The peak wavelength of a star’s spectrum is determined by analyzing the intensity of light emitted at different wavelengths. Here’s the process:

1. Collect the Light:

   • Use a spectrometer or diffraction grating to disperse the light from the star into its component wavelengths.

2. Measure Intensity Across Wavelengths:

   • A detector, such as a photodiode array or CCD camera, measures the intensity of light at each wavelength.
   • This produces a graph of intensity versus wavelength.

3. Locate the Peak:

   • Analyze the graph to find the wavelength (λmax) where the intensity is highest.
   • This can be done manually from the plotted graph or automatically using software that fits the data to a blackbody curve.

Example:

If the spectrum shows a graph with a maximum intensity at 600nm, the peak wavelength is 600nm. 

This value is then used in Wien’s Displacement Law to calculate the surface temperature of the star.

The maximum wavelength (or peak wavelength) of a star’s spectrum is not necessarily within the visible spectrum—it depends on the surface temperature of the star.

Key Points:

1. Stars Cooler Than the Sun:

   • If a star is relatively cool (e.g., $T\sim 3000\text{K}$), its peak wavelength lies in the infrared region, outside the visible spectrum. These stars often appear red because the visible part of their spectrum is strongest at longer wavelengths (red).

2. Stars Similar to the Sun:

   • For stars like the Sun ($T\sim 5800\text{K}$), the peak wavelength is around $500\text{nm}$, within the visible spectrum, near green. However, the Sun appears white or yellowish because it emits light across all visible wavelengths.

3. Hotter Stars:

   • Very hot stars ($T>10,000\text{K}$) have peak wavelengths in the ultraviolet region, beyond the visible spectrum. These stars often appear blue or white due to their strong emission in the shorter wavelengths of the visible spectrum.

Conclusion:

The maximum wavelength might or might not fall within the visible spectrum, depending on the star’s surface temperature. However, even when the peak is outside the visible range, the star can still emit enough visible light for us to perceive its color.

Using a reference object for validation involves comparing the observed spectrum of the star with the spectrum of an object of known temperature to confirm the accuracy of the measurements and calculations. Here’s how it works:

1. What is the Reference Object?

A reference object is typically a laboratory blackbody radiator or a standard light source with a well-defined temperature and known emission spectrum. For example:

• A tungsten filament lamp.
• A calibrated blackbody simulator.

These objects emit radiation following the blackbody curve for their given temperature.

2. Why Use a Reference Object?

The purpose is to ensure:

• The spectrometer is accurately measuring the spectrum.
• The peak wavelength and corresponding temperature derived from the star’s spectrum align with Wien’s Displacement Law.

3. Steps for Validation:

(a) Measure the Spectrum of the Reference Object:

• Use the same spectrometer to measure the spectrum of the reference object.
• Identify its peak wavelength (
  ${\lambda }_{\text{max<span style="background-color: transparent; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, "Helvetica Neue", Arial, "Noto Sans", sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; font-size: 16px;">).</span>}}$

(b) Compare with Known Values:

• Use Wien’s Displacement Law:
  
  $$T=\frac{b}{{\lambda }_{\text{max}}}$$
  • Calculate the temperature of the reference object from the measured
    ${\lambda }_{\text{max .}}$

  • Check if the calculated temperature matches the known temperature of the reference object.

(c) Calibrate if Necessary:

• If there’s a discrepancy, adjust or recalibrate the spectrometer to correct for errors (e.g., in wavelength measurement or intensity detection).

4. Apply the Same Process to the Star:

Once the spectrometer’s accuracy is validated with the reference object, it can reliably measure the spectrum of the star. The peak wavelength observed for the star can then be confidently used in Wien’s Displacement Law to calculate the star’s surface temperature.

Example:

• A laboratory blackbody with a known temperature of 3000K emits a peak wavelength of 967nm (near infrared).
• Measure its spectrum and ensure the spectrometer identifies a peak near this wavelength.
• Once validated, use the spectrometer on the star to measure its peak wavelength, ensuring accurate results.

Mpc stands for megaparsec, a unit of distance commonly used in astronomy to measure large-scale structures in the Universe.

1. Parsec (pc):

• A parsec is the distance at which 1 astronomical unit (AU) subtends an angle of 1 arcsecond (1″)

• 
  $1\text{pc}\approx 3.086\times 10^{16}\text{m <span style="font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, "Helvetica Neue", Arial, "Noto Sans", sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji"; background-color: transparent;">(about 3.26 light-years).</span><span style="background-color: transparent; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, "Helvetica Neue", Arial, "Noto Sans", sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol", "Noto Color Emoji";"> </span>}$

2. Megaparsec (Mpc):

• A megaparsec is one million parsecs (
  $1\text{Mpc}=10^6\text{pc}$
• $1 \, \text{Mpc} \approx 3.086 \times 10^{22} \, \text{m}$

  1Mpc≈3.086×1022m.

Why Mpc Is Useful:

• Distances between galaxies and other large cosmic structures are so vast that expressing them in smaller units like kilometers or light-years becomes impractical.
• For example, the Andromeda Galaxy is approximately 0.78Mpc away from the Milky Way.