Problem 56 - College Algebra

Question

Galileo's telescope brought about revolutionary changes in astronomy. A comparable leap in our ability to observe the universe took place as a result of the Hubble Space Telescope. The space telescope was able to see stars and galaxies whose brightness is \(\frac{1}{s i}\) of the faintest objects observable using ground-based telescopes. Use the fact that the brightness of a point source, such as a star, varies inversely as the square of its distance from an observer to show that the space telescope was able to see about seven times farther than a ground-based telescope.

Step-by-Step Solution

Answer

By using the inverse square law, it can be shown that the space telescope can observe objects approximately seven times farther away than a ground-based telescope.

Step 1: Write down Inverse Square Law

According to Inverse Square Law, the brightness \(B\) of a star is proportional to \(\frac{1}{d^2}\), where \(d\) is the distance from the observer.

Step 2: Set up equation

Let's denote \(d_g\) and \(d_s\) as the maximum observable distances for the ground-based telescope and the space telescope, respectively. Thus, according to the problem, we know that \(\frac{1}{d_s^2} = s_i \cdot \frac{1}{d_g^2}\), where \(s_i\) denotes the brightness difference between the two.

Step 3: Solve the equation

Solving the above equation for \(d_s\), we find that \(d_s = \sqrt{s_i} \cdot d_g\). Given \(s_i = \frac{1}{7}\), substituting this value in gives \(d_s = 7 \cdot d_g\).

Step 4: Conclusion

The result implies that the space telescope can observe objects approximately seven times farther away than a ground-based telescope, hence proving what was asked for.

Key Concepts

Brightness

In astronomy, brightness refers to how luminous a celestial object appears to an observer. It's an essential property since it allows scientists to infer many characteristics about stars and galaxies. To quantify this, astronomers use the Inverse Square Law. This law states that the brightness, or intensity, of a point light source (like a star) is inversely proportional to the square of the distance from the observer. That means as you double the distance from an object, its brightness reduces to one-fourth of its original value.

• Brightness is crucial in measuring and comparing the distances and sizes of celestial objects.
• This understanding helps scientists gauge how far or near objects are in space.
• The more distant an object is, the dimmer it appears.

By using these concepts, astronomers can predict and analyze the visibility of celestial bodies, enabling advancements in both observational and theoretical astronomy.

Distance

Distance plays a fundamental role in understanding the universe. In astronomy, it determines the visibility of celestial objects based on their inherent brightness. The Inverse Square Law is a fundamental concept when discussing the distance of stars and galaxies. This law reveals that an object’s brightness decreases with the square of the increase in distance. Thus, understanding how distance affects brightness allows astronomers to measure how far away stars and galaxies are.

• Accurately estimating distances is crucial for mapping the universe.
• It helps in determining the size and scale of the universe.
• This concept is key to studying the evolution of galaxies and stars.

When comparing different observation tools, like telescopes, astronomers use these calculations to determine their range and efficacy in capturing distant objects.

Space Telescope

Space telescopes, like the Hubble Space Telescope, have transformed our understanding of the cosmos. Unlike ground-based telescopes, these observatories are stationed in space, where they are free from the Earth’s atmospheric distortions and light pollution. These factors grant them the ability to observe fainter objects at greater distances. By applying the Inverse Square Law, it's evident that such telescopes can detect weaker brightness and observe objects significantly farther away.

• Space telescopes can provide clearer and more detailed images than their ground-based counterparts.
• They can observe wavelengths that are absorbed by the Earth's atmosphere.
• This capability leads to more precise measurements and groundbreaking discoveries.

The Hubble Space Telescope, for example, has been able to capture galaxies light-years away, vastly extending our vision into the depths of the universe and enhancing our understanding of cosmological phenomena.

Ground-Based Telescope

Ground-based telescopes have been the cornerstone of astronomical observations for centuries. These telescopes are situated on Earth and use various technologies to peer into the universe. However, they face limitations due to atmospheric interference, such as light pollution and atmospheric distortion, which can restrict the visibility of fainter stars and galaxies.

• Despite these limitations, new technologies like adaptive optics have improved their performance significantly.
• Ground-based observatories are more accessible and cost-effective compared to space telescopes.
• They often comprise a complementary part of broader observational strategies alongside space telescopes.

As a result, while space telescopes like the Hubble can see seven times farther under certain conditions, ground-based telescopes continue to play a vital role in astronomical research and discovery. They serve as important tools for initial observations and broader sky surveys.