The small magnitude of \(G\) shows that gravity is a relatively weak force. However, due to the massive size of celestial bodies and the vast distances in space, gravity has a significant effect.
Understanding \(G\)'s role is essential for precise calculations not only in planetary science but also in fields such as astrophysics and cosmology.

Satellite Motion

Phobos' movement around Mars is an example of satellite motion, a key aspect of orbital mechanics. A satellite's motion is dictated by the gravitational pull of the body around which it orbits. The orbit is typically elliptical but can be approximated as a circle if the eccentricity is small, as is common in many celestial bodies.
The factors affecting satellite motion include:

• Orbital radius: The average distance from the center of the planet to the satellite.
• Orbital period: The time it takes for a satellite to complete one full orbit.
• Central mass: The mass of the body around which the satellite orbits.

Understanding these elements allows scientists to predict satellite paths, plan space missions, and study planetary characteristics. By studying these forces and motions, scientists can determine the characteristics of not only satellites but also the planets they orbit.