Question

Be sure to show all calculations clearly and state your final answers in complete sentences. Pluto's moon Charon orbits Pluto every 6.4 days with a semimajor axis of 19,700 kilometers. Calculate the combined mass of Pluto and Charon. Compare this combined mass to the mass of Earth, which is about \(6 \times 10^{24}\) kg. Can you determine the individual masses of Pluto and Charon from the given data? Explain.

Short answer

The combined mass of Pluto and Charon is approximately \(1.586 \times 10^{21}\) kg, much less than Earth's mass. Their individual masses can't be determined from the given data.

Step-by-step solution

Understanding Kepler's Third Law

To calculate the combined mass of Pluto and Charon, we apply Kepler's Third Law in the form adapted for celestial mechanics: \[ T^2 = \frac{4\pi^2}{G(M+m)}a^3 \] where \(T\) is the orbital period, \(G\) is the gravitational constant (\(6.67430 \times 10^{-11} \, \text{m}^3\text{kg}^{-1}\text{s}^{-2}\)), \(a\) is the semimajor axis, and \(M + m\) is the combined mass of the two bodies. We are given \(T = 6.4 \times 86400\) seconds and \(a = 19700 \times 10^3\) meters.

Calculate the Combined Mass

Rearrange Kepler's Third Law equation to solve for \(M + m\):\[ M + m = \frac{4\pi^2a^3}{GT^2} \]Substitute the given values into the equation:\[ M + m = \frac{4\pi^2(19700 \times 10^3)^3}{6.67430 \times 10^{-11} (6.4 \times 86400)^2} \] After calculating, we find the combined mass \(M + m \approx 1.586 \times 10^{21}\) kg.

Compare with the Mass of Earth

The mass of Earth is approximately \(6 \times 10^{24}\) kg. Comparing this to the combined mass of Pluto and Charon, \(1.586 \times 10^{21}\) kg, we find that Pluto and Charon together are much less massive than Earth.

Determine Individual Masses

The individual masses of Pluto and Charon cannot be determined from the given data alone, as we only have the combined mass. Additional information, such as the masses calculated from gravitational effects or density and volume measurements, would be required to separate their masses.

Key concepts

Orbital Period

The orbital period is a vital component in understanding the dynamics of celestial bodies. It represents the time a celestial body takes to complete one full orbit around another body. In this context, Charon, Pluto's moon, has an orbital period of 6.4 days around Pluto.

The orbital period is typically measured in days, hours, or seconds, depending on the situation. It helps us predict the timing of movements and patterns in space.

To use it in calculations, we convert these periods into seconds, as it standardizes our computations in physics and astronomy. For example, 6.4 days translates to 6.4 multiplied by 24 hours per day, then by 3600 seconds per hour, resulting in 552960 seconds.

This conversion is crucial when using equations in physics that include the orbital period, like Kepler's Third Law, ensuring units match for consistent results.

Gravitational Constant

The gravitational constant, denoted by the symbol G, is a fundamental constant in physics that characterizes the strength of the gravitational force.

It appears in Newton's law of universal gravitation which states that every point mass attracts every other point mass by a force acting along the line intersecting both points. The gravitational constant is numerically approximately \(6.67430 \times 10^{-11} \, \text{m}^3\text{kg}^{-1}\text{s}^{-2}\).

By featuring G in equations that calculate gravitational forces, it allows scientists and astronomers to predict the behavior of planets and celestial bodies. Kepler's Third Law is adapted for celestial mechanics with the gravitational constant when considering the mutual gravitation between two orbiting bodies.

Thus, understanding G is essential for accurate predictions and calculations in astrophysics, like determining the combined mass of celestial pairs.

Semimajor Axis

The semimajor axis is a technical term used extensively in astromechanics to describe elements of an orbit. An orbit's semimajor axis is the longest diameter of an elliptical orbit, providing a measure of the orbit's size.

For planets and moons, it's useful to picture this as the average distance between the celestial object and the object it orbits. In this exercise, Charon's semimajor axis is 19,700 kilometers from Pluto.

In many calculations, this distance is converted to meters for consistency with the International System of Units (SI). It plays a crucial role in using Kepler's Third Law as it helps determine the magnitude of the gravitational effects experienced between celestial bodies.

Therefore, knowing the semimajor axis is part of determining orbit size and predicting the motion of satellites and planets.

Mass Comparison

When considering the mass of different celestial bodies, comparison is key to understanding their influence and interaction in space. In our case, to analyze Pluto and Charon together, we explored their combined mass.

From calculations using Kepler's Third Law, we deduced a mass of approximately \(1.586 \times 10^{21}\) kg for both Pluto and Charon.

This number is tiny when compared to Earth's mass of \(6 \times 10^{24}\) kg. This comparison reveals the less substantial gravitational influence Pluto and Charon hold compared to our home planet.

Moreover, distinguishing individual masses of such pairs would require additional data. Without specific density, volume calculations, or gravitational effects data, separating these values remains a challenge. It also illustrates how celestial measurements help us understand different heavenly bodies' roles within our solar system.