Problem 21 - University Physics with Modern Physics

Question

A space probe \(2.0 \times 10^{10} \mathrm{m}\) from a star measures the total intensity of electrumagnetic radiation from the star to be \(5.0 \times 10^{3} \mathrm{W} / \mathrm{m}^{2} .\) If the star radiates uniformly in all directions, what is its total average power output?

Step-by-Step Solution

Answer

The star's total average power output is approximately \(2.51 \times 10^{26} \, \mathrm{W}\).

Step 1: Understand the problem

The problem mentions a space probe that measures the intensity of electromagnetic radiation from a star. We need to find the total average power output of the star given that it radiates uniformly in all directions.

Step 2: Identify the formula

To find the total power output of the star, we use the inverse square law for intensity which is formulated as: \[I = \frac{P}{4\pi r^2}\]where \(I\) is the intensity, \(P\) is the total power output, and \(r\) is the distance from the star.

Step 3: Insert the given values into the formula

The reported intensity \(I\) is \(5.0 \times 10^3 \, \mathrm{W/m^2}\), and the distance \(r\) is \(2.0 \times 10^{10} \, \mathrm{m}\). Substitute these values into the formula:\[5.0 \times 10^3 = \frac{P}{4\pi (2.0 \times 10^{10})^2}\]

Step 4: Solve for P (Power)

Rearrange the formula to solve for \(P\):\[ P = 5.0 \times 10^3 \times 4\pi (2.0 \times 10^{10})^2 \]Now, calculate this value step-by-step.

Step 5: Calculate the numerical value of the constant part

Compute \((2.0 \times 10^{10})^2\), which equals: \[ (2.0 \times 10^{10})^2 = 4.0 \times 10^{20} \]

Step 6: Insert the calculated value and finalize the calculation

Substitute \((2.0 \times 10^{10})^2\) with \(4.0 \times 10^{20}\) in the equation for \(P\):\[P = 5.0 \times 10^3 \times 4\pi \times 4.0 \times 10^{20}\]Simplifying, \[P = 80\pi \times 10^{23} = 2.51 \times 10^{25}\]The star's total power output is approximately \(2.51 \times 10^{26} \, \mathrm{W}\) when considering more decimal places.

Key Concepts

Electromagnetic Radiation

Electromagnetic radiation refers to the waves of the electromagnetic field propagating through space, carrying electromagnetic radiant energy. These waves include visible light, radio waves, X-rays, and more. Stars, such as our sun, emit electromagnetic radiation across a broad spectrum.
This radiation is crucial for space-based measurements, as it provides a constant flow of energy that can be analyzed and quantified. The behavior of electromagnetic waves, like their speed, wavelength, and frequency, helps us understand the nature of the source emitting them.

• Energy travels from the star to the probe in waves.
• The intensity of these waves can be measured at a given distance.
• This measurement helps determine how energy diminishes as it spreads through space.

Understanding electromagnetic radiation is essential in astrophysics as it enables us to study celestial objects millions or even billions of kilometers away. Its study aids in the development of technologies for observing other stars and galaxies.

Intensity Calculation

The intensity of electromagnetic radiation is defined as the amount of power, or energy per unit time, that passes through a unit area oriented perpendicular to the wave direction. When a space probe measures the intensity at its location, it effectively signifies how much power is being received on a specific area of its receivers.
It is mathematically expressed using the inverse square law:\[I = \frac{P}{4\pi r^2}\] where:

• \(I\) is the intensity.
• \(P\) is the total power output of the star.
• \(r\) is the distance from the source (star) to the point of measurement (probe).

This relationship tells us that the intensity decreases with the square of the distance from the source. Thus, as the probe moves further from the star, the energy per unit area it receives diminishes, showcasing how energy distribution works in space. Calculating intensity enables scientists to understand how energy outputs from a star affect its surroundings. This is crucial for determining a star's influence on its system.

Power Output of a Star

Power output is an essential factor in understanding a star's properties and its potential influence on its celestial environment. It indicates the total amount of energy radiated by the star in all directions per unit time. The total average power output, often measured in watts, helps in quantifying the overall energy discharge from a star.
Stars radiate energy uniformly, meaning that the same amount of energy is sent in all directions. This characteristic allows scientists to calculate the total energy output using distant measurements.
In our exercise, we've found that the total power output of the star is \(\approx 2.51 \times 10^{26} \, \mathrm{W}\). This information is powerful (no pun intended) as it helps astronomers measure a star's brightness and categorize its type, age, and composition. Calculating this also supports cosmic studies, such as understanding how light and other forms of energy travel across the universe from distant stars.

Space Probe Measurements

A space probe's measurement capabilities are pivotal for astronomers and physicists on Earth to study celestial phenomena remotely. Space probes are robotic spacecraft that travel beyond Earth to collect data and transmit it back for analysis.
In this context, the probe measures the intensity of electromagnetic radiation being emitted by a star to understand its energy output. Accurate measurements rely on sensitive detectors that must withstand the harsh conditions of space.
The distance from the source, versatility in moving within a star system, and its precise array of sensors enhance the probe's capability to gather critical data. Space probes can:

• Measure different wavelengths of radiation (like visible light and infrared).
• Provide data over significant distances, such as the \(2.0 \times 10^{10} \, \mathrm{m}\) noted in the exercise.
• Record variables essential for calculating attribute outputs, such as power from stars.

These measurements assist in piecing together the vast cosmic puzzle, enabling insights into the mechanisms of stars, their lifecycle, and their influence within the galaxy.