Problem 36 - Astronomy: At Play in the Cosmos

Question

You observe that a star's wavelength of maximum intensity is a very reddish \(770 \mathrm{nm}\). What temperature is the surface of that star, in kelvins, and how does it compare to the Sun?

Step-by-Step Solution

Answer

The star's surface temperature is approximately 3763 K, cooler than the Sun's 5778 K.

Step 1: Understanding Wein's Law

The problem requires the use of Wein's Displacement Law, which relates the temperature of a star to the wavelength at which it emits light most intensely. Wein's Law states: \( \lambda_{max} = \frac{b}{T} \), where \( \lambda_{max} \) is the wavelength of maximum intensity (in meters), \( T \) is the temperature in kelvins, and \( b \) is Wein's constant, approximately \( 2.897 \times 10^{-3} \) m·K.

Step 2: Convert Wavelength Units

The given wavelength, \( \lambda_{max} = 770 \) nm, must be converted to meters to use in the formula. Recall that 1 nm = \( 10^{-9} \) m. Thus, \( \lambda_{max} = 770 \times 10^{-9} \) m.

Step 3: Calculate the Temperature

Using Wein's Law, solve for temperature \( T \) by rearranging the equation: \( T = \frac{b}{\lambda_{max}} \). Substitute \( b = 2.897 \times 10^{-3} \) m·K and \( \lambda_{max} = 770 \times 10^{-9} \) m to calculate \( T \).

Step 4: Perform the Calculation

Calculate \( T = \frac{2.897 \times 10^{-3}}{770 \times 10^{-9}} \), which simplifies to \( T = 3763 \) K (rounded to the nearest whole number).

Step 5: Compare to the Sun's Temperature

The approximate surface temperature of the Sun is about 5778 K. Compare this to the star's temperature of 3763 K, which is cooler than the Sun.

Key Concepts

Stellar Temperature

When we talk about stars, understanding their temperature is crucial to comprehend their characteristics and behavior. Stellar temperature not only indicates how hot a star is, but it also determines the color and type of light the star emits. A star's temperature is calculated in kelvins, a unit of measure used in physics to represent absolute temperature.

• The hotter the star, the shorter the wavelength of the maximum intensity light it emits. Generally, this means a hot star will appear bluish, while a cooler star will appear reddish.
• Stellar temperature plays a huge role in classifying stars in terms of spectral types, which range from very hot O-type stars to cooler M-type stars.

The surface temperature, therefore, is an important parameter that provides insights into the life cycle and age of a star.

Wavelength of Maximum Intensity

Stars emit light across a spectrum of wavelengths, however, they each have a particular wavelength where the intensity is highest. This is known as the wavelength of maximum intensity. It gives us a clue about the star's temperature due to the inverse relationship defined by Wein's Displacement Law.

• The wavelength where a star shines the brightest helps astronomers determine its temperature. Wein's law formula conveys this relationship: \( \lambda_{max} = \frac{b}{T} \), where \( \lambda_{max} \) is in meters and \( b \) is a constant with a value of approximately \( 2.897 \times 10^{-3} \) m·K.
• Stars that have a wavelength of maximum intensity leaning towards the blue spectrum tend to be hotter, while those leaning towards the red spectrum are cooler.

In learning and applying this relationship, we can decipher the physical properties of stars simply by analyzing the light they emit.

Surface Temperature Comparison

To better understand a star’s characteristics, comparing its surface temperature to known stars like the Sun is helpful. This comparison offers insight into how a star might behave or how it could evolve over time.

• For instance, in this exercise, the star with a reddish wavelength of maximum intensity of 770 nm is calculated to have a surface temperature of 3763 K.
• Comparatively, the Sun's surface temperature is approximately 5778 K, indicating that the star in question is cooler than our Sun.

This comparison not only helps classify stars but also aids in understanding their chemical composition, fusion processes, and future evolution possibilities.

Temperature Calculation

Calculating the temperature of a star can be as simple as applying a mathematical formula, once you know the wavelength of maximum intensity. This step involves using the language of physics to uncover the mysteries of distant stars.

• First, convert the given wavelength to meters if it's presented in nanometers, as in most astronomical data. One nanometer equals \( 10^{-9} \) meters.
• With the wavelength in the right units, use Wein's Law: \( T = \frac{b}{\lambda_{max}} \), substituting \( b = 2.897 \times 10^{-3} \) m·K and the converted \( \lambda_{max} \).
• The temperature calculation gives us a precise value, such as 3763 K for this particular problem, helping further scientific understanding and comparison across different stars.

Mastering this calculation is key to interpreting the light from the stars and contributes significantly to the field of astrophysics.