Problem 11 - A Graphical Approach to College Algebra

Question

The magnitude of a star is defined by the equation $$M=6-2.5 \log \frac{I}{I_{0}}$$ where \(I_{0}\) is the measure of a just-visible star and \(I\) is the actual intensity of the star being measured. The dimmest stars are of magnitude \(6,\) and the brightest are of magnitude 1. Determine the ratio of light intensities between a star of magnitude 1 and a star of magnitude 3.

Step-by-Step Solution

Answer

The magnitude 1 star is approximately 6.31 times brighter than the magnitude 3 star.

Step 1: Set Up the Magnitude Equations

First, write down the magnitude equation for both stars. For the star with magnitude 1, we have:\[ M_1 = 6 - 2.5 \log \frac{I_1}{I_0} \Rightarrow 1 = 6 - 2.5 \log \frac{I_1}{I_0} \] Similarly, for the star with magnitude 3, we get:\[ M_2 = 6 - 2.5 \log \frac{I_2}{I_0} \Rightarrow 3 = 6 - 2.5 \log \frac{I_2}{I_0} \]

Step 2: Solve for Logarithms

Solve the equations you set up in Step 1 for the logarithmic terms. For the magnitude 1 star: \[ 1 = 6 - 2.5 \log \frac{I_1}{I_0} \] \[ \Rightarrow \log \frac{I_1}{I_0} = \frac{5}{2.5} = 2 \], so \[ \frac{I_1}{I_0} = 10^2 \]For the magnitude 3 star:\[ 3 = 6 - 2.5 \log \frac{I_2}{I_0} \] \[ \Rightarrow \log \frac{I_2}{I_0} = \frac{3}{2.5} = 1.2 \], so \[ \frac{I_2}{I_0} = 10^{1.2} \]

Step 3: Calculate the Intensities

Now let's calculate the actual intensity ratios from the log results.\[ \frac{I_1}{I_0} = 10^2 = 100 \]\[ \frac{I_2}{I_0} = 10^{1.2} \approx 15.85 \]

Step 4: Determine Ratio of Intensities

Finally, calculate the ratio of intensities between the star of magnitude 1 and the star of magnitude 3. This is simply the ratio \( \frac{I_1}{I_2} \).\[ \frac{I_1}{I_2} = \frac{100}{15.85} \approx 6.31 \]

Step 5: Conclude the Solution

The ratio of the light intensities between a star of magnitude 1 and a star of magnitude 3 is approximately 6.31, meaning the magnitude 1 star is 6.31 times brighter than the magnitude 3 star.

Key Concepts

Understanding Intensity Ratio

When we talk about the intensity ratio in the context of stars, we are comparing how much light one star emits compared to another. In this exercise, we calculated the intensity ratio between stars of different magnitudes. The intensity ratio is determined by dividing the intensity of one star by the intensity of another. In our example, when we calculated the ratio \( \frac{I_1}{I_2} \), it told us how many times brighter the magnitude 1 star is compared to the magnitude 3 star.
The key takeaway is that the intensity ratio helps us understand brightness differences between stars. A higher ratio means a much brighter star compared to the dimmer one in our comparison.
This concept is important in astronomy, where comparing stellar brightness can reveal information about a star's properties.

Exploring Logarithmic Equations

Logarithmic equations are essential in understanding the relationship between magnitude and intensity in stars. The equation \( M=6-2.5 \log \frac{I}{I_{0}} \) shows how magnitude depends on intensity. Logarithms allow us to scale down large numbers into manageable ones for easier calculation.
When solving these equations, you're essentially reversing the process of exponentiation. For example, \( \log \frac{I_1}{I_0} = 2 \) means \( \frac{I_1}{I_0} = 10^2 \). This transformation is pivotal in calculating the intensity ratios.
The logarithmic scale is especially useful in astronomy because it accommodates the vast range of star brightnesses in a concise form. This makes it simpler to identify patterns and relationships in stellar data.

Decoding Light Intensity

Light intensity refers to the amount of light a star emits, which ultimately reaches the observer. It's a critical measure in determining a star's visibility and brightness.
In the magnitude equation, \( I \) stands for the intensity of the star, and \( I_0 \) represents a standard baseline intensity like that of a just-visible star.
Intensity affects how we perceive a star's brightness. The higher the intensity, the brighter the star appears. This link between intensity and brightness is foundational to the magnitude scale used in astronomy, helping scientists and observers to rank and compare stars.

Unpacking Stellar Brightness

Stellar brightness is how bright a star appears from Earth. It's an observable feature influenced by a star's light intensity.
The magnitude scale is the tool we use to quantify this brightness, with lower magnitudes indicating brighter stars. For example, a magnitude 1 star is much brighter than a magnitude 6 star.
The brightness of a star is not only a function of its emitted light but also its distance from Earth. However, in a controlled setting like this exercise, where distance is constant, it's the intensity that becomes the major factor. Understanding brightness variations through magnitudes helps astronomers categorize stars, analyze their properties, and determine their distances.