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Chapter 13: Problem 45

The Martian satellite Phobos travels in an approximately circular orbit of radius \(9.4 \times 10^{6} \mathrm{~m}\) with a period of \(7 \mathrm{~h} 39 \mathrm{~min}\). Calculate the mass of Mars from this information.

Short Answer

Expert verified
The mass of Mars is approximately \(6.39 \times 10^{23} \text{ kg}\).

Step by step solution

01

Understand the Given Data

We are given the radius of the orbit of Phobos as \( r = 9.4 \times 10^{6} \text{ m} \) and its orbital period as 7 hours and 39 minutes. First, convert the period into seconds: \( T = 7 \times 3600 + 39 \times 60 \text{ s} = 27540 \text{ s} \).
02

Use Kepler's Third Law Formula

Kepler's third law for a satellite orbiting a planet can be written as: \( T^2 = \frac{4 \pi^2}{G M} r^3 \), where \( G \) is the gravitational constant \( 6.674 \times 10^{-11} \text{ m}^3 \text{kg}^{-1} \text{s}^{-2} \), \( M \) is the mass of Mars, \( r \) is the radius of the orbit, and \( T \) is the period.
03

Rearrange the Formula

Rearrange the formula to solve for the mass of Mars \( M \): \[ M = \frac{4 \pi^2 r^3}{G T^2} \].
04

Substitute the Known Values

Substitute the known values into the formula: \( M = \frac{4 \pi^2 (9.4 \times 10^6)^3}{6.674 \times 10^{-11} \times 27540^2} \).
05

Calculate the Mass of Mars

Calculate \( M \) using the substituted values: \( M = \frac{4 \times 3.1416^2 \times (9.4 \times 10^6)^3}{6.674 \times 10^{-11} \times 27540^2} \approx 6.39 \times 10^{23} \text{ kg} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Orbital Mechanics
Orbital mechanics deals with the motion of objects in space under the influence of gravitational forces. When we talk about a satellite orbiting a planet, we are referring to how it travels along a curved path around the planet. This path is called an orbit. The properties of the orbit depend on several factors, including the mass of the planet and the satellite, as well as the distance between them.
  • Orbits can be elliptical or circular; for simplicity, we often assume them to be circular in calculations.
  • In circular orbits, the period and radius are key components in determining orbital parameters.
The study of how objects move under the influence of gravity is crucial for predicting the motion of natural celestial bodies and artificial satellites. Kepler's laws, especially Kepler's Third Law, are fundamental to such predictions.
Gravitational Constant
The gravitational constant, denoted by the symbol \( G \), is a crucial part of the universal law of gravitation. It measures the strength of gravity in the universe. The value of \( G \) is approximately \( 6.674 \times 10^{-11} \text{ m}^3 \text{kg}^{-1} \text{s}^{-2} \), a very small number that reflects the relative weakness of gravity compared to other forces in nature.
  • \( G \) allows us to calculate the gravitational attraction between two masses.
  • It is used extensively in orbital mechanics for calculating orbital periods and other dynamical properties of celestial bodies.
Understanding \( G \) is essential for problems involving planetary mass calculation, as it helps establish a relationship between the masses involved and the orbital characteristics of satellites.
Satellites
Satellites are objects that revolve around planets. They can be natural, like moons, or artificial, like the many human-made satellites in Earth's orbit. Understanding their behavior requires knowledge of several factors.
  • Satellites stay in orbit due to a balance between gravity pulling them towards the planet and their forward momentum.
  • The distance from the satellite to the planet and its orbital speed determine the nature of the orbit.
In the context of our exercise, Phobos is a satellite of Mars. By understanding its orbital period and radius, we can determine important information about Mars, such as its mass. This highlights the value of observing satellite motions in determining planetary characteristics.
Planetary Mass Calculation
Planetary mass calculation involves determining the mass of a planet using observations of satellite orbits. Kepler's Third Law is instrumental in this process. This law relates the period of orbit of a satellite to the radius of the orbit and the mass of the planet. The formula used is:\[M = \frac{4 \pi^2 r^3}{G T^2}\]Where:
  • \( M \) is the mass of the planet (Mars in this case).
  • \( r \) is the radius of the orbit of the satellite.
  • \( T \) is the orbital period of the satellite.
  • \( G \) is the gravitational constant.
By plugging the known values into this formula, we were able to calculate the mass of Mars. This is a typical example of how scientists and astronomers use observations of satellites to determine characteristics of celestial bodies they orbit.

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Most popular questions from this chapter

A very early, simple satellite consisted of an inflated spherical aluminum balloon \(30 \mathrm{~m}\) in diameter and of mass \(20 \mathrm{~kg}\). Suppose a meteor having a mass of \(7.0 \mathrm{~kg}\) passes within \(3.0\) \(\mathrm{m}\) of the surface of the satellite. What is the magnitude of the gravitational force on the meteor from the satellite at the closest approach?

One way to attack a satellite in Earth orbit is to launch a swarm of pellets in the same orbit as the satellite but in the opposite direction. Suppose a satellite in a circular orbit \(500 \mathrm{~km}\) above Earth's surface collides with a pellet having mass \(4.0 \mathrm{~g}\). (a) What is the kinetic energy of the pellet in the reference frame of the satellite just before the collision? (b) What is the ratio of this kinetic energy to the kinetic energy of a \(4.0 \mathrm{~g}\) bullet from a modern army rifle with a muzzle speed of \(950 \mathrm{~m} / \mathrm{s}\) ?

Miniature black holes. Left over from the big-bang beginning of the universe, tiny black holes might still wander through the universe. If one with a mass of \(1 \times 10^{11} \mathrm{~kg}\) (and a radius of only \(1 \times 10^{-16} \mathrm{~m}\) ) reached Earth, at what distance from your head would its gravitational pull on you match that of Earth's?

In a shuttle craft of mass \(m=3000 \mathrm{~kg}\), Captain Janeway orbits a planet of mass \(M=9.50 \times 10^{25} \mathrm{~kg}\), in a circular orbit of radius \(r=4.20 \times 10^{7} \mathrm{~m} .\) What are (a) the period of the orbit and (b) the speed of the shuttle craft? Janeway briefly fires a forwardpointing thruster, reducing her speed by \(2.00 \%\). Just then, what are (c) the speed, (d) the kinetic energy, (e) the gravitational potential energy, and (f) the mechanical energy of the shuttle craft? (g) What is the semimajor axis of the elliptical orbit now taken by the craft? (h) What is the difference between the period of the original circular orbit and that of the new elliptical orbit? (i) Which orbit has the smaller period?

Certain neutron stars (extremely dense stars) are believed to be rotating at about 1 rev/s. If such a star has a radius of \(20 \mathrm{~km}\), what must be its minimum mass so that material on its surface remains in place during the rapid rotation?
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