Problem 62 - 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 can see stars and galaxies whose brightness is \(\frac{1}{50}\) of the faintest objects now 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 can see about seven times farther than a ground-based telescope.

Step-by-Step Solution

Answer

The space telescope can see about seven times farther than a ground-based telescope.

Step 1: Understand the concept of brightness and distance

First, we need to recall the physicist law of the intensity of light which is also called the inverse square law, a rule which is applicable here. This implies that brightness \( B \) of a light is inversely proportional to the square of its distance \( d \) i.e. \( B = \frac{k}{d^2} \) where \( k \) is a proportionality constant.

Step 2: Setup the equation based on the relationship

Next, we find the ratio of the brightness of the faintest objects seen by the Hubble Space Telescope to that observable by a ground-based telescope. This would be \( \frac{B_{ground}}{B_{space}} = \frac{1}{50} \) and it can be written as \( \frac{\frac{k}{d_{ground}^2}}{\frac{k}{d_{space}^2}} = \frac{1}{50} \) where \( B_{ground} \) and \( B_{space} \) are the brightness for ground and space telescopes respectively and \( d_{ground} \) and \( d_{space} \) are their corresponding distances.

Step 3: Solve for the relationship between the distances

From the equation in Step 2, the \( k \) on both sides can be omitted and by cross multiplying, we can find the relationship between the distances. It would be \( d_{space} = d_{ground} \cdot \sqrt{50} \)

Step 4: Evaluate the square root of 50

From Step 3, we see that the distance by the Hubble Space Telescope is the ground distance multiplied by the square root of 50. Evaluating the square root of 50 gives a value approximately close to 7.

Key Concepts

Brightness and Distance

The relationship between brightness and distance is explained through the inverse square law. This law states that the brightness of a point source, like a star, decreases with the square of its distance from the observer. In mathematical terms, this can be expressed as \( B = \frac{k}{d^2} \), where \( B \) is brightness, \( d \) is distance, and \( k \) is a constant. As the distance between the observer and the source increases, the observed brightness decreases rapidly. This is crucial when studying astronomical objects because it helps astronomers understand how visible or faint an object might be based on its distance from Earth.
For example, compared to a ground-based telescope, the Hubble Space Telescope can spot stars whose brightness is just \( \frac{1}{50} \) of what the faintest stars visible from Earth-based observatories are. Using the inverse square law, it means that Hubble is capable of seeing objects significantly further away, offering a clearer picture of distant galaxies and stars.

Hubble Space Telescope

The Hubble Space Telescope (HST) has revolutionized our understanding of the universe. Placed in Earth's orbit, it is not hindered by the atmosphere, unlike ground-based telescopes. This makes Hubble capable of capturing exceptionally clear images of celestial bodies. Due to its location beyond the Earth's atmospheric distortions:

• It achieves much higher image resolution.
• It can detect very faint astronomy objects that are invisible from Earth.
• It can view in ultraviolet and near-infrared wavelengths that are otherwise absorbed by the atmosphere.

Through Hubble's lenses, we can study faint and distant galaxies, nebulae, and stars. For instance, it observes objects with brightness as low as \( \frac{1}{50} \) of what is possible from Earth, meaning it can observe around seven times farther into space. This allows astronomers to gather detailed data on cosmic phenomena, understand the structure of galaxies, and improve insight into the origins of the universe.

Astronomy Observations

Astronomy observations have dramatically evolved with advancements like the Hubble Space Telescope. Observing celestial phenomena is crucial as it leads to new discoveries and deeper understanding of the universe. Traditional ground-based telescopes have limitations due to atmospheric distortion, weather, and light pollution, but advancements in technology have mitigated many of these challenges.
The Hubble Space Telescope contributes significantly to these observations by:

• Providing a continuous, uninterrupted view of space regardless of weather conditions.
• Offering higher clarity and precision in measurements due to the absence of atmospheric interference.
• Enhancing the ability to track and study cosmic events over time.

Advancements like those provided by the Hubble are vital in fields such as cosmology, where observing the universe's most distant objects are essential. The data collected helps formulate theories about the universe's origin and evolution, supports the search for exoplanets, and plays a crucial role in mapping the cosmos.