Question

Jupiter's moon Io orbits Jupiter every 42.5 hours at an average distance of 422,000 kilometers from the center of Jupiter. Calculate the mass of Jupiter. (Hint: Io's mass is very small compared to Jupiter's.)

Short answer

Jupiter's mass is approximately \(1.898 \times 10^{27}\) kg.

Step-by-step solution

Understand Kepler's Third Law

Kepler's Third Law relates the orbital period of a moon around a planet to the mass of the planet. The formula is \[ T^2 = \left(\frac{4\pi^2}{G M}\right) r^3 \]where \(T\) is the orbital period, \(G\) is the gravitational constant \(6.674 \times 10^{-11} \text{Nm}^2/\text{kg}^2\), \(M\) is the mass of the planet, and \(r\) is the average radius of orbit.

Rearrange the Formula to Solve for Mass

We need to find the mass \(M\) of Jupiter. Rearrange the formula from Step 1:\[ M = \frac{4\pi^2 r^3}{G T^2} \]

Convert the Orbital Period to Seconds

Given that Io's Orbital period is 42.5 hours, convert this period into seconds:\[ T = 42.5 \times 3600 \text{ seconds} = 153000 \text{ seconds} \]

Substitute the Known Values

Substitute \(T = 153000\) seconds, \(r = 422000000\) meters, and \(G = 6.674 \times 10^{-11} \text{Nm}^2/\text{kg}^2\) into the formula:\[ M = \frac{4 \pi^2 (422000000)^3}{6.674 \times 10^{-11} (153000)^2} \]

Solve for the Mass of Jupiter

Calculate the mass by evaluating the expression:\[ M \approx \frac{4 \times (9.8696) \times (7.511848 \times 10^{23})}{6.674 \times 10^{-11} \times 2.3409 \times 10^{10}} \]\[ M \approx 1.898 \times 10^{27} \text{ kg} \]

Verify Units and Results

Ensure that calculations have been done correctly and all units align. The final mass should be in kilograms, representing Jupiter's estimated mass.

Key concepts

Orbital Mechanics

Orbital mechanics is a captivating field of study that helps us understand the motions of celestial bodies like moons and planets.
One of the key components is Kepler's Third Law, which links the orbital period of a moon or planet to the mass of the object it orbits. This interplay ensures moons like Io maintain a stable orbit around massive planets like Jupiter.
Scientists use the precise measurements of how long it takes a moon to complete its orbit (the orbital period) and how far away it is (the radius of orbit) to unlock information about the mass of the planet.
The relationship is governed by the formula:

• \[ T^2 = \left(\frac{4\pi^2}{G M}\right) r^3 \]
  

Where:

• \( T \) is the time it takes for one complete orbit (orbital period, usually in seconds),
• \( G \) is the gravitational constant,
• \( M \) is the mass of the planet, and
• \( r \) is the average distance from the moon to the planet.

This formula helps astronomers calculate unknown masses by measuring the observable orbit of a moon.

Mass Calculation

Calculating the mass of a planet, like Jupiter, involves some exciting math and physics.
We use Io's orbital details to estimate the mass of Jupiter, following Kepler's Third Law, which is adjusted to:

• \[ M = \frac{4\pi^2 r^3}{G T^2} \]
  

Here, the steps are quite methodical. First, we rearrange the formula to solve for mass \( M \).
Next, convert the time period of Io's orbit from hours to seconds since the metric system uses seconds in these calculations. For example, 42.5 hours is

• \( 42.5 \times 3600 = 153000 \) seconds.

Then using Io's average orbital distance \( r \), and the gravitational constant \( G \), we plug these values into the formula.
The final expression reveals Jupiter's mass, showcasing how gravity and orbital dynamics connect with mass calculation. By employing this equation, astronomers and scientists can determine planetary masses with remarkable accuracy!

Gravitational Constant

The gravitational constant, denoted as \( G \), is a vital component when studying celestial movements. It measures the strength of gravity and is a universal constant in physics. This constant plays an essential role when calculating gravitational forces and planetary masses.
The value of the gravitational constant is approximately \( 6.674 \times 10^{-11} \) m²/kg², ensuring that our calculations for forces and masses are consistent and accurate across various celestial bodies. In the context of Io's orbit around Jupiter:

• \( G \) helps connect the gravitational forces to the mass of Jupiter as seen in the formula for calculating mass.
• By knowing \( G \), we can determine how gravitational forces keep Io smoothly orbiting Jupiter.

Its precise determination allows for consistent scientific exploration and is fundamental when extending these calculations to different moons and planets across our solar system. Understanding \( G \) is a foundation upon which the mechanics of space and gravity are understood.