Question

A certain star is 14 million light-years from Earth. The intensity of the light that reaches Earth from the star is \(4 \times 10^{-21} \mathrm{W} / \mathrm{m}^{2} .\) At what rate does the star radiate EM energy?

Short answer

The star radiates energy at a rate of \( 8.81 \times 10^{36} \) watts.

Step-by-step solution

Understand the Problem

We have a star that is 14 million light-years away from Earth, and the intensity of its light reaching Earth is given as \(4 \times 10^{-21} \text{ W/m}^2\). We need to find the total energy radiated by the star, also known as its luminosity.

Use the Inverse Square Law for Intensity

The intensity \( I \) at a distance \( r \) from a source with luminosity \( L \) is given by the inverse square law: \[ I = \frac{L}{4 \pi r^2} \] where \( r \) is the distance to the source. We'll use this formula to find \( L\).

Convert Light-Years to Meters

First, we convert 14 million light-years to meters. The distance in meters \( r \) is \( 14 \times 10^6 \) light-years. Knowing 1 light-year \( \approx 9.461 \times 10^{15} \) meters, we calculate: \[ r = 14 \times 10^6 \times 9.461 \times 10^{15} \text{ m} = 1.32454 \times 10^{23} \text{ m}\]

Rearrange the Inverse Square Law

Solve for the luminosity \( L \) using the inverse square law formula: \[ L = I \times 4 \pi r^2 \] Insert the intensity and the distance: \( I = 4 \times 10^{-21} \text{ W/m}^2\) and \( r = 1.32454 \times 10^{23} \text{ m}\).

Calculate the Luminosity

Plug the values into the formula: \[ L = 4 \times 10^{-21} \times 4 \pi \times (1.32454 \times 10^{23})^2 \] Simplifying, we find: \[ L \approx 8.81 \times 10^{36} \text{ W}\] Thus, the star radiates energy at a rate of \( 8.81 \times 10^{36} \) watts.

Key concepts

Inverse Square Law

The inverse square law is a crucial concept when studying the intensity of light from a distant source like a star. It explains how light diminishes as it spreads out over a larger area. The principle states that the intensity of light or any other form of energy radiating from a point source is inversely proportional to the square of the distance from the source.
This means that as you move further away from a light source, the intensity decreases sharply. Mathematically, the relationship is expressed as:
\[ I = \frac{L}{4 \pi r^2} \]where \( I \) is the intensity, \( L \) is the luminosity or total power emitted by the star, and \( r \) is the distance between the light source and the observer.

• When the distance doubles, the intensity becomes one-fourth.
• This concept helps astronomers calculate how bright a star appears from Earth.

Understanding this law is essential for determining the energy output of celestial bodies.

Light-Year to Meter Conversion

Distances in space are vast, and astronomers use units like light-years to make them manageable. A light-year is the distance light travels in one year, which is about \( 9.461 \times 10^{15} \) meters. However, in calculations, we often need to convert these large units to meters.
When converting light-years into meters, multiply the number of light-years by the length of one light-year in meters. For instance, if a star is 14 million light-years away:
\[ r = 14 \times 10^6 \times 9.461 \times 10^{15} \text{ m} \]This provides a more tangible figure to use in calculations concerning distances in space.
These conversions allow for the application of the inverse square law and other mathematical models that operate with standard measurement units.

Electromagnetic Energy

Electromagnetic energy is the form of energy that stars emit. It encompasses a wide range of wavelengths, from gamma rays to radio waves. The energy radiated by stars spans this entire electromagnetic spectrum, but when calculating luminosity, we often consider just the visible spectrum.
This energy is crucial not only for understanding stellar properties but also for comprehending how stars interact with their environment. The emitted electromagnetic energy travels across space, following the inverse square law, losing intensity with distance.

• It is this energy that we observe as starlight.
• The energy's intensity upon reaching Earth helps us calculate the star's total energy output.

By measuring how much energy arrives on Earth, astronomers can infer the star's luminosity and other vital characteristics.

Intensity of Light

The intensity of light from a star refers to the power of the electromagnetic waves reaching us through a unit area. Given in watts per square meter (W/m²), it measures how much light energy hits a given area in space.
Understanding the intensity is key to determining a star’s luminosity when combined with the distance data using the inverse square law. The intensity observed from Earth is usually a minute fraction of the star’s entire energy output due to the vast distances.
For example, compensating for the further reduced intensity as a result of the star being 14 million light-years away, we used:\[ I = 4 \times 10^{-21} \text{ W/m}^2 \]
This tiny measurement highlights the effect of colossal space distances on the observed energy levels. Such calculations help astronomers determine various properties of stars, including size and temperature.
In turn, this information aids in building a comprehensive picture of the cosmos and its components.